├── demo
    ├── __init.py__
    ├── UVtruth.inr
    ├── demo_3dshift.png
    ├── demo_3dvortex.png
    ├── run010050000.tif
    ├── run010050010.tif
    ├── demo_particles.png
    └── inr.py
├── doc
    ├── dfd.png
    └── dfd_func.png
├── README.md
├── LICENSE
├── lib
    └── highorder.py
└── pytyphoon.py


/demo/__init.py__:
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 1 | ###
 2 | import re
 3 | ###
 4 | import numpy
 5 | ###
 6 | 
 7 | ### .INR MOTION/IMAGES ###
 8 | ##########################
 9 | 
10 | def readINR(fileName, printInfo=False):
11 |     """
12 |     Read an INR image file.
13 | 
14 |     Arguments:
15 |         * fileName    the .inr filename
16 |         * printInfo   print image info
17 | 
18 |     Note: to preserve memory continuity, image is loaded with dimensions in the following order:
19 |     0-V (channel), 1-Z (depth), 2-Y (height), 3-X (width).
20 |     """
21 |     # open file
22 |     with open(fileName, 'rb') as inrFile:
23 |         ##### HEADER #####
24 |         ##################
25 |         # read the header: chunk of 256 char
26 |         header = inrFile.read(256).decode('utf-8')
27 |         # the keys we look for
28 |         dimKey=['VDIM','ZDIM','YDIM', 'XDIM']
29 |         typeKey='TYPE'
30 |         sizeKey='PIXSIZE'
31 |         # default values
32 |         dataDim = []
33 |         dataType = ''
34 |         dataBitSize = 0
35 |         # read dimensions
36 |         for key in dimKey:
37 |             match = re.search(key+'=(?P<val>[0-9]+)', header)
38 |             if match:
39 |                 dataDim.append(int(match.group('val')))
40 |             else:
41 |                 print(str(key)+' not found !')
42 |         # read type
43 |         key = typeKey
44 |         match = re.search(key+'=(?P<val>[a-z]+)', header)
45 |         if match:
46 |             dataType = match.group('val')
47 |         else:
48 |             print(str(key)+' not found !')
49 |         # read pixel size (in BITS)
50 |         key = sizeKey
51 |         match = re.search(key+'=(?P<val>[0-9]+) bits', header)
52 |         if match:
53 |             dataBitSize = int(match.group('val'))
54 |         else:
55 |             print(str(key)+' not found !')
56 |         # print info
57 |         if printInfo:
58 |             print('File: '+fileName+'\nDimensions: '+str(dataDim)+'\nType: '\
59 |                   +str(dataType)+'\nPixel Size: '+str(dataBitSize))
60 |         ##### DATA #####
61 |         ################
62 |         # number of bytes to read
63 |         nByte = dataDim[0]*dataDim[1]*dataDim[2]*dataDim[3]*(dataBitSize//8)
64 |         # create a 1D float array
65 |         values = numpy.frombuffer(inrFile.read(nByte), dtype=numpy.dtype('{:s}{:d}'.format(dataType, dataBitSize)))
66 |         # resize to 4-D array
67 |         values.shape = dataDim
68 |         # clean shape from dimensions equal to 1 and return
69 |         return values.squeeze()
70 | 
71 | def readMotion(fileName, printInfo=False, reverse=False):
72 |     """
73 |     Read a motion from an INR file.
74 | 
75 |     Arguments:
76 | 
77 |         * fileName    the .inr file
78 |         * printInfo   print motion info.
79 |         * reverse     reverse the vertical axis and the sign of vertical component.
80 |     """
81 |     # read INR file
82 |     motion = readINR(fileName,printInfo)
83 |     v1 = motion[0,:,:]
84 |     v2 = motion[1,:,:]
85 |     # reverse reference maybe
86 |     if reverse:
87 |         v1 = numpy.flipud(v1)
88 |         v2 = -numpy.flipud(v2)
89 |     return v1,v2
90 | 


--------------------------------------------------------------------------------
/README.md:
--------------------------------------------------------------------------------
  1 | # PyTyphoon
  2 | Python implementation of [Typhoon motion estimator](http://www.pierrederian.net/typhoon.html): dense estimation of 2D/3D optical flow on wavelet bases, primarily aimed at **fluid motion estimation**.
  3 | 
  4 | ## Important remarks
  5 | At the moment, the wavelet-based data DFD term [(D&eacute;rian et al., 2013)] only is provided: the **high-order regularizers** [(Kadri-Harouna et al., 2013)] **are not included** in this implementation.
  6 | 
  7 | The reference implementation used in [(D&eacute;rian et al., 2015)] and [(D&eacute;rian et al., 2017)] is written in C++ and GPU-accelerated with CUDA, and contains the high-order regularizers. It is the property of Inria (FR) and the CSU Chico Research
  8 | Foundation (Ca, USA), and can be licensed from these institutions.
  9 | **This Python implementation is not the same as the
 10 | reference** for many reasons, and it is obviously much slower.
 11 | 
 12 | ## Requirements
 13 | - [Numpy, Scipy](https://scipy.org/);
 14 | - [PyWavelets](https://github.com/PyWavelets/pywt);
 15 | - [Pillow](https://pillow.readthedocs.io/) and [Matplotlib](https://matplotlib.org/) for the demos.
 16 | 
 17 | Tested with Anaconda Python 3.6.1, Numpy 1.12.1, Scipy 0.19.1, PyWavelet 0.5.2.
 18 | 
 19 | ## Usage
 20 | The `Typhoon` class can be imported from other modules/scripts to perform estimations as needed.
 21 | 
 22 | The script can also work as a standalone estimator in simple cases, e.g.:
 23 | ```
 24 | python pytyphoon.py -i0 path/to/im0.jpg -i1 path/to/im1.jpg -wav 'db3' --display
 25 | ```
 26 | will solve the problem for image pair (`im0.jpg`, `im1.jpg`) and wavelet Daubechies-3.
 27 | See `python pytyphoon.py -h` for the complete list of parameters.
 28 | 
 29 | ## How does it work?
 30 | 
 31 | Typhoon solves a **dense variational optical flow** problem, that is to say: (i) it provides one motion vector at every pixel of input images and (ii) it estimates the entire vector field altogether.
 32 | 
 33 | To do so, it looks for the motion field which minimizes the **displaced frame difference** (DFD):
 34 | 
 35 | ![DFD](doc/dfd.png)
 36 | 
 37 | This is achieved by minimizing the following functional:
 38 | 
 39 | ![DFD functional](doc/dfd_func.png)
 40 | 
 41 | where the integral is taken over the image.
 42 | The functional above is **non-linear with respect to the motion field**.
 43 | This has the advantage of better handling large displacement, but complicates the minimization process.
 44 | 
 45 | For the non-linear minimization to succeed, the solution should lie reasonable "close" to the first guess. This is where **wavelets bases** come into play: by providing a multiscale representation of the motion field, they enable to estimate the motion iteratively from its coarsest scales to the finests.
 46 | 
 47 | The minimization is handled by L-BFGS, which is efficient memory-wise and only requires the functional value and its gradient.
 48 | 
 49 | ## Demos
 50 | 
 51 | These demos are shipped with the project.
 52 | 
 53 | ### (2D) Synthetic particle images
 54 | Simple 2d estimation using synthetic particle images (256x256 pixels) originally created for the [FLUID project](http://fluid.irisa.fr/data-eng.htm) (image database #1). Run:
 55 | ```
 56 | python pytyphoon.py --demo particles
 57 | ```
 58 | ![Particle results](demo/demo_particles.png)
 59 | 
 60 | ### (3D) Homogeneous shift
 61 | Simple 3d estimation using synthetic images (64x64x64 pixels) obtained by filtering random normal noise at various scales. The displacements are integer shifts along each of the 3 dimensions. Run:
 62 | ```
 63 | python pytyphoon.py --demo 3dshift
 64 | ```
 65 | ![3dshift results](demo/demo_3dshift.png)
 66 | 
 67 | ### (3D) Column vortex
 68 | Simple 3d estimation using synthetic images (96x96x96 pixels) obtained by filtering random normal noise at various scales. The displacement field is a column vortex (first two axes) with an updraft increasing linearly along the third axis. Run:
 69 | ```
 70 | python pytyphoon.py --demo 3dvortex
 71 | ```
 72 | ![3dshift results](demo/demo_3dvortex.png)
 73 | 
 74 | ## References
 75 | - [(D&eacute;rian et al., 2017)]
 76 |     D&eacute;rian, P. & Almar, R.
 77 |     "Wavelet-based Optical Flow Estimation of Instant Surface Currents from Shore-based and UAV Video".
 78 |     _IEEE Transactions on Geoscience and Remote Sensing_, Vol. 55, pp. 5790-5797, 2017.
 79 | - [(D&eacute;rian et al., 2015)]
 80 |      D&eacute;rian, P.; Mauzey, C. F. and Mayor, S. D.
 81 |     "Wavelet-based optical flow for two-component wind field estimation from single aerosol lidar data".
 82 |     _Journal of Atmospheric and Oceanic Technology_, Vol. 32, pp. 1759-1778, 2015.
 83 | - [(D&eacute;rian et al., 2013)]
 84 |     D&eacute;rian, P.; H&eacute;as, P.; Herzet, C. & M&eacute;min, E.
 85 |     "Wavelets and Optical Flow Motion Estimation".
 86 |     _Numerical Mathematics: Theory, Method and Applications_, Vol. 6, pp. 116-137, 2013.
 87 | - [(Kadri-Harouna et al., 2013)] Kadri Harouna, S.; D&eacute;rian, P.; H&eacute;as, P. and     M&eacute;min, E.
 88 |    "Divergence-free Wavelets and High Order Regularization".
 89 |    _International Journal of Computer Vision_, Vol. 103, pp. 80-99, 2013.
 90 | 
 91 | [(D&eacute;rian et al., 2017)]: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7964775&isnumber=8049540
 92 | [(D&eacute;rian et al., 2015)]: http://journals.ametsoc.org/doi/abs/10.1175/JTECH-D-15-0010.1
 93 | [(D&eacute;rian et al., 2013)]: https://www.cambridge.org/core/journals/numerical-mathematics-theory-methods-and-applications/article/wavelets-and-optical-flow-motion-estimation/2A9D13B316F000F0530AD42621B42FFD
 94 | [(Kadri-Harouna et al., 2013)]: https://link.springer.com/article/10.1007/s11263-012-0595-7
 95 | 
 96 | ## Todo
 97 | - **NetCDF for output results?** in standalone mode;
 98 | - support of masks;
 99 | - some regularizers;
100 | - alternative penalization functions;
101 | - divergence-free wavelets;
102 | - ...
103 | 


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--------------------------------------------------------------------------------
/lib/highorder.py:
--------------------------------------------------------------------------------
  1 | """[WIP] High-order regularization terms for PyTyphoon.
  2 | 
  3 | Note:
  4 |     - the computation of the regularization terms in HighOrderRegularizerConv
  5 |         is a convolution-based one, likely closer to what was introduced in [1].
  6 |     - the computation of the regularization terms in HighOrderRegularizerMat
  7 |         is a matrix-based one as in [2], [3].
  8 |     - largely untested, use with care.
  9 |         
 10 | References:
 11 |     [1] BEYLKIN, Gregory.
 12 |     On the representation of operators in bases of compactly supported wavelets.
 13 |     SIAM Journal on Numerical Analysis, 1992, vol. 29, no 6, p. 1716-1740.
 14 |     [2] KADRI-HAROUNA, S., DÉRIAN, Pierre, HÉAS, Patrick, et al.
 15 |     Divergence-free wavelets and high order regularization.
 16 |     International journal of computer vision, 2013, vol. 103, no 1, p. 80-99.
 17 |     [3] DÉRIAN, Pierre.
 18 |     Wavelets and Fluid Motion Estimation.
 19 |     PhD thesis, MATISSE doctoral school, Université Rennes 1, 2012.
 20 | """
 21 | # Standard
 22 | import logging
 23 | 
 24 | # Third-party
 25 | import numpy as np
 26 | import pywt
 27 | import scipy.ndimage as ndimage
 28 | import scipy.optimize as optimize
 29 | 
 30 | 
 31 | LOGGER = logging.getLogger(__name__)
 32 | 
 33 | 
 34 | def connection_coefficients(wav, order):
 35 |     """Find the connection coefficients of the wavelet at given order.
 36 | 
 37 |     :param wav: a pywt.Wavelet;
 38 |     :param order: the derivation order;
 39 |     :param return: a vector of coefficients.
 40 | 
 41 |     This is the evaluation of L2 dot-products of the form:
 42 |          \int[ Phi(x) (d^(n)/dx^n){Phi}(x) ]dx
 43 |     where Phi is the mother wavelet.
 44 | 
 45 |     Written by P. DERIAN 2018-01-09.
 46 |     Updated by P. DERIAN 2019-02-15: using reconstruction filter.
 47 |     """
 48 |     ctol = 1e-15  # Tolerance for coefficients
 49 |     etol = 1e-4  # Tolerance for eigenvalues    
 50 |     # Get the low-pass reconstruction filter
 51 |     lo = wav.rec_lo
 52 |     len_lo = len(lo)
 53 |     # Create the matrix
 54 |     dim = 2*len_lo - 3
 55 |     matrix = np.zeros((dim, dim))
 56 |     for m in range(dim):
 57 |         for n in range(dim):
 58 |             tmp = 0.
 59 |             # for each filter value
 60 |             for p, lo_p in enumerate(lo):
 61 |                 idx = m - 2*n + p + len_lo - 2
 62 |                 if (idx>=0 and idx<len_lo):
 63 |                     tmp += lo_p*lo[idx]
 64 |             # store only if above threshold
 65 |             if np.abs(tmp)>ctol:
 66 |                 matrix[n,m] = tmp
 67 |     # Find coefficient vector: solve eigenvalue problem
 68 |     eval, evec = np.linalg.eig(matrix)
 69 |     # Check if any REAL eigenvalue matches our order
 70 |     sigma = 1./float(2**order)
 71 |     ev_found = False
 72 |     for i, ev in enumerate(eval):
 73 |         if (np.abs(np.real(ev)-sigma)<etol) and (np.abs(np.imag(ev)<1e-14)):
 74 |             ev_found = True
 75 |             break
 76 |     # [TODO] warn if not found?
 77 |     coeffs = None
 78 |     if ev_found:
 79 |         # Get the associated eigen vector
 80 |         coeffs = np.real(evec[:,i])
 81 |         # If the derivation order is odd, force mid value to be exactly zero
 82 |         # as it should be by construction.
 83 |         if order % 2:
 84 |             coeffs[dim//2] = 0.
 85 |         # Apply Beylkin normalization
 86 |         # [TODO] document
 87 |         norm_factors = np.array([float(-len_lo + 2 + p) for p in range(dim)])**order
 88 |         coeffs /= np.sum(norm_factors*coeffs)
 89 |         tmp = (np.prod(np.arange(1, order+1)))*np.power(-1.,order)
 90 |         coeffs *= tmp
 91 |     return coeffs
 92 | 
 93 |     
 94 | def mass_matrix(order, wav, levels, size):
 95 |     """Compute the mass matrix.
 96 |     
 97 |     :param order: total derivation order;
 98 |     :param wav: a pywt.Wavelet;
 99 |     :param levels: wavelet decomposition levels;
100 |     :param size: size of the matrix;
101 |     :return: a (size, size) mass matrix.
102 |     """
103 |     # Check that size is compatible with level
104 |     if size % (2**levels):
105 |         raise ValueError("size {} is not compatible with levels {}".format(size, levels))
106 |     # Swap the wavelet for the decomposition
107 |     wavswap = pywt.Wavelet(
108 |             name='{}_swap'.format(wav.name),
109 |             filter_bank=wav.inverse_filter_bank,
110 |             )
111 |     # Compute connection coefficients
112 |     coeffs = connection_coefficients(wav, order)
113 |     # Fill the square matrix, periodizing coefficients
114 |     matrix = np.zeros((size, size), dtype=np.float64) 
115 |     mid = (coeffs.size - 1)//2
116 |     for p in range(size):
117 |         for k, c in enumerate(coeffs, -mid):
118 |             matrix[p, (p + k) % size] += c
119 |     # Perform anisotropic (aka fully-separable transform)
120 |     result = pywt.fswavedecn(matrix, wavswap, mode='periodization', levels=levels)
121 |     result = result.coeffs
122 |     return result
123 |             
124 | 
125 | class HighOrderRegularizerConv:
126 |     """Implements high-order wavelet-based regularization terms,
127 |     here implemented with convolutions.
128 | 
129 |     Written by P. DERIAN 2018-01-09.
130 |     """
131 |     ORDER_MAX = 5  # Max order for coefficients computation.
132 | 
133 |     def __init__(self, wav):
134 |         """Instance constructor.
135 | 
136 |         Written by P. DERIAN 2018-01-09.
137 |         Updated by P. DERIAN 2019-02-15: fixed biorthogonal case.
138 |         """
139 |         self.mode = 'periodization'
140 |         # The main wavelet
141 |         self.wav = wav
142 |         # Get a swapped version (primal / dual filters are swapped)
143 |         # Note: if the wav is orthogonal, wavswap is in practice the same as wav.
144 |         self.wavswap = pywt.Wavelet(
145 |             name='{}_swap'.format(self.wav.name),
146 |             filter_bank=self.wav.inverse_filter_bank,
147 |             )
148 |         # Compute the connection coefficients up to order max. 
149 |         self.coeffs = {k: connection_coefficients(wav, k) for k in range(self.ORDER_MAX)}
150 |         # The set of available regularizers
151 |         self.regularizers = {'l2norm': self._l2norm_gradient,
152 |                              'hornschunck': self._hornschunck_gradient,
153 |                              'div': self._div_gradient,
154 |                              'curl': self._curl_gradient,
155 |                              'laplacian': self._laplacian_gradient,
156 |                              'graddiv': self._graddiv_gradient,
157 |                              'gradcurl': self._gradcurl_gradient,
158 |                              }
159 | 
160 |     def evaluate(self, C1, C2, regul_type='l2norm'):
161 |         """Evaluate the value and gradient of given regularizer.
162 | 
163 |         :param C1: wavelet coefficients.
164 |         :param C2: wavelet coefficients.
165 |         :param regul_type:
166 |         :return: value, (grad1, grad2)
167 | 
168 |         Written by P. DERIAN 2018-01-09.
169 |         Updated by P. DERIAN 2019-02-15: fixed biorthogonal case.
170 |         """
171 |         # Infer levels for further decomposition
172 |         levels = len(C1) - 1
173 |         # Rebuild the fields
174 |         U1 = pywt.waverecn(C1, self.wav, mode=self.mode)
175 |         U2 = pywt.waverecn(C2, self.wav, mode=self.mode)
176 |         # Evaluate gradient
177 |         [grad1, grad2] = self.regularizers[regul_type](U1, U2)
178 |         # Decompose gradient to complete its computation 
179 |         # Note: use wavswap here!
180 |         grad1 = pywt.wavedecn(grad1, self.wavswap, level=levels, mode=self.mode)
181 |         grad2 = pywt.wavedecn(grad2, self.wavswap, level=levels, mode=self.mode)
182 |         # Compute the functional value
183 |         result = 0.
184 |         for c, g in zip([C1, C2], [grad1, grad2]):
185 |             # Add contribution of approx
186 |             result += np.dot(c[0].ravel(), g[0].ravel())
187 |             # And details
188 |             for cd, gd in zip(c[1:], g[1:]):
189 |                 for k in cd.keys():
190 |                     result += np.dot(cd[k].ravel(), gd[k].ravel())
191 |         return 0.5*result, (grad1, grad2)
192 | 
193 |     def inner_product(self, A, B, order_d1, order_d2):
194 |         """
195 |         """
196 |         # Infer levels for further decomposition
197 |         levels = len(A) - 1
198 |         # Rebuild the field
199 |         Aa = pywt.waverecn(A, self.wav, mode=self.mode)
200 |         # Evaluate the convolution
201 |         tmp = self.convolve_separable(Aa, self.coeffs[order_d1], self.coeffs[order_d2])
202 |         # Decompose with wavswap
203 |         C = pywt.wavedecn(tmp, self.wavswap, level=levels, mode=self.mode)
204 |         # Compute the inner product
205 |         result = np.dot(B[0].ravel(), C[0].ravel())
206 |         for bd, cd in zip(B[1:], C[1:]):
207 |             for k in cd.keys():
208 |                 result += np.dot(bd[k].ravel(), cd[k].ravel())
209 |         return result
210 |            
211 |     def norm(self, A, order_d1, order_d2):
212 |         """Evaluates the l2-norm of coefficents A with derivation orders:
213 |             \int[ |d1d2{A}(x)|^2 ]dx
214 |             
215 |         WARNING: 
216 |             
217 |         Written by P. DERIAN 2019-02-19
218 |         """
219 |         LOGGER.warning("norm() has normalization issues...")
220 |         order_total = order_d1 + order_d2
221 |         sign = (-1)**order_total
222 |         return sign * self.inner_product(A, A, 2*order_d1, 2*order_d2)
223 |            
224 |     def _l2norm_gradient(self, U1, U2):
225 |         """Evaluates the gradient of l2-norm (order 0) regularizer:
226 |             0.5\int[ |U1(x)|^2 + |U2(x)|^2 ]dx
227 | 
228 |         Written by P. DERIAN 2018-01-09.
229 |         """
230 |         c0 = self.coeffs[0]
231 |         result = [self.convolve_separable(U, c0, c0) for U in [U1, U2]]
232 |         return result
233 | 
234 |     def _hornschunck_gradient(self, U1, U2):
235 |         """Evaluates the gradient of Horn&Schunck (order 1) regularizer:
236 |             0.5\int[ |grad{U1}(x)|^2 + |grad{U2}(x)|^2 ]dx
237 | 
238 |         Written by P. DERIAN 2018-01-09.
239 |         """
240 |         c0 = self.coeffs[0]
241 |         c2 = self.coeffs[2]
242 |         result = [-(self.convolve_separable(U, c2, c0) + self.convolve_separable(U, c0, c2))
243 |                   for U in [U1, U2]]
244 |         return result
245 | 
246 |     def _div_gradient(self, U1, U2):
247 |         """Evaluates the gradient of divergence (order 1) regularizer:
248 |             0.5\int[ |div{U1, U2}(x)|^2 ]dx, with div{U1, U2} = d1(U1) + d2(U2)
249 | 
250 |         Written by P. DERIAN 2019-02-19.
251 |         """
252 |         c0 = self.coeffs[0]
253 |         c1 = self.coeffs[1]
254 |         c2 = self.coeffs[2]
255 |         grad1 = -(self.convolve_separable(U2, c1, c1) + self.convolve_separable(U1, c2, c0))
256 |         grad2 = -(self.convolve_separable(U1, c1, c1) + self.convolve_separable(U2, c0, c2))
257 |         return [grad1, grad2]
258 |       
259 |     def _curl_gradient(self, U1, U2):
260 |         """Evaluates the gradient of curl (order 1) regularizer:
261 |             0.5\int[ |curl{U1, U2}(x)|^2 ]dx, with curl{U1, U2} = d1(U2) - d2(U1)
262 | 
263 |         Written by P. DERIAN 2019-02-19.
264 |         """
265 |         c0 = self.coeffs[0]
266 |         c1 = self.coeffs[1]
267 |         c2 = self.coeffs[2]
268 |         grad1 = self.convolve_separable(U2, c1, c1) - self.convolve_separable(U1, c0, c2)
269 |         grad2 = self.convolve_separable(U1, c1, c1) - self.convolve_separable(U2, c2, c0)
270 |         return [grad1, grad2]    
271 |       
272 |     def _laplacian_gradient(self, U1, U2):
273 |         """Evaluates the gradient of laplacian (order 2) regularizer:
274 |             0.5\int[ |laplacian{U1}(x)|^2 + |laplacian{U2}(x)|^2 ]dx ,
275 |             with laplacian{Ui} = d1^2(Ui) + d2^2(Ui)
276 |             
277 |         Written by P. DERIAN 2019-02-19.
278 |         """    
279 |         c0 = self.coeffs[0]
280 |         c2 = self.coeffs[2]
281 |         c4 = self.coeffs[4]
282 |         grad1 = (self.convolve_separable(U1, c4, c0) + 
283 |                  self.convolve_separable(U1, c0, c4) +
284 |                  2.*self.convolve_separable(U1, c2, c2))
285 |         grad2 = (self.convolve_separable(U2, c4, c0) + 
286 |                  self.convolve_separable(U2, c0, c4) +
287 |                  2.*self.convolve_separable(U2, c2, c2))
288 |         return [grad1, grad2]
289 | 
290 |     def _graddiv_gradient(self, U1, U2):
291 |         """Evaluates the gradient of curl (order 1) regularizer:
292 |             0.5\int[ |grad{div{U1, U2}}(x)|^2 ]dx, with div{U1, U2} = d1(U1) + d2(U2)
293 | 
294 |         Written by P. DERIAN 2019-02-19.
295 |         """
296 |         c0 = self.coeffs[0]
297 |         c1 = self.coeffs[1]
298 |         c2 = self.coeffs[2]
299 |         c3 = self.coeffs[3]
300 |         c4 = self.coeffs[4]
301 |         grad1 = (self.convolve_separable(U1, c4, c0) +
302 |                  self.convolve_separable(U1, c2, c2) + 
303 |                  self.convolve_separable(U2, c3, c1) + 
304 |                  self.convolve_separable(U2, c1, c3))
305 |         grad2 = (self.convolve_separable(U2, c0, c4) +
306 |                  self.convolve_separable(U2, c2, c2) + 
307 |                  self.convolve_separable(U1, c3, c1) + 
308 |                  self.convolve_separable(U1, c1, c3))
309 |         return [grad1, grad2]    
310 |         
311 |     def _gradcurl_gradient(self, U1, U2):
312 |         """Evaluates the gradient of curl (order 1) regularizer:
313 |             0.5\int[ |grad{curl{U1, U2}}(x)|^2 ]dx, with curl{U1, U2} = d1(U2) - d2(U1)
314 | 
315 |         Written by P. DERIAN 2019-02-19.
316 |         """
317 |         c0 = self.coeffs[0]
318 |         c1 = self.coeffs[1]
319 |         c2 = self.coeffs[2]
320 |         c3 = self.coeffs[3]
321 |         c4 = self.coeffs[4]
322 |         grad1 = (self.convolve_separable(U1, c0, c4) +
323 |                  self.convolve_separable(U1, c2, c2) - 
324 |                  self.convolve_separable(U2, c3, c1) - 
325 |                  self.convolve_separable(U2, c1, c3))
326 |         grad2 = (self.convolve_separable(U2, c4, c0) +
327 |                  self.convolve_separable(U2, c2, c2) - 
328 |                  self.convolve_separable(U1, c3, c1) - 
329 |                  self.convolve_separable(U1, c1, c3))
330 |         return [grad1, grad2]    
331 |             
332 |     @staticmethod
333 |     def convolve_separable(x, filter1, filter2, origin1=0, origin2=0):
334 |         """Separable convolution of x by filter1 along first axis and filter2 along second axis.
335 | 
336 |         :param filter1: filter along 1st axis (rows);
337 |         :param filter2: filter along 2nd axis (columns);
338 | 
339 |         Written by P. DERIAN 2018-01-09.
340 |         """
341 |         tmp = ndimage.filters.convolve1d(x, filter1, axis=0, mode='wrap', origin=origin1)
342 |         return ndimage.filters.convolve1d(tmp, filter2, axis=1, mode='wrap', origin=origin2)
343 | 
344 | 
345 | class HighOrderRegularizerMat:
346 |     """Implements high-order wavelet-based regularization terms,
347 |     here implemented with matrices.
348 | 
349 |     Written by P. DERIAN 2018-02-18.
350 |     """
351 |     ORDER_MAX = 5  # Max order for coefficients computation.
352 | 
353 |     def __init__(self, wav, levels, image_shape):
354 |         """Instance constructor.
355 | 
356 |         Written by P. DERIAN 2018-02-19.
357 |         """
358 |         # Keep reference to the wavelet
359 |         self.wav = wav
360 |         self.mode = 'periodization'
361 |         # Compute mass matrices
362 |         width, height = image_shape[:2]
363 |         self.width_matrices = {o: mass_matrix(o, wav, levels, width) for o in range(self.ORDER_MAX)}
364 |         self.height_matrices = {o: mass_matrix(o, wav, levels, height) for o in range(self.ORDER_MAX)}        
365 |         # The set of available regularizers
366 |         self.regularizers = {'l2norm': self._l2norm_gradient,
367 |                              'hornschunck': self._hornschunck_gradient,
368 |                              'div': self._divergence_gradient,
369 |                              'curl': self._curl_gradient,
370 |                              'laplacian': self._laplacian_gradient,
371 |                              'graddiv': self._graddiv_gradient,
372 |                              'gradcurl': self._gradcurl_gradient,
373 |                              }
374 |  
375 |     def evaluate(self, C1, C2, regul_type='l2norm'):
376 |         """Evaluate the value and gradient of given regularizer.
377 | 
378 |         :param C1: wavelet coefficients;
379 |         :param C2: wavelet coefficients;
380 |         :param regul_type: name of the relarizer;
381 |         :return: value, (grad1, grad2).
382 | 
383 |         Written by P. DERIAN 2019-02-18.
384 |         """
385 |         grad1, grad2 = self.regularizers[regul_type](C1, C2)
386 |         value = 0.5*(np.dot(C1.ravel(), grad1.ravel()) + np.dot(C2.ravel(), grad2.ravel()))
387 |         return value, [grad1, grad2] 
388 | 
389 |     def _l2norm_gradient(self, C1, C2):
390 |         """Evaluates the gradient of l2-norm (order 0) regularizer:
391 |             0.5\int[ |U1(x)|^2 + |U2(x)|^2 ]dx
392 | 
393 |         Written by P. DERIAN 2019-02-18.
394 |         """
395 |         # Note: transpose operators are omitted in even-order matrices
396 |         # as they are symmetric by construction.
397 |         return (self._triple_product(self.height_matrices[0], C, self.width_matrices[0])
398 |                 for C in [C1, C2])
399 | 
400 |     def _hornschunck_gradient(self, C1, C2):
401 |         """Evaluates the gradient of Horn&Schunck (order 1) regularizer:
402 |             0.5\int[ |grad{U1}(x)|^2 + |grad{U2}(x)|^2 ]dx
403 | 
404 |         Written by P. DERIAN 2019-02-18.
405 |         """
406 |         return (-(self._triple_product(self.height_matrices[2], C, self.width_matrices[0]) +
407 |                   self._triple_product(self.height_matrices[0], C, self.width_matrices[2]))
408 |                 for C in [C1, C2])
409 |    
410 |     def _divergence_gradient(self, C1, C2):
411 |         """Evaluates the gradient of Divergence (order 1) regularizer:
412 |             0.5\int[ |div{U1, U2}(x)|^2 ]dx, with div{1, U2} = d1(U1) + d2(U2)
413 | 
414 |         Written by P. DERIAN 2019-02-18.
415 |         """
416 |         n0h = self.height_matrices[0]
417 |         n0w = self.width_matrices[0]
418 |         n1h = self.height_matrices[1]
419 |         n1w = self.width_matrices[1]
420 |         n2h = self.height_matrices[2]
421 |         n2w = self.width_matrices[2]
422 |         grad1 = -(self._triple_product(n1h, C2, n1w.T) + self._triple_product(n2h, C1, n0w))
423 |         grad2 = -(self._triple_product(n1h, C1, n1w.T) + self._triple_product(n0h, C2, n2w))
424 |         return grad1, grad2
425 | 
426 |     def _curl_gradient(self, C1, C2):
427 |         """Evaluates the gradient of curl (order 1) regularizer:
428 |             0.5\int[ |curl{U1, U2}(x)|^2 ]dx, with curl{U1, U2} = d1(U2) - d2(U1)
429 | 
430 |         Written by P. DERIAN 2019-02-19.
431 |         """
432 |         n0h = self.height_matrices[0]
433 |         n0w = self.width_matrices[0]
434 |         n1h = self.height_matrices[1]
435 |         n1w = self.width_matrices[1]
436 |         n2h = self.height_matrices[2]
437 |         n2w = self.width_matrices[2]
438 |         grad1 = self._triple_product(n1h, C2, n1w.T) - self._triple_product(n0h, C1, n2w.T)
439 |         grad2 = self._triple_product(n1h, C1, n1w.T) - self._triple_product(n2h, C2, n0w.T)
440 |         return grad1, grad2
441 |     
442 |     def _laplacian_gradient(self, U1, U2):
443 |         """Evaluates the gradient of laplacian (order 2) regularizer:
444 |             0.5\int[ |laplacian{U1}(x)|^2 + |laplacian{U2}(x)|^2 ]dx ,
445 |             with laplacian{Ui} = d1^2(Ui) + d2^2(Ui)
446 |             
447 |         Written by P. DERIAN 2019-02-19.
448 |         """        
449 |         n0h = self.height_matrices[0]
450 |         n0w = self.width_matrices[0]
451 |         n2h = self.height_matrices[2]
452 |         n2w = self.width_matrices[2]
453 |         n4h = self.height_matrices[4]
454 |         n4w = self.width_matrices[4]
455 |         grad1 = (self._triple_product(n4h, U1, n0w) + 
456 |                  self._triple_product(n0h, U1, n4w) +
457 |                  2.*self._triple_product(n2h, U1, n2w))
458 |         grad2 = (self._triple_product(n4h, U2, n0w) + 
459 |                  self._triple_product(n0h, U2, n4w) +
460 |                  2.*self._triple_product(n2h, U2, n2w))
461 |         return grad1, grad2
462 | 
463 |     def _graddiv_gradient(self, U1, U2):
464 |         """Evaluates the gradient of curl (order 1) regularizer:
465 |             0.5\int[ |grad{div{U1, U2}}(x)|^2 ]dx, with div{U1, U2} = d1(U1) + d2(U2)
466 | 
467 |         Written by P. DERIAN 2019-02-19.
468 |         """
469 |         n0h = self.height_matrices[0]
470 |         n0w = self.width_matrices[0]
471 |         n1h = self.height_matrices[1]
472 |         n1w = self.width_matrices[1]
473 |         n2h = self.height_matrices[2]
474 |         n2w = self.width_matrices[2]
475 |         n3h = self.height_matrices[3]
476 |         n3w = self.width_matrices[3]
477 |         n4h = self.height_matrices[4]
478 |         n4w = self.width_matrices[4]
479 |         grad1 = (self._triple_product(n4h, U1, n0w) + 
480 |                  self._triple_product(n2h, U1, n2w) +
481 |                  self._triple_product(n1h, U2, n3w.T) +
482 |                  self._triple_product(n3h, U2, n1w.T))
483 |         grad2 = (self._triple_product(n0h, U2, n4w) + 
484 |                  self._triple_product(n2h, U2, n2w) +
485 |                  self._triple_product(n1h, U1, n3w.T) +
486 |                  self._triple_product(n3h, U1, n1w.T))
487 |         return grad1, grad2
488 | 
489 |     def _gradcurl_gradient(self, U1, U2):
490 |         """Evaluates the gradient of curl (order 1) regularizer:
491 |             0.5\int[ |grad{curl{U1, U2}}(x)|^2 ]dx, with curl{U1, U2} = d1(U2) - d2(U1)
492 | 
493 |         Written by P. DERIAN 2019-02-19.
494 |         """
495 |         n0h = self.height_matrices[0]
496 |         n0w = self.width_matrices[0]
497 |         n1h = self.height_matrices[1]
498 |         n1w = self.width_matrices[1]
499 |         n2h = self.height_matrices[2]
500 |         n2w = self.width_matrices[2]
501 |         n3h = self.height_matrices[3]
502 |         n3w = self.width_matrices[3]
503 |         n4h = self.height_matrices[4]
504 |         n4w = self.width_matrices[4]
505 |         grad1 = (self._triple_product(n0h, U1, n4w) + 
506 |                  self._triple_product(n2h, U1, n2w) -
507 |                  self._triple_product(n1h, U2, n3w.T) -
508 |                  self._triple_product(n3h, U2, n1w.T))
509 |         grad2 = (self._triple_product(n4h, U2, n0w) + 
510 |                  self._triple_product(n2h, U2, n2w) -
511 |                  self._triple_product(n1h, U1, n3w.T) -
512 |                  self._triple_product(n3h, U1, n1w.T))
513 |         return grad1, grad2
514 |         
515 |     @staticmethod
516 |     def _triple_product(A, B, C):
517 |         """compute A*B*C.
518 |         """
519 |         tmp = np.matmul(B, C)
520 |         return np.matmul(A, tmp)   
521 | 
522 |         
523 | ### Demonstrations ###
524 | 
525 | if __name__=="__main__":
526 |     # Standard library
527 |     import matplotlib.pyplot as pyplot
528 |     # Custom
529 |     import sys
530 |     sys.path.append('..')
531 |     import demo.inr as inr
532 |     import matplotlib.pyplot as plt
533 | 
534 |     # Filters for gradient computation (finite differences, periodic bc)
535 |     # see https://en.wikipedia.org/wiki/Finite_difference_coefficient
536 |     d1_filters = {
537 |         2: [-1./2., 0., 1./2.],
538 |         4: [1./12., -2./3., 0., 2/3., -1./12.],
539 |         6: [-1./60., 3./20., -3./4., 0., 3./4., -3./20., 1./60.],
540 |         8: [1./280., -4./105., 1./5., -4./5., 0., 4./5., -1./5., 4./105., -1./280.],
541 |         }
542 |     d2_filters = {
543 |         2: [1., -2., 1.],
544 |         4: [-1./12., 4./3., -5./2., 4/3., -1./12.],
545 |         6: [1./90., -3./20., 3./2., -49./18., 3./2., -3./20., 1./90.],   
546 |         8: [-1./560., 8./315., -1./5., 8./5., -205./72., 8./5., -1./5., 8./315., -1./560.],
547 |         }
548 |     
549 |     def l2norm(U1, U2):
550 |         return 0.5*np.sum(U1**2 + U2**2)
551 |  
552 |     def hornschunk(U1, U2, d1):
553 |         result = 0.
554 |         g11 = ndimage.filters.convolve1d(U1, d1, axis=0, mode='wrap', origin=0)
555 |         g12 = ndimage.filters.convolve1d(U1, d1, axis=1, mode='wrap', origin=0)
556 |         result += np.sum(g11**2 + g12**2)
557 |         g21 = ndimage.filters.convolve1d(U2, d1, axis=0, mode='wrap', origin=0)
558 |         g22 = ndimage.filters.convolve1d(U2, d1, axis=1, mode='wrap', origin=0)
559 |         result += np.sum(g21**2 + g22**2)
560 |         return 0.5*result
561 |   
562 |     def divergence(U1, U2, d1):
563 |         g11 = ndimage.filters.convolve1d(U1, d1, axis=0, mode='wrap', origin=0)
564 |         g22 = ndimage.filters.convolve1d(U2, d1, axis=1, mode='wrap', origin=0)
565 |         result = np.sum((g11 + g22)**2)
566 |         return 0.5*result       
567 |       
568 |     def curl(U1, U2, d1):
569 |         g12 = ndimage.filters.convolve1d(U1, d1, axis=1, mode='wrap', origin=0)
570 |         g21 = ndimage.filters.convolve1d(U2, d1, axis=0, mode='wrap', origin=0)
571 |         result = np.sum((g21 - g12)**2)
572 |         return 0.5*result         
573 |       
574 |     def laplacian(U1, U2, d2):  
575 |         result = 0.
576 |         g11 = ndimage.filters.convolve1d(U1, d2, axis=1, mode='wrap', origin=0)
577 |         g12 = ndimage.filters.convolve1d(U1, d2, axis=1, mode='wrap', origin=0)
578 |         result += np.sum(g11**2 + g12**2)
579 |         g21 = ndimage.filters.convolve1d(U2, d2, axis=0, mode='wrap', origin=0)
580 |         g22 = ndimage.filters.convolve1d(U2, d2, axis=1, mode='wrap', origin=0)  
581 |         result += np.sum(g21**2 + g22**2)
582 |         return 0.5*result
583 | 
584 |     def graddiv(U1, U2, d1):
585 |         g11 = ndimage.filters.convolve1d(U1, d1, axis=0, mode='wrap', origin=0)
586 |         g22 = ndimage.filters.convolve1d(U2, d1, axis=1, mode='wrap', origin=0)
587 |         div = g11 + g22
588 |         g1 = ndimage.filters.convolve1d(div, d1, axis=0, mode='wrap', origin=0)
589 |         g2 = ndimage.filters.convolve1d(div, d1, axis=1, mode='wrap', origin=0)
590 |         return 0.5*np.sum(g1**2 + g2**2)
591 |         
592 |     def gradcurl(U1, U2, d1):
593 |         g12 = ndimage.filters.convolve1d(U1, d1, axis=1, mode='wrap', origin=0)
594 |         g21 = ndimage.filters.convolve1d(U2, d1, axis=0, mode='wrap', origin=0)
595 |         curl = g21 - g12
596 |         g1 = ndimage.filters.convolve1d(curl, d1, axis=0, mode='wrap', origin=0)
597 |         g2 = ndimage.filters.convolve1d(curl, d1, axis=1, mode='wrap', origin=0)
598 |         return 0.5*np.sum(g1**2 + g2**2)
599 |       
600 |     def demo_mass_matrix():
601 |         """
602 | 
603 |         Written by P. DERIAN 2019-02-18.
604 |         """    
605 |         size = 256  # Image size [px]
606 |         levels = 0  # Decomposition levels
607 |         wav = pywt.Wavelet('db10')
608 |         N0 = mass_matrix(0, wav, levels, size)
609 |         N1 = mass_matrix(1, wav, levels, size)
610 |         N2 = mass_matrix(2, wav, levels, size)
611 |         
612 |         # Display matrices
613 |         eps = 1e-15
614 |         fig, axes = plt.subplots(1, 3)
615 |         # axes[0].imshow(abs(N0) > eps)
616 |         # axes[1].imshow(abs(N1) > eps)
617 |         # axes[2].imshow(abs(N2) > eps)
618 |         axes[0].imshow(N0, cmap='RdYlBu_r', vmin=-1, vmax=1)
619 |         axes[1].imshow(N1, cmap='RdYlBu_r', vmin=-1, vmax=1)
620 |         axes[2].imshow(N2, cmap='RdYlBu_r', vmin=-1, vmax=1)        
621 |         plt.show()
622 |             
623 |     def demo_highorder_conv():
624 |         """
625 | 
626 |         Written by P. DERIAN 2018-01-09.
627 |         """         
628 |         # Parameters
629 |         wav = pywt.Wavelet('bior6.8')
630 |         wav_levels = 3  # Note: levels have no influence with the convolution-based form.
631 |         fd_order = 8  # Order of finite-differences filters for truth
632 |         U2, U1 = inr.readMotion('../demo/UVtruth.inr')
633 |         # Wavelet-based computations
634 |         hor_conv = HighOrderRegularizerConv(wav)
635 |         C1 = pywt.wavedecn(U1, hor_conv.wav, level=wav_levels, mode=hor_conv.mode)
636 |         C2 = pywt.wavedecn(U2, hor_conv.wav, level=wav_levels, mode=hor_conv.mode)
637 |         l2norm_hor, _ = hor_conv.evaluate(C1, C2, 'l2norm')
638 |         hornschunk_hor, _ = hor_conv.evaluate(C1, C2, 'hornschunck')
639 |         divergence_hor, _ = hor_conv.evaluate(C1, C2, 'div')
640 |         curl_hor, _ = hor_conv.evaluate(C1, C2, 'curl')
641 |         laplacian_hor, _ = hor_conv.evaluate(C1, C2, 'laplacian')
642 |         graddiv_hor, _ = hor_conv.evaluate(C1, C2, 'graddiv')
643 |         gradcurl_hor, _ = hor_conv.evaluate(C1, C2, 'gradcurl')
644 |         # Compute the truth
645 |         # Note: not real "truth" since gradients are computed by finite differences.
646 |         d1 = d1_filters[fd_order]
647 |         d2 = d2_filters[fd_order]
648 |         l2norm_truth = l2norm(U1, U2)
649 |         hornschunk_truth = hornschunk(U1, U2, d1)
650 |         divergence_truth = divergence(U1, U2, d1)
651 |         curl_truth = curl(U1, U2, d1)
652 |         laplacian_truth = laplacian(U1, U2, d2)
653 |         graddiv_truth = graddiv(U1, U2, d1)
654 |         gradcurl_truth = gradcurl(U1, U2, d1)
655 |         # Print out
656 |         print('\n** Convolution-based high-order computations')
657 |         print('Wavelet: {}'.format(wav))
658 |         print('Truth: centered finite-differences, order-{} accuracy'.format(fd_order))
659 |         print('High-order terms:')
660 |         for label, test, truth in zip(
661 |             ['l2-norm', 'H&S', 'divergence', 'curl', 'laplacian', 'graddiv', 'gradcurl'],
662 |             [l2norm_hor, hornschunk_hor, divergence_hor, curl_hor, laplacian_hor, graddiv_hor, gradcurl_hor],
663 |             [l2norm_truth, hornschunk_truth, divergence_truth, curl_truth, laplacian_truth, graddiv_truth, gradcurl_truth],
664 |             ):
665 |             print('\t{:>10}: rel. err={:.3f} % (wav={:.3f}, ref={:.3f})'.format(
666 |                 label, 100.*np.abs(test-truth)/truth, test, truth))
667 |             
668 |     def demo_highorder_mat():
669 |         """
670 | 
671 |         Written by P. DERIAN 2019-02-18.
672 |         """
673 |         # Parameters
674 |         wav = pywt.Wavelet('bior6.8')
675 |         wav_levels = 2  # Note: levels have no influence with the convolution-based form.
676 |         fd_order = 8  # Order of finite-differences filters for truth
677 |         U2, U1 = inr.readMotion('../demo/UVtruth.inr')
678 |         # Wavelet-based computations
679 |         hor_mat = HighOrderRegularizerMat(wav, wav_levels, U1.shape)
680 |         C1, _ = pywt.coeffs_to_array(pywt.wavedec2(
681 |             U1, hor_mat.wav, level=wav_levels, mode=hor_mat.mode))
682 |         C2, _ = pywt.coeffs_to_array(pywt.wavedec2(
683 |             U2, hor_mat.wav, level=wav_levels, mode=hor_mat.mode))
684 |         l2norm_hor, _ = hor_mat.evaluate(C1, C2, 'l2norm')
685 |         hornschunk_hor, _ = hor_mat.evaluate(C1, C2, 'hornschunck')
686 |         divergence_hor, _ = hor_mat.evaluate(C1, C2, 'div')
687 |         curl_hor, _ = hor_mat.evaluate(C1, C2, 'curl')
688 |         laplacian_hor, _ = hor_mat.evaluate(C1, C2, 'laplacian')
689 |         graddiv_hor, _ = hor_mat.evaluate(C1, C2, 'graddiv')
690 |         gradcurl_hor, _ = hor_mat.evaluate(C1, C2, 'gradcurl')
691 |         # Note: not actual "truth" since gradients are computed by finite differences.
692 |         d1 = d1_filters[fd_order]
693 |         d2 = d2_filters[fd_order]
694 |         l2norm_truth = l2norm(U1, U2)
695 |         hornschunk_truth = hornschunk(U1, U2, d1)
696 |         divergence_truth = divergence(U1, U2, d1)
697 |         curl_truth = curl(U1, U2, d1)
698 |         laplacian_truth = laplacian(U1, U2, d2)
699 |         graddiv_truth = graddiv(U1, U2, d1)
700 |         gradcurl_truth = gradcurl(U1, U2, d1)
701 |         # Print out
702 |         print('\n** Matrix-based high-order computations')
703 |         print('Wavelet: {}'.format(wav))
704 |         print('Truth: centered finite-differences, order-{} accuracy'.format(fd_order))
705 |         print('High-order terms:')        
706 |         for label, test, truth in zip(
707 |             ['l2-norm', 'H&S', 'divergence', 'curl', 'laplacian', 'grad(div)', 'grad(curl)'],
708 |             [l2norm_hor, hornschunk_hor, divergence_hor, curl_hor, laplacian_hor, graddiv_hor, gradcurl_hor],
709 |             [l2norm_truth, hornschunk_truth, divergence_truth, curl_truth, laplacian_truth, graddiv_truth, gradcurl_truth],
710 |             ):
711 |             print('\t{:>10}: rel. err={:.3f} % (wav={:.3f}, ref={:.3f})'.format(
712 |                 label, 100.*np.abs(test-truth)/truth, test, truth))
713 | 
714 |     def compare_highorders(name='laplacian'):
715 |         """
716 |         """
717 |         # Parameters
718 |         wav = pywt.Wavelet('bior6.8')
719 |         wav_levels = 3  # Note: levels have no influence with the convolution-based form.
720 |         fd_order = 8  # Order of finite-differences filters for truth
721 |         U2, U1 = inr.readMotion('../demo/UVtruth.inr')
722 |         # Regularizers
723 |         hor_conv = HighOrderRegularizerConv(wav)
724 |         hor_mat = HighOrderRegularizerMat(wav, wav_levels, U1.shape)
725 |         # Wavelet coeffs
726 |         C1 = pywt.wavedecn(U1.astype(np.float64), wav, level=wav_levels, mode='periodization')
727 |         C2 = pywt.wavedecn(U2.astype(np.float64), wav, level=wav_levels, mode='periodization')
728 |         C1c, _ = pywt.coeffs_to_array(C1)
729 |         C2c, _ = pywt.coeffs_to_array(C2)
730 |         # Compute terms
731 |         value_conv, grad_conv = hor_conv.evaluate(C1, C2, name)
732 |         value_mat, grad_mat = hor_mat.evaluate(C1c, C2c, name)
733 |         # Display
734 |         print('\n** Comparison of convolution vs Matrix high-order computations')
735 |         print('{}: conv={:.3f}, mat={:.3f}'.format(name, value_conv, value_mat))
736 |         fig, axes = plt.subplots(2, 4)
737 |         vmin = -0.5
738 |         vmax = 0.5
739 |         # Conv-based
740 |         g1_conv, _ = pywt.coeffs_to_array(grad_conv[0])
741 |         g2_conv, _ = pywt.coeffs_to_array(grad_conv[1])
742 |         axes[0, 0].imshow(g1_conv, vmin=vmin, vmax=vmax)
743 |         axes[1, 0].imshow(g2_conv, vmin=vmin, vmax=vmax)
744 |         # Matrix-based
745 |         g1_mat, g2_mat = grad_mat
746 |         axes[0, 1].imshow(g1_mat, vmin=vmin, vmax=vmax)
747 |         axes[1, 1].imshow(g2_mat, vmin=vmin, vmax=vmax)
748 |         # Errors
749 |         axes[0, 2].imshow(np.log10(np.abs(g1_conv-g1_mat)/abs(g1_mat)), cmap='RdYlBu_r', vmin=-5, vmax=5)
750 |         axes[1, 2].imshow(np.log10(np.abs(g2_conv-g2_mat)/abs(g2_mat)), cmap='RdYlBu_r', vmin=-5, vmax=5.)
751 |         # Ratios
752 |         axes[0, 3].imshow(np.abs(g1_conv/g1_mat), cmap='RdYlBu_r', vmin=0.5, vmax=2.)
753 |         axes[1, 3].imshow(np.abs(g2_conv/g2_mat), cmap='RdYlBu_r', vmin=0.5, vmax=2.)
754 |         plt.show()
755 |         
756 |     def check_conv():
757 |         """
758 |         """
759 |         # Data parameters
760 |         fine_scale = 7
761 |         nx = 1
762 |         ny = 3
763 |         sizex = nx * (2**fine_scale)
764 |         sizey = ny * (2**fine_scale)
765 |         wx = 3.  # x-freq
766 |         wy = 10.  # y-freq 
767 |         # Wavelet parameters
768 |         wav = pywt.Wavelet('db10')
769 |         wav_levels = 3  # Note: levels have no influence with the convolution-based form.
770 |         # The grid
771 |         x, dx = np.linspace(0, 2.*np.pi*nx, sizex, endpoint=False, retstep=True)
772 |         y, dy = np.linspace(0, 2.*np.pi*ny, sizey, endpoint=False, retstep=True)
773 |         yy, xx = np.meshgrid(y, x)
774 |         print(yy.size, np.sqrt(yy.size))
775 |         # The data
776 |         f = np.cos(wx*xx) + np.sin(wy*yy)
777 |         fx = -wx*np.sin(wx*xx)
778 |         fy = wy*np.cos(wy*yy)
779 |         fxx = -(wx**2)*np.cos(wx*xx)
780 |         fyy = -(wy**2)*np.sin(wy*yy)
781 |         fxy = np.zeros_like(xx)
782 |         # Theoretical norms
783 |         l2t = {
784 |             'f': 4. * (np.pi**2) * nx *ny,
785 |             'fx': 2 * (np.pi**2) * nx * ny * (wx**2),
786 |             'fy': 2 * (np.pi**2) * nx * ny * (wy**2),
787 |             'fxx': 2 * (np.pi**2) * nx * ny * (wx**4),
788 |             'fyy': 2 * (np.pi**2) * nx * ny * (wy**4),
789 |             'fxy': 0.,
790 |         }
791 |         # Rectangle quadrature formulas
792 |         def l2norm_rectangle(field):
793 |             return (dx*dy)*np.square(field).sum()
794 |         l2r_f = l2norm_rectangle(f)
795 |         l2r_fx = l2norm_rectangle(fx)
796 |         l2r_fy = l2norm_rectangle(fy)
797 |         l2r_fxx = l2norm_rectangle(fxx)
798 |         l2r_fyy = l2norm_rectangle(fyy)
799 |         l2r_fxy = l2norm_rectangle(fxy)
800 |         # Wavelet
801 |         hor_conv = HighOrderRegularizerConv(wav)
802 |         cf = pywt.wavedecn(f, hor_conv.wav, level=wav_levels, mode=hor_conv.mode)
803 |         l2w_f = hor_conv.norm(cf, 0, 0)
804 |         l2w_fx = hor_conv.norm(cf, 1, 0)
805 |         l2w_fy = hor_conv.norm(cf, 0, 1)
806 |         l2w_fxx = hor_conv.norm(cf, 2, 0)
807 |         l2w_fyy = hor_conv.norm(cf, 0, 2)
808 |         l2w_fxy = hor_conv.norm(cf, 1, 1)
809 |         # Print out
810 |         for label, l2r, l2w in zip(
811 |             ['f', 'fx', 'fy', 'fxx', 'fyy', 'fxy'],
812 |             [l2r_f, l2r_fx, l2r_fy, l2r_fxx, l2r_fyy, l2r_fxy],
813 |             [l2w_f, l2w_fx, l2w_fy, l2w_fxx, l2w_fyy, l2w_fxy],
814 |             ):
815 |             print('{:>3}: exact={:.3f}, rectangle approx.={:.3f} ; wavelet approx.={:.3f}'.format(
816 |                 label,
817 |                 l2t[label],
818 |                 l2r,
819 |                 l2w,
820 |             ))
821 |             print('\trect/exact={:.3f} ; wav/exact={:.3f}'.format(
822 |                 l2r / l2t[label],
823 |                 l2w / l2t[label],
824 |             ))
825 |         # Display
826 |         fig, ax  = plt.subplots()
827 |         ax.imshow(f, extent=[y[0], y[-1], x[0], x[-1]], origin='lower')
828 |         ax.set_xlabel('y')
829 |         ax.set_ylabel('x')
830 |         ax.set_title('f(x, y)')
831 |         plt.show()
832 |         
833 |     
834 |     # demo_mass_matrix()
835 |     demo_highorder_conv()
836 |     demo_highorder_mat()
837 |     compare_highorders('l2norm')
838 |     compare_highorders('hornschunck')
839 |     compare_highorders('div')
840 |     compare_highorders('curl')
841 |     compare_highorders('laplacian')
842 |     check_conv()


--------------------------------------------------------------------------------
/pytyphoon.py:
--------------------------------------------------------------------------------
  1 | """Python implementation of the Typhoon algorithm solving dense 2D optical flow problems.
  2 | 
  3 | Description: This module provides pure Python implementation of Typhoon.  At the moment,
  4 |     only the data term [1] is provided. The high-order regularizers [2] are not implemented
  5 |      in this module.
  6 | 
  7 | Disclaimer: The reference implementation used in [3], [4] is written in C++ and
  8 |     GPU-accelerated with CUDA. It is the property of Inria (FR) and the CSU Chico Research
  9 |     Foundation (Ca, USA). This Python implementation is *not* exactly the same as the
 10 |     reference for many reasons, and it is obviously much slower.
 11 | 
 12 | Dependencies:
 13 |     - numpy, scipy;
 14 |     - Pillow;
 15 |     - pywavelets.
 16 | 
 17 | References:
 18 |     [1] Data-term & basic algorithm:
 19 |         Derian, P.; Heas, P.; Herzet, C. & Memin, E.
 20 |         "Wavelets and Optical Flow Motion Estimation".
 21 |         Numerical Mathematics: Theory, Method and Applications, Vol. 6, pp. 116-137, 2013.
 22 |     [2] High-order regularization terms:
 23 |         Kadri Harouna, S.; Derian, P.; Heas, P. & Memin, E.
 24 |         "Divergence-free Wavelets and High Order Regularization".
 25 |         International Journal of Computer Vision, Vol. 103, pp. 80-99, 2013.
 26 |     [3] Application to wind estimation from lidar images:
 27 |         Derian, P.; Mauzey, C. F. & Mayor, S. D.
 28 |         "Wavelet-based optical flow for two-component wind field estimation from single aerosol lidar data"
 29 |         Journal of Atmospheric and Oceanic Technology, Vol. 32, pp. 1759-1778, 2015.
 30 |     [4] Application to near-shore surface current estimation from shore and UAV cameras:
 31 |         Derian, P. & Almar, R.
 32 |         "Wavelet-based Optical Flow Estimation of Instant Surface Currents from Shore-based and UAV Video"
 33 |         IEEE Transactions on Geoscience and Remote Sensing, Vol. 55, pp. 5790-5797, 2017.
 34 | 
 35 | Written by P. DERIAN 2018-01-09.
 36 | www.pierrederian.net - contact@pierrederian.net
 37 | """
 38 | __version__ = 0.1
 39 | ###
 40 | import numpy
 41 | import pywt
 42 | import scipy
 43 | import scipy.ndimage as ndimage
 44 | import scipy.optimize as optimize
 45 | ###
 46 | 
 47 | class OpticalFlowCore:
 48 |     """Core functions for optical flow.
 49 | 
 50 |     Written by P. DERIAN 2018-01-09.
 51 |     Updated by P. DERIAN 2018-01-11: generic n-d version, added __str__().
 52 |     """
 53 | 
 54 |     def __init__(self, shape, dtype=numpy.float32, interpolation_order=3, sigma_blur=0.5):
 55 |         """Constructor.
 56 | 
 57 |         :param shape: the grid shape;
 58 |         :param dtype: numpy.float32 or numpy.float64.
 59 |         :param interpolation_order: spline interpolation order>0, faster when lower.
 60 |         :param sigma_blur: sigma of gaussian blur applied before spatial gradient computation.
 61 | 
 62 |         Written by P. DERIAN 2018-01-09.
 63 |         Updated by P. DERIAN 2018-01-11: generic n-d version, changed boundary condition.
 64 |         """
 65 |         self.dtype = dtype
 66 |         ### grid coordinates
 67 |         self.shape = shape
 68 |         self.ndim = len(self.shape)
 69 |         # 1D
 70 |         self.x = tuple(numpy.arange(s, dtype=self.dtype) for s in self.shape)
 71 |         # N-D
 72 |         self.X = numpy.indices(self.shape, dtype=self.dtype)
 73 |         self.Xcoords = numpy.vstack((X.ravel() for X in self.X))
 74 |         ### Misc parameters
 75 |         self.sigma_blur = sigma_blur #gaussian blur sigma before spatial gradient computation
 76 |         self.interpolation_order = interpolation_order #pixel interpolation order
 77 |         self.boundary_condition = 'mirror' #boundary condition
 78 |         # Note: setting the bc to 'constant', 'reflect' or 'wrap' seems to cause issues...?
 79 |         # while 'nearest', 'mirror' are OK.
 80 |         ### Buffer
 81 |         self.buffer = numpy.zeros(numpy.prod(self.shape), dtype=dtype)
 82 | 
 83 |     def __str__(self):
 84 |         """
 85 |         Written by P. DERIAN 2018-10-11.
 86 |         """
 87 |         param_str = '\n\tshape={}'.format(self.shape)
 88 |         param_str += '\n\tdtype={}'.format(self.dtype.__name__)
 89 |         param_str += '\n\tinterpolation order={}'.format(self.interpolation_order)
 90 |         param_str += '\n\tsigma blur={}'.format(self.sigma_blur)
 91 |         return self.__class__.__name__ + param_str
 92 | 
 93 |     def DFD(self, im0, im1, U):
 94 |         """Compute the DFD.
 95 | 
 96 |         :param im0: the first (grayscale) image;
 97 |         :param im1: the second (grayscale) image;
 98 |         :param U: displacements (U1, U2, ...) along the first, second, ... axis;
 99 |         :return: the DFD.
100 | 
101 |         Written by P. DERIAN 2018-01-09.
102 |         Updated by P. DERIAN 2018-01-11: generic n-d version.
103 |         """
104 |         # warp im1
105 |         map_coords = self.Xcoords.copy()
106 |         for n, Ui in enumerate(U):
107 |             map_coords[n] += Ui.ravel()
108 |         im1_warp = ndimage.interpolation.map_coordinates(im1, map_coords,
109 |                                                          order=self.interpolation_order,
110 |                                                          mode=self.boundary_condition)
111 |         # difference
112 |         return im1_warp.reshape(self.shape) - im0
113 | 
114 |     def DFD_gradient(self, im0, im1, U):
115 |         """Compute the displaced frame difference (DFD) functional value and its gradients.
116 | 
117 |         :param im0: the first (grayscale) image;
118 |         :param im1: the second (grayscale) image;
119 |         :param U: displacements (U1, U2, ...) along the first, second, ... axis;
120 |         :return: dfd, (grad1, grad2, ...)
121 |             - DFD: the value of the functional;
122 |             - (grad1, grad2, ...): the gradient of the DFD functional w.r.t. component U1, U2, ....
123 | 
124 |         Written by P. DERIAN 2018-01-09.
125 |         Updated by P. DERIAN 2018-01-11: generic n-d version.
126 |         """
127 |         # warp im1->buffer
128 |         map_coords = self.Xcoords.copy()
129 |         for n, Ui in enumerate(U):
130 |             map_coords[n] += Ui.ravel()
131 |         ndimage.interpolation.map_coordinates(im1, map_coords,
132 |                                               output=self.buffer,
133 |                                               order=self.interpolation_order,
134 |                                               mode=self.boundary_condition)
135 |         im1_warp = self.buffer.reshape(self.shape)
136 |         # difference
137 |         dfd = im1_warp - im0
138 |         # spatial gradients
139 |         # [TODO] use derivative of gaussians?
140 |         if self.sigma_blur>0:
141 |             im1_warp = ndimage.filters.gaussian_filter(im1_warp, self.sigma_blur)
142 |         grad = numpy.gradient(im1_warp)
143 |         grad = (g*dfd for g in grad)
144 |         # return
145 |         return 0.5*numpy.sum(dfd**2), grad
146 | 
147 | class Typhoon:
148 |     """Implements the Typhoon algorithm: dense optical flow estimation on wavelet bases.
149 | 
150 |     Written by P. DERIAN 2018-01-09.
151 |     Updated by P. DERIAN 2018-01-10: added default values and solve_fast().
152 |     """
153 | 
154 |     DEFAULT_WAV = 'haar' #default wavelet name
155 |     DEFAULT_MODE = 'zero' #default wavelet signal extension mode
156 | 
157 |     def __init__(self, shape=None):
158 |         """Instance constructor with optional shape.
159 | 
160 |         :param shape: optional shape of the problem.
161 | 
162 |         Written by P. DERIAN 2018-01-09.
163 |         """
164 |         self.core = OpticalFlowCore(shape) if (shape is not None) else None
165 | 
166 |     def solve(self, im0, im1, wav=None, mode=None,
167 |               levels_decomp=3, levels_estim=None, U0=None,
168 |               interpolation_order=3, sigma_blur=0.5,
169 |               ):
170 |         """Solve the optical flow problem for given images and wavelet.
171 | 
172 |         :param im0: the first (grayscale) image;
173 |         :param im1: the second (grayscale) image;
174 |         :param wav: the name of the wavelet;
175 |         :param mode: the signal extension mode ('zero', 'periodization'), see [a].
176 |         :param levels_decomp: number of decomposition levels;
177 |         :param levels_estim: number of estimation levels (<=levels_decomp);
178 |         :param U0: optional first guess for U as (U1_0, U2_0, ...).
179 |         :param interpolation_order: spline interpolation order>0, faster when lower.
180 |         :param sigma_blur: sigma of gaussian blur applied before spatial gradient computation.
181 |         :return: U=(U1, U2, ...) the estimated displacement along the first, second, ... axes.
182 | 
183 |         Notes:
184 |             - without explicit regularization terms, it is necessary to set
185 |               levels_estim<levels_decomp in order to close the estimation problem.
186 |             - 'periodization' mode is much faster, but creates larger errors near edges.
187 | 
188 |         References:
189 |             [a] https://pywavelets.readthedocs.io/en/latest/ref/signal-extension-modes.html
190 | 
191 |         Written by P. DERIAN 2018-01-09.
192 |         Updated by P. DERIAN 2018-01-11: generic n-d version, checked levels, checked shapes.
193 |         Updated by P. DERIAN 2018-06-11: fixed first-guess motion (U0) which was not paded when needed.
194 |         """
195 |         ### Wavelets
196 |         # check arguments
197 |         if (wav is None) or (wav not in pywt.wavelist(kind='discrete')):
198 |             wav = self.DEFAULT_WAV
199 |             print('[!] set wavelet to default "{}".'.format(wav))
200 |         if (mode is None) or (mode not in pywt.Modes.modes):
201 |             mode = self.DEFAULT_MODE
202 |             print('[!] set mode to default "{}".'.format(mode))
203 |         # set final values
204 |         self.wav = pywt.Wavelet(wav)
205 |         self.wav_boundary_condition = mode
206 |         # check levels_decomp argument w.r.t. pywt
207 |         levels_max = min([pywt.dwt_max_level(s, self.wav) for s in im0.shape])
208 |         if levels_decomp>levels_max:
209 |             print('[!] too many decomposition levels ({}) requested for given wavelet/shape, set to {}.'.format(
210 |                   levels_decomp, levels_max))
211 |             levels_decomp = levels_max
212 |         # check levels_estim argument w.r.t levels_decomp, if not None
213 |         if (levels_estim is not None) and (levels_estim>levels_decomp):
214 |             print('[!] too many estimation levels ({}) requested for decomposition, set to {}.'.format(
215 |                   levels_estim, levels_decomp))
216 |             levels_estim = levels_decomp
217 |         # set the final levels values
218 |         self.levels_decomp = levels_decomp
219 |         self.levels_estim = levels_estim if (levels_estim is not None) else self.levels_decomp-1
220 | 
221 |         ### Images and core
222 |         # make sure the images have size compatible with decomposition levels, padding
223 |         # if necessary with zeros
224 |         block_size = 2**self.levels_decomp
225 |         # how much is missing in each axis
226 |         self.pad_size = tuple((block_size - (s%block_size))%block_size for s in im0.shape)
227 |         # the slice to crop back the original area
228 |         self.crop_slice = tuple(slice(None, -p if p else None, None) for p in self.pad_size)
229 |         # padd if necessary
230 |         if any(self.pad_size):
231 |             padding = [(0, p) for p in self.pad_size]
232 |             im0 = numpy.pad(im0, padding, mode='constant')
233 |             im1 = numpy.pad(im1, padding, mode='constant')
234 |         # create a new core if the image shape is not compatible
235 |         if (self.core is None) or (not numpy.testing.assert_equal(self.core.shape, im0.shape)):
236 |             self.core = OpticalFlowCore(im0.shape, interpolation_order=interpolation_order,
237 |                                         sigma_blur=sigma_blur)
238 |         print(self.core)
239 |         # and the images
240 |         self.im0 = im0.astype(self.core.dtype)
241 |         self.im1 = im1.astype(self.core.dtype)
242 | 
243 |         ### Motion fields
244 |         # get a sequence of the proper length
245 |         if U0 is None:
246 |             U0 = [None,]*self.core.ndim
247 |         # for each component
248 |         U = []
249 |         for Ui in U0:
250 |             # if None, initialize with zeros
251 |             if Ui is None:
252 |                 Ui = numpy.zeros(self.core.shape, dtype=self.core.dtype)
253 |             # else pad the given field if necessary
254 |             elif any(self.pad_size):
255 |                 padding = [(0, p) for p in self.pad_size]
256 |                 Ui = numpy.pad(Ui, padding, mode='constant')
257 |             U.append(Ui)
258 |         # as a tuple
259 |         U = tuple(U)
260 |         # the corresponding wavelet coefficients
261 |         self.C_list = [pywt.wavedecn(Ui, self.wav, level=self.levels_decomp,
262 |                                      mode=self.wav_boundary_condition) for Ui in U]
263 |         # which we reshape as arrays to get the slices for future manipulations.
264 |         self.slices = tuple(pywt.coeffs_to_array(Ci)[1] for Ci in self.C_list)
265 | 
266 |         ### Solve
267 |         print('Decomposition over {} scales of details, {} estimated'.format(
268 |             self.levels_decomp, self.levels_estim))
269 |         # for each level
270 |         for level in range(self.levels_estim+1):
271 |             print('details ({})'.format(level) if level else 'approx. (0)')
272 |             # the initial condition, as array (Note: flattened by l-bfgs)
273 |             C_array = (pywt.coeffs_to_array(Ci[:level+1])[0] for Ci in self.C_list)
274 |             C_array = numpy.vstack((Ci[numpy.newaxis,...] for Ci in C_array))
275 |             # so that C_array[i] contains all coefficients of component i.
276 |             # we remember the shape for future manipulations.
277 |             C_shape = C_array.shape
278 |             # create the cost function for this step
279 |             f_g = self.create_cost_function(level, C_shape)
280 |             # minimize
281 |             C_array, min_value, optim_info = optimize.fmin_l_bfgs_b(f_g,
282 |                                                                     C_array.astype(numpy.float64),
283 |                                                                     factr=1000.,
284 |                                                                     iprint=0)
285 |             print('\tl-bfgs completed with status {warnflag} - {nit} iterations, {funcalls} calls'.format(**optim_info))
286 |             print('\tcurrent functional value: {:.2f}'.format(min_value))
287 |             # store result in main coefficients
288 |             C_array = C_array.reshape(C_shape)
289 |             C_list = (pywt.array_to_coeffs(Ci, self.slices[i][:level+1], output_format='wavedecn')
290 |                       for i, Ci in enumerate(C_array))
291 |             for n, Ci in enumerate(C_list):
292 |                 for l in range(level+1):
293 |                     self.C_list[n][l] = Ci[l]
294 | 
295 |         ### Rebuild field and return
296 |         # cropping the relevant area
297 |         U = tuple(pywt.waverecn(Ci, self.wav, mode=self.wav_boundary_condition)[self.crop_slice]
298 |                   for Ci in self.C_list)
299 |         return U
300 | 
301 |     def solve_fast(self, im0, im1, wav='haar',
302 |                    levels_decomp=3, levels_estim=None, U0=None):
303 |         """Fastest estimation (but lower accuracy).
304 | 
305 |         :param im0: the first (grayscale) image;
306 |         :param im1: the second (grayscale) image;
307 |         :param wav: the name of the wavelet;
308 |         :param levels_decomp: number of decomposition levels;
309 |         :param levels_estim: number of estimation levels (<=levels_decomp);
310 |         :param U0: optional first guess for U as (U1_0, U2_0, ...).
311 |         :return: U=(U1, U2, ...) the estimated displacement along the first, second, ... axes.
312 | 
313 |         Note: uses linear (order 1) interpolation, no blurring and 'periodization' mode.
314 | 
315 |         Written by P. DERIAN 2018-01-11.
316 |         """
317 |         return self.solve(im0, im1, wav=wav, mode='periodization',
318 |                           levels_decomp=levels_decomp, levels_estim=levels_estim,
319 |                           interpolation_order=1, sigma_blur=0.)
320 | 
321 |     def create_cost_function(self, step, shape):
322 |         """The cost function takes wavelet coefficient as input parameters;
323 |         and returns (f, grad).
324 | 
325 |         :param step: 0 (coarse), 1 (first level of details, etc);
326 |         :param shape: the shape to reshape x.
327 |         :return: the cost function for l-bfgs.
328 | 
329 |         Written by P. DERIAN 2018-01-09.
330 |         """
331 |         def f_g(x):
332 |             """Compute the coast function and its gradient for given input x.
333 | 
334 |             :param x: input point, ndarray of float64.
335 |             :return: f, g
336 |                 - f the function value, scalar;
337 |                 - g the gradient, ndarray of same size and type as x.
338 | 
339 |             Notes:
340 |                 - We cast from (at input) and to (at output) float64, as l-bgs wants 8-byte floats.
341 |                 - There are losts of shape manipulations, some of them could possibly be
342 |                   avoided. This is due to l-bfgs using 1d arrays whereas pywt relies on lists
343 |                   of (tupples of) arrays. Here it was chosen to be as explicit and clear
344 |                   as possible, possibly sacrificing some speed along the way.
345 | 
346 |             Written by P. DERIAN 2018-01-09.
347 |             Updated by P. DERIAN 2018-01-11: generic n-d version.
348 |             """
349 |             ### rebuild motion field
350 |             # reshape 1d vector to 3d array
351 |             x = x.reshape(shape).astype(self.core.dtype)
352 |             # extract coefficients, complement with finer scales and reshape for pywt
353 |             # Note: ideally we would not complement, as finer scales are zeros. This is made
354 |             #   necessary by pywt.
355 |             C_list = (pywt.array_to_coeffs(xi, self.slices[i][:step+1],
356 |                                            output_format='wavedecn')
357 |                       + self.C_list[i][step+1:] for i, xi in enumerate(x))
358 |             # rebuild motion field
359 |             U = (pywt.waverecn(Ci, self.wav, mode=self.wav_boundary_condition) for Ci in C_list)
360 |             ### evaluate DFD and gradient
361 |             func_value, G = self.core.DFD_gradient(self.im0, self.im1, U)
362 |             # decompose gradient over wavelet basis, keep coefficients only up to current step
363 |             G_list = (pywt.wavedecn(Gi, self.wav, level=self.levels_decomp,
364 |                                     mode=self.wav_boundary_condition)[:step+1]
365 |                       for Gi in G)
366 |             # reshape as array, flatten and concatenate for l-bfgs
367 |             G_array = numpy.hstack(
368 |                 (pywt.coeffs_to_array(Gi)[0].ravel() for Gi in G_list))
369 |             ### evaluate regularizer and its gradient
370 |             # [TODO]
371 |             ### return DFD (+ regul) value, and its gradient w.r.t. x
372 |             return func_value, G_array.astype(numpy.float64)
373 |         return f_g
374 | 
375 |     @staticmethod
376 |     def solve_pyramid(im0, im1, levels_pyr=1, solve_fast=False,
377 |                       wav='haar', levels_decomp=3, levels_estim=None,
378 |                       **kwargs):
379 |         """Wrapper for pyramidal estimation.
380 | 
381 |         When very large displacements are involved, the usual pyramidal approach
382 |         can be employed to help achieving a correct estimation.
383 | 
384 |         :param im0: the first (grayscale) image;
385 |         :param im1: the second (grayscale) image;
386 |         :param levels_pyr: number of pyramid levels;
387 |         :param fast: if True, use faster (but less accurate) estimation;
388 |         :param wav: the name of the wavelet;
389 |         :param levels_decomp: number of decomposition levels;
390 |         :param levels_estim: number of estimation levels (<=levels_decomp);
391 |         :param **kwargs: remaining arguments passed to Typhoon.solve().
392 |         :return: U, typhoon
393 |             U=(U1, U2, ...) the estimated displacement along the first, second, ... axes.
394 |             typhoon the instance of Typhoon used for the last (finest) pyramid level.
395 | 
396 |         Written by P. DERIAN 2018-01-11.
397 |         """
398 |         downscale = 0.5
399 |         upscale = 1./downscale
400 |         ### create the pyramid of images
401 |         pyr_im0 = [im0,]
402 |         pyr_im1 = [im1,]
403 |         for l in range(levels_pyr):
404 |             # filter image and interpolate
405 |             tmp_im = ndimage.gaussian_filter(pyr_im0[0], 2./3.)
406 |             pyr_im0.insert(0, ndimage.interpolation.zoom(tmp_im, downscale))
407 |             tmp_im = ndimage.gaussian_filter(pyr_im1[0], 2./3.)
408 |             pyr_im1.insert(0, ndimage.interpolation.zoom(tmp_im, downscale))
409 |         ### for each level:
410 |         for l, (im0, im1) in enumerate(zip(pyr_im0, pyr_im1)):
411 |             # if not the first, use previous motion as first guess
412 |             if l:
413 |                 U0 = [ndimage.interpolation.zoom(Ui, upscale) for Ui in U]
414 |             else:
415 |                 U0 = None
416 |             typhoon = Typhoon()
417 |             if solve_fast:
418 |                 U = typhoon.solve_fast(im0, im1, wav=wav,
419 |                                        levels_decomp=levels_decomp,
420 |                                        levels_estim=levels_estim)
421 |             else:
422 |                 U = typhoon.solve(im0, im1, U0=U0, wav=wav,
423 |                                   levels_decomp=levels_decomp,
424 |                                   levels_estim=levels_estim,
425 |                                   **kwargs)
426 |         ### return the last estimates
427 |         return U, typhoon
428 | 
429 | ### Helpers ###
430 | 
431 | def RMSE(Ua, Ub):
432 |     """Root Mean Squared Error (RMSE).
433 | 
434 |     :param Ua: (Ua1, Ua2, ...) first vector field;
435 |     :param Ub: (Ub1, Ub2, ...) second vector field.
436 |     :return: the RMSE.
437 | 
438 |     Written by P. DERIAN 2018-01-09.
439 |     """
440 |     return numpy.sqrt(numpy.mean(numpy.sum((
441 |         numpy.asarray(Ua) - numpy.asarray(Ub))**2, axis=0)))
442 | 
443 | ### Demonstrations ###
444 | 
445 | if __name__=="__main__":
446 |     ###
447 |     import argparse
448 |     import os.path
449 |     import sys
450 |     import time
451 |     ###
452 |     import matplotlib
453 |     import matplotlib.pyplot as pyplot
454 |     from PIL import Image
455 |     ###
456 |     import demo.inr as inr
457 |     ###
458 | 
459 |     def print_versions():
460 |         print("\n* Module versions:")
461 |         print('Python:', sys.version)
462 |         print('Numpy:', numpy.__version__)
463 |         print('Scipy:', scipy.__version__)
464 |         print('PyWavelet:', pywt.__version__)
465 |         print('Matplotlib:', matplotlib.__version__)
466 |         print('PyTyphoon (this):', __version__)
467 | 
468 |     def demo_particles():
469 |         """Demo with the synthetic particle images.
470 | 
471 |         Written by P. DERIAN 2018-01-09.
472 |         """
473 |         print('\n* PyTyphoon {} ({}) – "particles" demo'.format(__version__, __file__))
474 | 
475 |         ### load data
476 |         im0 = numpy.array(Image.open('demo/run010050000.tif').convert(mode='L')).astype(float)/255.
477 |         im1 = numpy.array(Image.open('demo/run010050010.tif').convert(mode='L')).astype(float)/255.
478 | 
479 |         # Note:
480 |         #   - U1, V1 are vertical (1st axis) components;
481 |         #   - U2, V2 are horizontal (2nd axis) components.
482 |         #   - inr.ReadMotion() returns (horizontal, vertical).
483 |         V2, V1 = inr.readMotion('demo/UVtruth.inr')
484 | 
485 |         ### solve OF
486 |         typhoon = Typhoon()
487 |         tstart = time.clock()
488 |         U1, U2 = typhoon.solve(im0, im1, wav='db2', mode='periodization',
489 |                                levels_decomp=3, levels_estim=1)
490 |         tend = time.clock()
491 |         print('Estimation completed in {:.2f} s.'.format(tend-tstart))
492 | 
493 |         ### post-process & display
494 |         rmse = RMSE((U1, U2), (V1, V2))
495 |         dpi = 72.
496 |         fig, axes = pyplot.subplots(2,4, figsize=(800./dpi, 450./dpi))
497 |         # images
498 |         for ax, var, label in zip(axes[:,0], [im0, im1], ['image #0', 'image #1']):
499 |             pi = ax.imshow(var, vmin=0., vmax=1., interpolation='nearest', cmap='gray')
500 |             ax.set_title(label)
501 |             ax.set_xticks([])
502 |             ax.set_yticks([])
503 |         # velocity fields
504 |         for ax, var, label in zip(axes[:,1:-1].flat,
505 |                                   [U1, V1, U2, V2],
506 |                                   ['estim U1', 'true U1', 'estim U2', 'true U2']):
507 |             pf = ax.imshow(var, vmin=-3., vmax=3., interpolation='nearest', cmap='RdYlBu_r')
508 |             ax.set_title(label)
509 |             ax.set_xticks([])
510 |             ax.set_yticks([])
511 |         # error maps
512 |         for ax, var, label in zip(axes[:,-1],
513 |                                   [V1-U1, V2-U2],
514 |                                   ['abs(error U1)', 'abs(error U2)']):
515 |             pe = ax.imshow(numpy.abs(var), vmin=0, vmax=0.3, interpolation='nearest')
516 |             ax.set_title(label)
517 |             ax.set_xticks([])
518 |             ax.set_yticks([])
519 |         # colormaps
520 |         pyplot.subplots_adjust(bottom=0.25, top=0.94, left=.05, right=.95)
521 |         axf = fig.add_axes([1.25/8., 0.16, 2.5/8., 0.03])
522 |         pyplot.colorbar(pf, cax=axf, orientation='horizontal')
523 |         axf.set_title('displacement (pixel)', fontdict={'fontsize':'medium'})
524 |         axe = fig.add_axes([4.5/8., 0.16, 2.5/8., 0.03])
525 |         pyplot.colorbar(pe, cax=axe, orientation='horizontal')
526 |         axe.set_title('error (pixel)', fontdict={'fontsize':'medium'})
527 |         # labels
528 |         pyplot.figtext(
529 |             0.05, 0.015, 'PyTyphoon {} "particles" demo\n"{}" wavelet, {} scales decomp., {} scales estim., no regularizer.'.format(
530 |                 __version__, typhoon.wav.name, typhoon.levels_decomp, typhoon.levels_estim),
531 |             size='medium', ha='left', va='bottom')
532 |         pyplot.figtext(
533 |             0.95, 0.015, 'RMSE={:.2f} pixel'.format(rmse),
534 |             size='medium', ha='right', va='bottom')
535 |         pyplot.show()
536 | 
537 |     def demo_3dshift(seed=123456789):
538 |         """Demo of 3D estimation with simple shift.
539 | 
540 |         :param seed: seed for images generation. Pass None for random images.
541 | 
542 |         Synthetic images are generated by filtering gaussian noise.
543 |         The true motion is a simple integer shift in all directions.
544 |         Estimation is performed with the basic "fast" solver.
545 | 
546 |         Written by P. DERIAN 2018-10-12.
547 |         """
548 |         print('\n* PyTyphoon {} ({}) – "3dshift" demo'.format(__version__, __file__))
549 | 
550 |         ### create synthetic images
551 |         print('Generating images...')
552 |         # sum of random normal noise filtered by different gaussian kernels.
553 |         size = 64
554 |         sigma_list = [1., 3., 9.]
555 |         im0 = numpy.zeros((size, size, size))
556 |         numpy.random.seed(seed) #seed the generator for repeatability
557 |         for sigma in sigma_list:
558 |             tmp = numpy.random.randn(size, size, size)
559 |             im0 += ndimage.gaussian_filter(tmp, sigma, mode='wrap')
560 |         # rescale to [0, 1]
561 |         min = im0.min()
562 |         max = im0.max()
563 |         im0 = (im0-min)/(max-min)
564 |         # second image: simply shift
565 |         shift = (-1, 2, 3)
566 |         im1 = numpy.roll(im0, shift=shift, axis=(0,1,2))
567 | 
568 |         ### perform estimation
569 |         typhoon = Typhoon()
570 |         tstart = time.clock()
571 |         U = typhoon.solve_fast(im0, im1, wav='db2', levels_decomp=3, levels_estim=1)
572 |         tend = time.clock()
573 |         print('Estimation completed in {:.2f} s.'.format(tend-tstart))
574 | 
575 |         ### post-process and display
576 |         truth_label = 'true shift: ({}) pixel'.format(', '.join('{:.2f}'.format(s) for s in shift))
577 |         estim_label = 'estimated mean displacement: ({}) pixel'.format(', '.join('{:.2f}'.format(Ui.mean()) for Ui in U))
578 |         dpi = 72.
579 |         fig, axes = pyplot.subplots(3, 5, figsize=(800./dpi, 450./dpi))
580 |         # show the 3 component in the middle plane along each axis
581 |         slice_all = slice(None, None, None)
582 |         for i in range(3):
583 |             # slice of the mid-plane along this axis
584 |             slice_mid = [slice_all, slice_all, slice_all]
585 |             slice_mid[i] = size//2
586 |             # for each image
587 |             for j, (ax, imj) in enumerate(zip(axes[i,:2], [im0, im1])):
588 |                 ax.imshow(imj[slice_mid], interpolation='nearest', cmap='gray',
589 |                           vmin=0, vmax=1)
590 |                 ax.set_xticks([])
591 |                 ax.set_yticks([])
592 |                 ax.set_title('im{} @ mid-ax{}'.format(j, i+1))
593 |             # for each component
594 |             for j, (ax, Uj) in enumerate(zip(axes[i,2:], U), 1):
595 |                 pc = ax.imshow(Uj[slice_mid], interpolation='nearest', cmap='RdYlBu_r',
596 |                                vmin=-3, vmax=3)
597 |                 ax.set_xticks([])
598 |                 ax.set_yticks([])
599 |                 ax.set_title('U{} @ mid-ax{}'.format(j, i+1))
600 |         pyplot.subplots_adjust(left=.1, right=.9, bottom=.14, top=.9, hspace=.3)
601 |         # colorbar
602 |         cax = fig.add_axes([.75, 0.05, .2, 0.03])
603 |         cb = pyplot.colorbar(pc, cax=cax, orientation='horizontal')
604 |         cb.ax.set_title('displacement (pixel)', fontdict={'fontsize':'medium'})
605 |         # labels
606 |         pyplot.figtext(0.5, 0.98, truth_label+' - '+estim_label,
607 |                        ha='center', va='top', size='medium')
608 |         pyplot.figtext(
609 |             0.05, 0.015, 'PyTyphoon {} "3dshift" demo\n"{}"wavelet, {} scales decomp., {} scales estim., no regularizer, "fast" solver.'.format(
610 |                 __version__, typhoon.wav.name, typhoon.levels_decomp, typhoon.levels_estim),
611 |             size='medium', ha='left', va='bottom')
612 |         pyplot.show()
613 | 
614 |     def demo_3dvortex(seed=123456789):
615 |         """Demo of 3D estimation with vortex motion.
616 | 
617 |         :param seed: seed for images generation. Pass None for random images.
618 | 
619 |         Synthetic images are generated by filtering gaussian noise.
620 |         Estimation is performed with the basic "fast" solver.
621 | 
622 |         Written by P. DERIAN 2018-10-12.
623 |         """
624 |         print('\n* PyTyphoon {} ({}) – "3dvortex" demo'.format(__version__, __file__))
625 | 
626 |         ### create synthetic images
627 |         print('Generating images...')
628 |         # sum of random normal noise filtered by different gaussian kernels.
629 |         size = 96
630 |         margin = 4 #add a margin on every dimension
631 |         sigma_list = [1., 3., 9.] #a list of gaussian blur sigma
632 |         size_tmp = size + 2*margin
633 |         im0 = numpy.zeros((size_tmp, size_tmp, size_tmp))
634 |         numpy.random.seed(seed) #seed the generator for repeatability
635 |         for sigma in sigma_list:
636 |             tmp = numpy.random.randn(size_tmp, size_tmp, size_tmp)
637 |             im0 += ndimage.gaussian_filter(tmp, sigma, mode='wrap')
638 |         # rescale to [0, 1]
639 |         min = im0.min()
640 |         max = im0.max()
641 |         im0 = (im0-min)/(max-min)
642 |         ### create synthetic motion
643 |         # the coordinates
644 |         X1, X2, X3 = numpy.indices(im0.shape, dtype=float)
645 |         # tubular vortex: from streamfunction
646 |         sigma = float(size)/3. #make sure it decays enough near edges
647 |         alpha = float(size) #amplitude factor
648 |         beta = 3. #amplitude max of 3rd component
649 |         x0 = [size_tmp//2 for _ in range(3)] #coordinates of center of volume
650 |         SF = alpha*numpy.exp((-1./sigma**2)*( (X1-x0[0])**2 + (X2-x0[1])**2))
651 |         G1, G2, _ = numpy.gradient(SF)
652 |         # first 2 components are vortex, third increase linearly from zero to beta
653 |         U_truth = (G2,
654 |                    -G1,
655 |                    (X3 - margin)*(beta/float(size))) #divide by n_adv due to multiple steps
656 |         # displacement coordinates
657 |         n_adv = 30 #number of advection steps
658 |         map_coords = numpy.vstack((X.ravel() for X in (X1, X2, X3))) #for interpolation
659 |         for n, Ui in enumerate(U_truth):
660 |             map_coords[n] -= (1./float(n_adv))*Ui.ravel()
661 |         # very crude advection
662 |         im1 = im0
663 |         for _ in range(n_adv):
664 |             im1 = ndimage.interpolation.map_coordinates(im1, map_coords, order=3,
665 |                                                         mode='constant').reshape(im0.shape)
666 |         # finally retain only the working area
667 |         if margin>0:
668 |             im0 = im0[margin:-margin, margin:-margin, margin:-margin]
669 |             im1 = im1[margin:-margin, margin:-margin, margin:-margin]
670 |             U_truth = tuple(Ui[margin:-margin, margin:-margin, margin:-margin] for Ui in U_truth)
671 | 
672 |         ### perform estimation
673 |         # instance typhoon so we can use its internal coordinates for simplicity.
674 |         typhoon = Typhoon()
675 |         tstart = time.clock()
676 |         U = typhoon.solve_fast(im0, im1, wav='db3', levels_decomp=3, levels_estim=0)
677 |         tend = time.clock()
678 |         print('Estimation completed in {:.2f} s.'.format(tend-tstart))
679 | 
680 |         ### post-process and display
681 |         # show the 3 component in the middle plane along each axis
682 |         rmse = RMSE(U_truth, U)
683 |         dpi = 72.
684 |         fig, axes = pyplot.subplots(3, 5, figsize=(800./dpi, 450./dpi))
685 |         # for each axis
686 |         slice_all = slice(None, None, None)
687 |         for i in range(3):
688 |             # slice of the mid-plane along this axis
689 |             slice_mid = [slice_all, slice_all, slice_all]
690 |             slice_mid[i] = size//2
691 |             # for each image
692 |             for j, (ax, imj) in enumerate(zip(axes[i,:2], [im0, im1])):
693 |                 ax.imshow(imj[slice_mid], interpolation='nearest', cmap='gray',
694 |                           vmin=0, vmax=1)
695 |                 ax.autoscale(tight=True)
696 |                 ax.set_xticks([])
697 |                 ax.set_yticks([])
698 |                 ax.set_title('im{} @ mid-ax{}'.format(j, i+1))
699 |             # for each component
700 |             for j, (ax, Uj) in enumerate(zip(axes[i,2:], U), 1):
701 |                 pc = ax.imshow(Uj[slice_mid], interpolation='nearest', cmap='RdYlBu_r',
702 |                                vmin=-3, vmax=3)
703 |                 ax.set_aspect('equal')
704 |                 ax.set_xticks([])
705 |                 ax.set_yticks([])
706 |                 ax.set_title('U{} @ mid-ax{}'.format(j, i+1))
707 |         pyplot.subplots_adjust(left=.1, right=.9, bottom=.14, top=.95, hspace=.3)
708 |         # colorbar
709 |         cax = fig.add_axes([.565, 0.05, .2, 0.03])
710 |         cb = pyplot.colorbar(pc, cax=cax, orientation='horizontal')
711 |         cb.ax.set_title('displacement (pixel)', fontdict={'fontsize':'medium'})
712 |         # labels
713 |         pyplot.figtext(
714 |             0.05, 0.015, 'PyTyphoon {} "3dvortex" demo\n"{}"wavelet, {} scales decomp., {} scales estim., no regularizer, "fast" solver.'.format(
715 |                 __version__, typhoon.wav.name, typhoon.levels_decomp, typhoon.levels_estim),
716 |             size='medium', ha='left', va='bottom')
717 |         pyplot.figtext(
718 |             0.95, 0.015, 'RMSE={:.2f} pixel'.format(rmse),
719 |             size='medium', ha='right', va='bottom')
720 |         pyplot.show()
721 | 
722 |     def main(argv):
723 |         """Entry point.
724 | 
725 |         Written by P. DERIAN 2018-01-11.
726 |         """
727 |         ### Arguments
728 |         # create parser
729 |         parser = argparse.ArgumentParser(
730 |             description="Python implementation of Typhoon motion estimator.",
731 |             )
732 |         # estimation parameters
733 |         parser.add_argument('-i0', dest="im0", type=str, default='',
734 |                             help="first image")
735 |         parser.add_argument('-i1', dest="im1", type=str, default='',
736 |                             help="second image")
737 |         parser.add_argument('-w', '--wav', dest="wav", type=str, default=None,
738 |                             help="wavelet name")
739 |         parser.add_argument('-d', '--decomp', dest="levels_decomp", type=int, default=3,
740 |                             help="number of decomposition levels")
741 |         parser.add_argument('-e', '--estim', dest="levels_estim", type=int, default=None,
742 |                             help="number of estimation levels")
743 |         parser.add_argument('-p', '--pyr', dest="levels_pyr", type=int, default=0,
744 |                             help="pyramidal estimation levels")
745 |         parser.add_argument('--order', dest="interpolation_order", type=int, default=3,
746 |                             help="image interpolation order")
747 |         parser.add_argument('--blur', dest="sigma_blur", type=float, default=0.5,
748 |                             help="gaussian blur sigma")
749 |         parser.add_argument('--mode', dest="mode", type=str, default=None,
750 |                             help="signal extension mode")
751 |         parser.add_argument('--fast', dest="solve_fast", action='store_true',
752 |                             help="faster estimation (less accurate)")
753 |         parser.add_argument('--display', dest="display_result", action='store_true',
754 |                             help="display estimation results")
755 |         # misc arguments
756 |         parser.add_argument('--demo', dest="demo_name", type=str, default='',
757 |                             help="name of the demo")
758 |         parser.add_argument('--version', dest="print_version", action='store_true',
759 |                             help="print module versions")
760 |         #parser.add_argument('-q', "--quiet", dest="verbose", action='store_false',
761 |         #                    help="set quiet mode")
762 |         # set defaults and parse
763 |         parser.set_defaults(verbose=True, print_versions=False, display_result=False,
764 |                             solve_fast=False)
765 |         args = parser.parse_args(argv)
766 | 
767 |         ### Versions
768 |         if args.print_version:
769 |             print_versions()
770 |             return
771 | 
772 |         ### Demos
773 |         # list of available demos
774 |         demos = {'particles': demo_particles,
775 |                  '3dshift': demo_3dshift,
776 |                  '3dvortex': demo_3dvortex,
777 |                  }
778 |         default_demo = lambda : print("[!] Unkown demo '{}', valid demos: {}".format(
779 |             args.demo_name, list(demos.keys())))
780 |         # start a demo
781 |         if args.demo_name:
782 |             demos.get(args.demo_name, default_demo)()
783 |             return
784 | 
785 |         ### Single estimation
786 |         if args.im0 and args.im1:
787 |             print('\nEstimation:\n\t{}\n\t{}'.format(args.im0, args.im1))
788 |             ### load images
789 |             im0 = numpy.array(Image.open(args.im0).convert(mode='L')).astype(numpy.float32)/255.
790 |             im1 = numpy.array(Image.open(args.im1).convert(mode='L')).astype(numpy.float32)/255.
791 |             ### Solve problem
792 |             tstart = time.clock()
793 |             (U1, U2), typhoon = Typhoon.solve_pyramid(im0, im1, levels_pyr=args.levels_pyr,
794 |                                                       solve_fast=args.solve_fast,
795 |                                                       wav=args.wav, mode=args.mode,
796 |                                                       levels_decomp=args.levels_decomp,
797 |                                                       levels_estim=args.levels_estim,
798 |                                                       interpolation_order=args.interpolation_order,
799 |                                                       sigma_blur=args.sigma_blur)
800 |             tend = time.clock()
801 |             print('Estimation completed in {:.2f} s.'.format(tend-tstart))
802 |             # export
803 |             # [TODO] retrieve estim parameters and save as well.
804 |             # display
805 |             # [TODO] move to function?
806 |             if args.display_result:
807 |                 dpi = 72.
808 |                 fig, axes = pyplot.subplots(2,3, figsize=(800./dpi, 600./dpi))
809 |                 # images
810 |                 for ax, var, label in zip(axes[0,1:], [im0, im1], ['image #0', 'image #1']):
811 |                     pi = ax.imshow(var, vmin=0., vmax=1., interpolation='nearest', cmap='gray')
812 |                     ax.set_title(label)
813 |                     ax.set_xticks([])
814 |                     ax.set_yticks([])
815 |                 axes[0,0].remove()
816 |                 # velocity fields
817 |                 pf = []
818 |                 for ax, var, label in zip(axes[1,1:], [U1, U2], ['estim U1', 'estim U2']):
819 |                     pf.append(ax.imshow(var, interpolation='nearest', cmap='RdYlBu_r'))
820 |                     ax.set_title(label)
821 |                     ax.set_xticks([])
822 |                     ax.set_yticks([])
823 |                 # colorbars?
824 |                 # [TODO]
825 |                 # quiver - note that array axes order is reversed
826 |                 ax = axes[1,0]
827 |                 ax.set_aspect('equal')
828 |                 qstep = 8
829 |                 ax.quiver(typhoon.core.x[1][::qstep], typhoon.core.x[0][::qstep],
830 |                           U2[::qstep, ::qstep], U1[::qstep, ::qstep],
831 |                           pivot='middle', color='b', units='xy', angles='xy')
832 |                 ax.set_xlim(typhoon.core.x[1][0], typhoon.core.x[1][-1])
833 |                 ax.set_ylim(typhoon.core.x[0][0], typhoon.core.x[0][-1])
834 |                 ax.invert_yaxis()
835 |                 ax.set_title('motion field')
836 |                 ax.set_xticks([])
837 |                 ax.set_yticks([])
838 |                 pyplot.subplots_adjust(left=.05, right=.95, top=.95, bottom=.1)
839 |                 # labels
840 |                 pyplot.figtext(0.05, 0.015, 'PyTyphoon {}'.format(__version__),
841 |                                size='medium', ha='left', va='bottom')
842 |                 pyplot.figtext(0.05, 0.95,
843 |                                'im0: {}\nim1: {}\nlevels pyramid: {}\nwavelet: {}\nwav. decomp. lvls: {}\nwav. estim. lvls: {}'.format(
844 |                                     os.path.basename(args.im0),
845 |                                     os.path.basename(args.im1),
846 |                                     args.levels_pyr,
847 |                                     typhoon.wav.name,
848 |                                     typhoon.levels_decomp,
849 |                                     typhoon.levels_estim),
850 |                                size='medium', ha='left', va='top')
851 |                 pyplot.show()
852 | 
853 |     main(sys.argv[1:])
854 | 


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