Riesz Pyramid Creation and Reconstruction in Python

riesz_pyramid.py

"""
riesz_pyramid.py
Conversion between Riesz and Laplacian image pyramids
Based on the data structures and methodoligies described in:

Riesz Pyramids for Fast Phase-Based Video Magnification
Neal Wadhwa, Michael Rubinstein, Fredo Durand and William T. Freeman
Computational Photography (ICCP), 2014 IEEE International Conference on 

Copyright (c) 2016 Jack Doerner

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
""" 

import numpy, math
import scipy, scipy.signal

#riesz_band_filter = numpy.asarray([[-0.5, 0, 0.5]])
#riesz_band_filter = numpy.asarray([[-0.2,-0.48, 0, 0.48,0.2]])
riesz_band_filter = numpy.asarray([[-0.12,0,0.12],[-0.34, 0, 0.34],[-0.12,0,0.12]])

def laplacian_to_riesz(pyr):
	newpyr = {'I':pyr[:-1], 'R1':[], 'R2':[]}
	for ii in range(len(pyr) - 1):
		newpyr['R1'].append( scipy.signal.convolve2d(pyr[ii], riesz_band_filter, mode='same', boundary='symm') )
		newpyr['R2'].append( scipy.signal.convolve2d(pyr[ii], riesz_band_filter.T, mode='same', boundary='symm') )
	newpyr['base'] = pyr[-1]
	return newpyr

def riesz_to_spherical(pyr):
	newpyr = {'A':[],'theta':[],'phi':[],'Q':[],'base':pyr['base']}
	for ii in range(len(pyr['I']) ):
		I = pyr['I'][ii]
		R1 = pyr['R1'][ii]
		R2 = pyr['R2'][ii]
		A = numpy.sqrt(I*I + R1*R1 + R2*R2)
		theta = numpy.arctan2(R2,R1)
		Q = R1 * numpy.cos(theta) + R2 * numpy.sin(theta)
		phi = numpy.arctan2(Q,I)

		newpyr['A'].append( A )
		newpyr['theta'].append( theta )
		newpyr['phi'].append( phi )
		newpyr['Q'].append( Q )
	return newpyr


def riesz_spherical_to_laplacian(pyr):
	newpyr = []
	for ii in range(len(pyr['A'])):
		newpyr.append( pyr['A'][ii] * numpy.cos( pyr['phi'][ii] ) )
	newpyr.append(pyr['base'])
	return newpyr

rp_boundary.py

import numpy

def symmetrical_boundary(img):
	#manually set up a symmetrical boundary condition so we can use fftconvolve
	#but avoid edge effects
	(h,w) = img.shape
	imgsymm = numpy.empty((h*2, w*2))
	imgsymm[h/2:-(h+1)/2, w/2:-(w+1)/2] = img.copy()
	imgsymm[0:h/2, 0:w/2] = img[0:h/2, 0:w/2][::-1,::-1].copy()
	imgsymm[-(h+1)/2:, -(w+1)/2:] = img[-(h+1)/2:, -(w+1)/2:][::-1,::-1].copy()
	imgsymm[0:h/2, -(w+1)/2:] = img[0:h/2, -(w+1)/2:][::-1,::-1].copy()
	imgsymm[-(h+1)/2:, 0:w/2] = img[-(h+1)/2:, 0:w/2][::-1,::-1].copy()
	imgsymm[h/2:-(h+1)/2, 0:w/2] = img[:, 0:w/2][:,::-1].copy()
	imgsymm[h/2:-(h+1)/2, -(w+1)/2:] = img[:, -(w+1)/2:][:,::-1].copy()
	imgsymm[0:h/2, w/2:-(w+1)/2] = img[0:h/2, :][::-1,:].copy()
	imgsymm[-(h+1)/2:, w/2:-(w+1)/2] = img[-(h+1)/2:, :][::-1,:].copy()
	return imgsymm

rp_laplacian_like.py

"""
rp_laplacian_like.py
Conversion between image and laplacian-like pyramids
Based on the data structures and methodoligies described in:

Riesz Pyramids for Fast Phase-Based Video Magnification
Neal Wadhwa, Michael Rubinstein, Fredo Durand and William T. Freeman
Computational Photography (ICCP), 2014 IEEE International Conference on 

Copyright (c) 2016 Jack Doerner

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
""" 

import numpy, cv2, scipy.signal
from rp_boundary import *

lowpass = numpy.asarray([
	[-0.0001,   -0.0007,  -0.0023,  -0.0046,  -0.0057,  -0.0046,  -0.0023,  -0.0007,  -0.0001],
	[-0.0007,   -0.0030,  -0.0047,  -0.0025,  -0.0003,  -0.0025,  -0.0047,  -0.0030,  -0.0007],
	[-0.0023,   -0.0047,   0.0054,   0.0272,   0.0387,   0.0272,   0.0054,  -0.0047,  -0.0023],
	[-0.0046,   -0.0025,   0.0272,   0.0706,   0.0910,   0.0706,   0.0272,  -0.0025,  -0.0046],
	[-0.0057,   -0.0003,   0.0387,   0.0910,   0.1138,   0.0910,   0.0387,  -0.0003,  -0.0057],
	[-0.0046,   -0.0025,   0.0272,   0.0706,   0.0910,   0.0706,   0.0272,  -0.0025,  -0.0046],
	[-0.0023,   -0.0047,   0.0054,   0.0272,   0.0387,   0.0272,   0.0054,  -0.0047,  -0.0023],
	[-0.0007,   -0.0030,  -0.0047,  -0.0025,  -0.0003,  -0.0025,  -0.0047,  -0.0030,  -0.0007],
	[-0.0001,   -0.0007,  -0.0023,  -0.0046,  -0.0057,  -0.0046,  -0.0023,  -0.0007,  -0.0001]
])

highpass = numpy.asarray([
	[0.0000,   0.0003,   0.0011,   0.0022,   0.0027,   0.0022,   0.0011,   0.0003,   0.0000],
	[0.0003,   0.0020,   0.0059,   0.0103,   0.0123,   0.0103,   0.0059,   0.0020,   0.0003],
	[0.0011,   0.0059,   0.0151,   0.0249,   0.0292,   0.0249,   0.0151,   0.0059,   0.0011],
	[0.0022,   0.0103,   0.0249,   0.0402,   0.0469,   0.0402,   0.0249,   0.0103,   0.0022],
	[0.0027,   0.0123,   0.0292,   0.0469,  -0.9455,   0.0469,   0.0292,   0.0123,   0.0027],
	[0.0022,   0.0103,   0.0249,   0.0402,   0.0469,   0.0402,   0.0249,   0.0103,   0.0022],
	[0.0011,   0.0059,   0.0151,   0.0249,   0.0292,   0.0249,   0.0151,   0.0059,   0.0011],
	[0.0003,   0.0020,   0.0059,   0.0103,   0.0123,   0.0103,   0.0059,   0.0020,   0.0003],
	[0.0000,   0.0003,   0.0011,   0.0022,   0.0027,   0.0022,   0.0011,   0.0003,   0.0000]
])

def getsize(img):
	h, w = img.shape[:2]
	return w, h

def build_laplacian(img, minsize=2, convolutionThreshold=500, dtype=numpy.float64):
	img = dtype(img)
	levels = []
	while (min(img.shape) > minsize):

		if (img.size < convolutionThreshold):
			convolutionFunction = scipy.signal.convolve2d
		else:
			convolutionFunction = scipy.signal.fftconvolve

		w, h = getsize(img)
		symmimg = symmetrical_boundary(img)
		
		hp_img = convolutionFunction(symmimg, highpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]
		lp_img = convolutionFunction(symmimg, lowpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]
		levels.append(hp_img)

		img = cv2.pyrDown(lp_img)

	levels.append(img)
	return levels

def collapse_laplacian(levels, convolutionThreshold=500):
	img = levels[-1]
	for ii in range(len(levels)-2,-1,-1):
		lev_img = levels[ii]

		img = cv2.pyrUp(img, dstsize=getsize(lev_img))

		if (img.size < convolutionThreshold):
			convolutionFunction = scipy.signal.convolve2d
		else:
			convolutionFunction = scipy.signal.fftconvolve

		w, h = getsize(img)
		symmimg = symmetrical_boundary(img)
		symmlev = symmetrical_boundary(lev_img)

		img = convolutionFunction(symmimg, lowpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]
		img += convolutionFunction(symmlev, highpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]
	return img

Thank you for uploading these codes. They are super helpful to me.
However, one thing that is strange for me is the use of "cv2.pyrDown" and "cv2.pyrUp" in "build_laplacian" and "collapse_laplacian" functions. As far as I know, "cv2.pyrDown" and "cv2.pyrUp" include Gaussian kernel convolution, which potentially breaks the original design of the lowpass/ highpass kernels by Wadhwa et al.

"cv2.pyrDown" and "cv2.pyrUp" include Gaussian kernel convolution, which potentially breaks the original design of the lowpass/ highpass kernels by Wadhwa et al.

That's correct. Instead of using pyrUp/pyrDown, one should use subsampling without interpolation, as well as upsampling with zero induced even cols/rows. To compensate for the lost energy, the lowpass is multiplied with 2.

Not tested but for rp_laplacian_like.py one should replace:

Row 76:
lp_img = convolutionFunction(symmimg, 2.0*lowpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]

Row 79:
img = lp_img[::2, ::2]

Row 89:
img = cv2.resize(img, dstsize=getsize(lev_img), interpolation=cv2.INTER_AREA)
img[::2, ::2] = 0.0

Row 100:
img = convolutionFunction(symmimg, 2.0*lowpass, mode='same')[h/2:-(h+1)/2,w/2:-(w+1)/2]

That should do the trick.

@t-fukiage: Sorry I didn't initially notice your comment! I'm glad you've found this all helpful. Your observation likely explains why I originally had problems with this code.

@tschnz: Thank you for posting these corrections. I wrote this code for a film I was working on a few years ago, not expecting it ever to be used for anything else. I was never sure I got everything right, mathematically-speaking. By stunning coincidence, I actually shot another film a few months ago for which I once again need these techniques, so you've been a great help to me as well.

As a side-note, I use Reisz pyramids for motion interpolation (i.e. slow-motion) instead of motion magnification as described in the original paper. I find it works dramatically better than any commercial solution for certain kinds of motion (in particular, fire and liquids). I've been meaning to write it up for years, but I still haven't gotten around to it.

@jackdoerner Thank you for writing that up. I initially didn't understand it from what I read in the paper so I got the supplemental Matlab code, a C++ application (with pyrDown/pyrUp) and this Gist. All in all I got it now. Glad I could help. Out of curiosity: How did you interpolate the frames? Any papers or sources referencing to that technique? Sounds interesting!

@jackdoerner Thank you for sharing this code!
I was able to make it run if casting all divisions inside indexes to int, for example imgsymm[int(h/2):int(-(h+1)/2). I am not sure if something has changed in python (3.9) that gives me the error TypeError: slice indices must be integers or None or have an index method

@tschnz There still appears to be some problems with the changes you made to the code: I am able to create what appears to be correct laplace or rectangular/cylindrical/spherical riesz pyramid and transform the riesz back to laplace. The problem appears when trying to reconstruct the original image from the laplace pyramid. This code makes a blurry img2 rather than a clone of the original img:
img2=rp_laplacian_like.collapse_laplacian(rp_laplacian_like.build_laplacian(img))

@t-fukiage Did you manage to change the code and was able to use it?

@evenlund
The following code is a modified version of rp_laplacian_like.py that I used.
Not sure if it still works as it was tested more than 3 years ago.
Hope this helps.

import numpy, scipy.signal

lowpass = numpy.asarray([
	[-0.0001,   -0.0007,  -0.0023,  -0.0046,  -0.0057,  -0.0046,  -0.0023,  -0.0007,  -0.0001],
	[-0.0007,   -0.0030,  -0.0047,  -0.0025,  -0.0003,  -0.0025,  -0.0047,  -0.0030,  -0.0007],
	[-0.0023,   -0.0047,   0.0054,   0.0272,   0.0387,   0.0272,   0.0054,  -0.0047,  -0.0023],
	[-0.0046,   -0.0025,   0.0272,   0.0706,   0.0910,   0.0706,   0.0272,  -0.0025,  -0.0046],
	[-0.0057,   -0.0003,   0.0387,   0.0910,   0.1138,   0.0910,   0.0387,  -0.0003,  -0.0057],
	[-0.0046,   -0.0025,   0.0272,   0.0706,   0.0910,   0.0706,   0.0272,  -0.0025,  -0.0046],
	[-0.0023,   -0.0047,   0.0054,   0.0272,   0.0387,   0.0272,   0.0054,  -0.0047,  -0.0023],
	[-0.0007,   -0.0030,  -0.0047,  -0.0025,  -0.0003,  -0.0025,  -0.0047,  -0.0030,  -0.0007],
	[-0.0001,   -0.0007,  -0.0023,  -0.0046,  -0.0057,  -0.0046,  -0.0023,  -0.0007,  -0.0001]
])

highpass = numpy.asarray([
	[0.0000,   0.0003,   0.0011,   0.0022,   0.0027,   0.0022,   0.0011,   0.0003,   0.0000],
	[0.0003,   0.0020,   0.0059,   0.0103,   0.0123,   0.0103,   0.0059,   0.0020,   0.0003],
	[0.0011,   0.0059,   0.0151,   0.0249,   0.0292,   0.0249,   0.0151,   0.0059,   0.0011],
	[0.0022,   0.0103,   0.0249,   0.0402,   0.0469,   0.0402,   0.0249,   0.0103,   0.0022],
	[0.0027,   0.0123,   0.0292,   0.0469,  -0.9455,   0.0469,   0.0292,   0.0123,   0.0027],
	[0.0022,   0.0103,   0.0249,   0.0402,   0.0469,   0.0402,   0.0249,   0.0103,   0.0022],
	[0.0011,   0.0059,   0.0151,   0.0249,   0.0292,   0.0249,   0.0151,   0.0059,   0.0011],
	[0.0003,   0.0020,   0.0059,   0.0103,   0.0123,   0.0103,   0.0059,   0.0020,   0.0003],
	[0.0000,   0.0003,   0.0011,   0.0022,   0.0027,   0.0022,   0.0011,   0.0003,   0.0000]
])



def build_laplacian(img, minsize=2, dtype=numpy.float64):
    img = dtype(img)
    levels = []
    while (min(img.shape) > minsize):
        convolutionFunction = scipy.signal.convolve2d

        hp_img = convolutionFunction(img, highpass, mode='same',boundary='fill')
        lp_img = convolutionFunction(img, lowpass, mode='same',boundary='fill')
        levels.append(hp_img)
        img = lp_img[0::2,0::2]

    levels.append(img)
    return levels


def collapse_laplacian(levels):
    img = levels[-1]
    for ii in range(len(levels)-2,-1,-1):
        lev_img = levels[ii]

        upimg = numpy.zeros((img.shape[0]*2,img.shape[1]*2))
        upimg[0::2,0::2]=img.copy()*4

        img = upimg[0:lev_img.shape[0],0:lev_img.shape[1]]

        convolutionFunction = scipy.signal.convolve2d

        img = convolutionFunction(img, lowpass, mode='same',boundary='fill')
        img += convolutionFunction(lev_img, highpass, mode='same',boundary='fill')
    return img

@t-fukiage That was working really nice - Thank you!

I got some small deviations close to the edges, but not a whole lot. I have experimented a bit with the borders, but the 'fill' method appears to be best. I also included the scipy.signal.fftconvolve method to speed up the processing for larger images and that appears to have similar edge deviations.

@evenlund
Good to hear that the code worked.

If you want to handle boundary with convolution, the following modification would do the trick:

def build_laplacian(img, minsize=2, dtype=numpy.float64):
    img = dtype(img)
    levels = []
    while (min(img.shape) > minsize):
        # convolutionFunction = scipy.signal.convolve2d
        # hp_img = convolutionFunction(img, highpass, mode='same',boundary='fill')
        # lp_img = convolutionFunction(img, lowpass, mode='same',boundary='fill')
        hp_img = scipy.signal.convolve2d(numpy.pad(img, (highpass.shape[0]-1)//2, mode='reflect'), highpass, mode='valid')
        lp_img = scipy.signal.convolve2d(numpy.pad(img, (lowpass.shape[0]-1)//2, mode='reflect'), lowpass, mode='valid')

        levels.append(hp_img)
        img = lp_img[0::2,0::2]

    levels.append(img)
    return levels


def collapse_laplacian(levels):
    img = levels[-1]
    for ii in range(len(levels)-2,-1,-1):
        lev_img = levels[ii]

        upimg = numpy.zeros((img.shape[0]*2,img.shape[1]*2))
        upimg[0::2,0::2]=img.copy()*4

        img = upimg[0:lev_img.shape[0],0:lev_img.shape[1]]

        # convolutionFunction = scipy.signal.convolve2d
        # img = convolutionFunction(img, lowpass, mode='same',boundary='fill')
        # img += convolutionFunction(lev_img, highpass, mode='same',boundary='fill')

        img = scipy.signal.convolve2d(numpy.pad(img, (lowpass.shape[0]-1)//2, mode='reflect'), lowpass, mode='valid')
        img += scipy.signal.convolve2d(numpy.pad(lev_img, (highpass.shape[0]-1)//2, mode='reflect'), highpass, mode='valid')
   
    return img

I used numpy.pad function to apply mirror (reflection) padding. The original code by @jackdoerner includes a function to apply symmetric padding, but the symmetric padding can cause another artifact in my modified code. (e.g., it amplifies energy at the top-left corner)