Image reconstruction based on Zernike moments using mahotas and opencv

Question

I heard about mahotas following this tutorial in the hope of finding a good implementation of Zernike polynomials in python. It couldn't be easier. However, I need to compare the Euclidean difference between the original image and the one reconstructed from the Zernike moments. I asked mahotas' author if he could possibly add the reconstruction functionality to his library, but he doesn't have time to build it.

How can I reconstruct an image in OpenCV using the Zernike moments provided by mahotas?

Answer 1

Based on the code that fireant mentioned in his answer I developed the following code for reconstruction. I also found the research papers [A. Khotanzad and Y. H. Hong, “Invariant image recognition by Zernike moments”] and [S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments”] very useful.

The function _slow_zernike_poly constructs 2-D Zernike basis functions. In the zernike_reconstruct function, we project the image on to the basis functions returned by _slow_zernike_poly and calculate the moments. Then we use the reconstruction formula.

Below is an example reconstruction done using this code:

Input image

Real part of the reconstructed image using order 12

'''
Copyright (c) 2015
Dhanushka Dangampola <[email protected]>

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 as np
from math import atan2
from numpy import cos, sin, conjugate, sqrt

def _slow_zernike_poly(Y,X,n,l):
    def _polar(r,theta):
        x = r * cos(theta)
        y = r * sin(theta)
        return 1.*x+1.j*y

    def _factorial(n):
        if n == 0: return 1.
        return n * _factorial(n - 1)
    y,x = Y[0],X[0]
    vxy = np.zeros(Y.size, dtype=complex)
    index = 0
    for x,y in zip(X,Y):
        Vnl = 0.
        for m in range( int( (n-l)//2 ) + 1 ):
            Vnl += (-1.)**m * _factorial(n-m) /  \
                ( _factorial(m) * _factorial((n - 2*m + l) // 2) * _factorial((n - 2*m - l) // 2) ) * \
                ( sqrt(x*x + y*y)**(n - 2*m) * _polar(1.0, l*atan2(y,x)) )
        vxy[index] = Vnl
        index = index + 1

    return vxy

def zernike_reconstruct(img, radius, D, cof):

    idx = np.ones(img.shape)

    cofy,cofx = cof
    cofy = float(cofy)
    cofx = float(cofx)
    radius = float(radius)    

    Y,X = np.where(idx > 0)
    P = img[Y,X].ravel()
    Yn = ( (Y -cofy)/radius).ravel()
    Xn = ( (X -cofx)/radius).ravel()

    k = (np.sqrt(Xn**2 + Yn**2) <= 1.)
    frac_center = np.array(P[k], np.double)
    Yn = Yn[k]
    Xn = Xn[k]
    frac_center = frac_center.ravel()

    # in the discrete case, the normalization factor is not pi but the number of pixels within the unit disk
    npix = float(frac_center.size)

    reconstr = np.zeros(img.size, dtype=complex)
    accum = np.zeros(Yn.size, dtype=complex)

    for n in range(D+1):
        for l in range(n+1):
            if (n-l)%2 == 0:
                # get the zernike polynomial
                vxy = _slow_zernike_poly(Yn, Xn, float(n), float(l))
                # project the image onto the polynomial and calculate the moment
                a = sum(frac_center * conjugate(vxy)) * (n + 1)/npix
                # reconstruct
                accum += a * vxy
    reconstr[k] = accum
    return reconstr

if __name__ == '__main__':

    import cv2
    import pylab as pl
    from matplotlib import cm

    D = 12

    img = cv2.imread('fl.bmp', 0)

    rows, cols = img.shape
    radius = cols//2 if rows > cols else rows//2

    reconst = zernike_reconstruct(img, radius, D, (rows/2., cols/2.))

    reconst = reconst.reshape(img.shape)

    pl.figure(1)
    pl.imshow(img, cmap=cm.jet, origin = 'upper')
    pl.figure(2)    
    pl.imshow(reconst.real, cmap=cm.jet, origin = 'upper')

Answer 2

It's not that difficult, I think you can code it yourself. First, remember the inverse of each moment/matrix aka basis image is the transpose of that matrix, as they are orthogonal. Then look at the code the author of that library uses to test that function. This is simpler than the code in the library so you can read and understand how it works (of course much slower too). You need to get those matrices for each moment which are basis images. You can modify _slow_znl to get the values forx,y calculated inside the main loop for x,y,p in zip(X,Y,P): and store in a matrix with the same size as the input image. Pass a white image to _slow_zernike and get all the moment matrices up to the radial degree you want. To reconstruct an image using the coefficients, just use the transpose of those matrices as you would do with say Haar transformation.