Question

我在this教程之后听说过mahotas，希望在python中找到一个很好的Zernike多项式实现。这可不容易。但是，我需要比较原始图像和Zernike矩重建的图像之间的欧几里德差异。我是asked mahotas的作者，如果他可以将重建功能添加到他的库中，但他没有时间来构建它。

如何使用mahotas提供的Zernike时刻在OpenCV中重建图像？

Answer 1

基于他在答案中提到的code，我开发了以下重建代码。我还发现研究论文[A. Khotanzad and Y. H. Hong, “Invariant image recognition by Zernike moments”]和[S.-K. Hwang and W.-Y. Kim, “A novel approach to the fast computation of Zernike moments”]非常有用。

函数 _slow_zernike_poly 构造2-D Zernike基函数。在 zernike_reconstruct 函数中，我们将图像投影到 _slow_zernike_poly 返回的基函数并计算矩。然后我们使用重建公式。

以下是使用此代码完成的重建示例：

输入图片

使用顺序12重建图像的实部

'''
Copyright (c) 2015
Dhanushka Dangampola <dhanushkald@gmail.com>

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

它并不难，我想你可以自己编写代码。首先，记住每个矩/矩阵的逆，也就是基础图像是该矩阵的转置，因为它们是正交的。然后查看该库的作者用来测试该函数的code。这比库中的代码更简单，因此您可以阅读并理解它的工作原理（当然也慢得多）。你需要为每个时刻得到那些基础图像的矩阵。您可以修改_slow_znl以获取在主循环x,y内计算的for x,y,p in zip(X,Y,P):的值，并将其存储在与输入图像大小相同的矩阵中。将白色图像传递给_slow_zernike，并将所有矩阵矩阵调整到所需的径向度。要使用系数重建图像，只需使用那些矩阵的转置，就像使用Haar转换一样。

基于Zernike矩的图像重建使用mahotas和opencv

2 个答案: