反卷积似乎不起作用（fft2; scipy）

Question

我在空间域中将信号 f 与内核 h 进行卷积，然后在频域中进行反卷积。由于我知道内核 h，理论上我应该能够毫无问题地恢复 f，但是，它不起作用。这就是我所拥有的：

    from scipy.fft import fft, ifft, fftfreq, fft2, ifft2, fftshift, ifftshift
    from scipy.ndimage import convolve

    d = 11  # diameter of image
    f = np.zeros((d, d))
    r = 1  # radius of line
    f[d//2-1: d//2+r+1] = 1  # image with a line
    f[d//2, d//2] = 0  # and 0 at the center
    f[0, 0] = 2  # and a 2 at top left corner
    
    print("\nf =")
    print(f)
    
    h = np.zeros_like(f)
    h[d//2-1, d//2-1] = h[d//2, d//2] = h[d//2+1, d//2+1] = 1  # identity filter
    print("\nh =")
    print(h)

    g = convolve(f, h)  # conv spatial domain
    G = fft2(g)
    
    H = fft2(h)  # fft of filter

    f_approx = fftshift(ifft2(G/H))  # deconvolution
    
    print("\nf^approx =")
    print(np.round(f_approx.real, 2))

这是输出：

f =
[[2. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 0. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]

h =
[[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.]]

f^approx =
[[ 0.   -0.67 -0.   -0.   -0.   -0.   -0.   -0.   -0.   -0.    1.33]
 [-0.67  2.    1.33  0.   -0.   -0.   -0.    0.   -0.    0.   -0.  ]
 [-0.    1.33  0.   -0.67 -0.   -0.    0.    0.   -0.    0.   -0.  ]
 [-0.    0.   -0.67 -2.   -0.67  0.   -0.   -0.   -0.   -0.   -0.  ]
 [ 1.    1.    1.    0.33  3.    2.33  1.    1.    1.    1.    1.  ]
 [ 1.    1.    1.    1.    2.33  0.    0.33  1.    1.    1.    1.  ]
 [ 1.    1.    1.    1.    1.    0.33 -1.    0.33  1.    1.    1.  ]
 [-0.    0.    0.    0.   -0.    0.   -0.67  2.    1.33 -0.    0.  ]
 [-0.   -0.    0.   -0.   -0.   -0.   -0.    1.33 -0.   -0.67 -0.  ]
 [ 0.    0.    0.   -0.    0.   -0.    0.    0.   -0.67 -2.   -0.67]
 [ 1.33 -0.   -0.   -0.    0.    0.    0.   -0.   -0.   -0.67  2.  ]]

为什么我没有得到原始的 f ？是我使用的功能不正确，还是方法有问题？谢谢

PS：顺便说一句，如果 h 是恒等式，那么反卷积工作正常，问题是当过滤器是其他东西时。

Answer 1

谢谢 @chris-luengo。另外，感谢 ahmed 。

我想这就是这样：

    from scipy.fft import fft, ifft, fftfreq, fft2, ifft2, fftshift, ifftshift
    from scipy.ndimage import convolve

    d = 11  # diameter of image
    f = np.zeros((d, d))
    r = 1  # radius of line
    f[d//2-1: d//2+r+1] = 1  # image with a line
    f[d//2, d//2] = 0  # and 0 at the center
    f[0, 0] = 2  # and a 2 at top left corner
    
    print("\nf =")
    print(f)
    
    h = np.zeros_like(f)
    h[d//2-1, d//2-1] = h[d//2, d//2] = h[d//2+1, d//2+1] = 1
    print("\nh =")
    print(h)

    g = convolve(f, h, mode="wrap")  # conv spatial domain
    G = fft2(g)
    
    H = fft2(ifftshift(h))  # fft of filter

    f_approx = ifft2(G/H)  # deconvolution
    
    print("\nf^approx =")
    print(np.round(f_approx.real, 2))

我希望它对某人有用;）