我正在编写一些代码，利用相位相关性a la Reddy & Chatterji 1996恢复测试图像相对于模板的旋转、缩放和平移。我对原始测试图像进行FFT，以找到缩放因子和旋转Angular ，但我需要对 * 旋转和缩放 * 测试图像进行FFT，以获得平移。
现在，我可以在空间域中应用旋转和缩放，然后进行FFT，但这似乎有点低效-是否可以在频域中 * 直接 * 获得旋转/缩放图像的傅立叶系数？

**编辑1：**好吧，我按照用户1816548的建议做了一个尝试。对于90度的倍数，我可以得到看起来模糊的旋转，尽管图像极性会有奇怪的变化。而不是90度的倍数给予我得到的结果相当滑稽。
**编辑2：**我对图像进行了零填充，并且在旋转时包裹了FFT的边缘。我非常确定我是围绕FFT的直流分量旋转的，但对于不是90 °倍数的Angular ，我仍然会得到奇怪的结果。
示例输出：

可执行Numpy/Scipy代码：

import numpy as np
from scipy.misc import lena
from scipy.ndimage.interpolation import rotate,zoom
from scipy.fftpack import fft2,ifft2,fftshift,ifftshift
from matplotlib.pyplot import subplots,cm

def testFourierRotation(angle):

    M = lena()
    newshape = [2*dim for dim in M.shape]
    M = procrustes(M,newshape)

    # rotate, then take the FFT
    rM = rotate(M,angle,reshape=False)
    FrM = fftshift(fft2(rM))

    # take the FFT, then rotate
    FM = fftshift(fft2(M))
    rFM = rotatecomplex(FM,angle,reshape=False)
    IrFM = ifft2(ifftshift(rFM))

    fig,[[ax1,ax2,ax3],[ax4,ax5,ax6]] = subplots(2,3)

    ax1.imshow(M,interpolation='nearest',cmap=cm.gray)
    ax1.set_title('Original')
    ax2.imshow(rM,interpolation='nearest',cmap=cm.gray)
    ax2.set_title('Rotated in spatial domain')
    ax3.imshow(abs(IrFM),interpolation='nearest',cmap=cm.gray)
    ax3.set_title('Rotated in Fourier domain')
    ax4.imshow(np.log(abs(FM)),interpolation='nearest',cmap=cm.gray)
    ax4.set_title('FFT')
    ax5.imshow(np.log(abs(FrM)),interpolation='nearest',cmap=cm.gray)
    ax5.set_title('FFT of spatially rotated image')
    ax6.imshow(np.log(abs(rFM)),interpolation='nearest',cmap=cm.gray)
    ax6.set_title('Rotated FFT')
    fig.tight_layout()

    pass

def rotatecomplex(a,angle,reshape=True):
    r = rotate(a.real,angle,reshape=reshape,mode='wrap')
    i = rotate(a.imag,angle,reshape=reshape,mode='wrap')
    return r+1j*i

def procrustes(a,target,padval=0):
    b = np.ones(target,a.dtype)*padval
    aind = [slice(None,None)]*a.ndim
    bind = [slice(None,None)]*a.ndim
    for dd in xrange(a.ndim):
        if a.shape[dd] > target[dd]:
            diff = (a.shape[dd]-target[dd])/2.
            aind[dd] = slice(np.floor(diff),a.shape[dd]-np.ceil(diff))
        elif a.shape[dd] < target[dd]:
            diff = (target[dd]-a.shape[dd])/2.
            bind[dd] = slice(np.floor(diff),target[dd]-np.ceil(diff))
    b[bind] = a[aind]
    return b