Python傅里叶分析：高斯滤波后逆变换图像偏移问题求助

我用Python对自制形状图像做傅里叶分析，已经拿到2D傅里叶变换结果并应用了高斯滤波，但滤波后的逆傅里叶变换图像出现偏移。试过仅取实部解决，没有效果。以下是代码和运行结果，求解决思路。

#importing necessary libraries 
from PIL import Image, ImageDraw
import matplotlib.pyplot as plt
import numpy as np
from scipy.ndimage import gaussian_filter


#function to apply Gaussian filter to the Fourier transform
def apply_gaussian_filter(fourier_transform, sigma):
    return gaussian_filter(fourier_transform, sigma)

##code to produce an image of a box and perform Fourier analysis on it

#defining the size bounds of the box
w, h = 65, 65
box = [(50, 29), (w-50, h-30)]

#creating the image with a white background
im = Image.new("I", (w,h))
img = ImageDraw.Draw(im)

#producing the box shape
img.rectangle(box, fill = "white", outline = "white")

#convering image to numpy library
im_array = np.array(im)

#taking the 2D Fourier transform
fourier_box = np.fft.fft2(im_array)

#shifting the zero frequency component to the center
fourier_box_shifted = np.fft.fftshift(fourier_box)

#applying Gaussian filter to the Fourier transform of the box
sigma_box = 5
filtered_fourier_box = apply_gaussian_filter(np.abs(fourier_box_shifted), sigma_box)

##code to produce an image of a circle and perform Fourier analysis on it

#defining the size bounds of the shape
w_1, h_1 = 65, 65
circle = [(10, 10), (w_1-10, h_1-10)]

#creating the image with a white background
im_1 = Image.new("I", (w_1,h_1))
img_1 = ImageDraw.Draw(im_1)

#producing the circle shape
img_1.ellipse(circle, fill = "white", outline = "white")

#convering image to numpy library
im_1_array = np.array(im_1)

#taking the 2D Fourier transform
fourier_circle = np.fft.fft2(im_1_array)

#shifting the zero frequency component to the center
fourier_circle_shifted = np.fft.fftshift(fourier_circle)

#applying Gaussian filter to the Fourier transform of the circle
sigma_circle = 5
filtered_fourier_circle = apply_gaussian_filter(np.abs(fourier_circle_shifted), sigma_circle)

#applying inverse Fourier transform to the filtered Fourier data
inverse_filtered_box = np.fft.ifft2(np.fft.ifftshift(filtered_fourier_box)).real
inverse_filtered_circle = np.fft.ifft2(np.fft.ifftshift(filtered_fourier_circle)).real

#adjusting size of the graphs 
plt.figure(figsize=(13, 8))

#displaying the Fourier transform of the box 
plt.subplot(231)  
plt.imshow(np.log(np.abs(fourier_box_shifted) + 1))
plt.title('2D Fourier Transform Box')

#displaying the Fourier transform of the box with Gaussian filter
plt.subplot(232) 
plt.imshow(np.log(filtered_fourier_box + 1))
plt.title('Gaussian Filtered 2D Fourier Transform Box')

#displaying the inverse of the filtered Fourier box
plt.subplot(233)  
plt.imshow(np.abs(np.real(inverse_filtered_box)))
plt.title('Inverse Filtered Box')

#displaying the Fourier transform of the circle
plt.subplot(234)
plt.imshow(np.log(np.abs(fourier_circle_shifted) + 1))
plt.title('2D Fourier Transform Circle')

#displaying the Fourier transform of the circle with Gaussian filter
plt.subplot(235)
plt.imshow(np.log(filtered_fourier_circle + 1))
plt.title('Gaussian Filtered 2D Fourier Transform Circle')

#displaying the inverse of the filtered Fourier circle
plt.subplot(236)  
plt.imshow(np.abs(np.real(inverse_filtered_circle)))
plt.title('Inverse Filtered Circle')

问题根源 #

你在滤波时只对傅里叶变换的幅值（np.abs(fourier_box_shifted)）做了高斯滤波，完全丢失了相位信息！傅里叶变换由幅值和相位共同构成：幅值决定图像的明暗、频率分布，而相位直接决定图像中各元素的位置。仅保留幅值会彻底破坏位置关联，导致逆变换图像偏移、失真。

修正方案 #

必须对傅里叶变换的复数形式做高斯滤波，同时保留幅值和相位。具体修改：

1. 无需修改滤波函数，gaussian_filter原生支持对复数数组的实部、虚部分别处理
2. 去掉滤波步骤中的np.abs，直接传入移位后的复数傅里叶变换结果

修正后的关键代码片段 #

# 直接对复数傅里叶变换做滤波，不再取幅值
filtered_fourier_box = apply_gaussian_filter(fourier_box_shifted, sigma_box)
filtered_fourier_circle = apply_gaussian_filter(fourier_circle_shifted, sigma_circle)

# 逆变换后取实部（原图像为实数，逆变换虚部极小，可忽略）
inverse_filtered_box = np.fft.ifft2(np.fft.ifftshift(filtered_fourier_box)).real
inverse_filtered_circle = np.fft.ifft2(np.fft.ifftshift(filtered_fourier_circle)).real

额外说明 #

• 原图像是实数，其傅里叶变换满足共轭对称性，高斯核是对称核，滤波后逆变换的虚部会非常小，取实部是合理操作。
• 如果虚部数值仍不可忽略，可使用np.around做近似，或结合np.abs与实部处理，但核心前提是必须保留相位信息。

内容的提问来源于stack exchange，提问作者Elisa