16×16图像2D离散傅里叶变换（DFT）实现出错，请求排查错误原因

我此前从事音频采样工作，现转向图像处理领域，正尝试为16×16尺寸的图像实现离散傅里叶变换（DFT）。我自制了图像解码器，图像文件中的每个像素以括号包裹的RGB值形式存储，例如纯红色像素表示为(255,0,0)，每行包含16个像素，共16行。我处理的目标图像是粉红到红色的渐变图像，其编码格式如下：

(255,0,239)(255,0,223)(255,0,207)(255,0,191)(255,0,175)(255,0,159)(255,0,143)(255,0,127)(255,0,111)(255,0,95)(255,0,79)(255,0,63)(255,0,47)(255,0,31)(255,0,15)(255,0,0) 
(255,0,223)(255,0,210)(255,0,197)(255,0,184)(255,0,171)(255,0,158)(255,0,145)(255,0,132)(255,0,119)(255,0,106)(255,0,93)(255,0,80)(255,0,67)(255,0,54)(255,0,41)(255,0,15) 
(255,0,207)(255,0,196)(255,0,185)(255,0,174)(255,0,163)(255,0,152)(255,0,141)(255,0,130)(255,0,119)(255,0,108)(255,0,97)(255,0,86)(255,0,75)(255,0,64)(255,0,53)(255,0,31) 
(255,0,191)(255,0,182)(255,0,173)(255,0,164)(255,0,155)(255,0,146)(255,0,137)(250,0,121)(250,0,112)(250,0,103)(250,0,94)(250,0,85)(250,0,76)(255,0,67)(255,0,58)(255,0,47) 
(255,0,175)(255,0,168)(255,0,161)(255,0,154)(255,0,147)(255,0,140)(255,0,133)(255,0,126)(255,0,119)(255,0,112)(255,0,105)(255,0,98)(255,0,91)(255,0,84)(255,0,77)(255,0,63) 
(255,0,159)(255,0,154)(255,0,149)(255,0,144)(255,0,139)(255,0,134)(255,0,129)(255,0,124)(255,0,119)(255,0,114)(255,0,109)(255,0,104)(255,0,99)(255,0,94)(255,0,89)(255,0,79) 
(255,0,143)(255,0,140)(255,0,137)(255,0,134)(255,0,131)(255,0,128)(255,0,125)(255,0,122)(255,0,119)(255,0,116)(255,0,113)(255,0,110)(255,0,107)(255,0,104)(255,0,101)(255,0,95) 
(255,0,127)(255,0,126)(255,0,125)(255,0,124)(255,0,123)(255,0,122)(255,0,121)(255,0,120)(255,0,119)(255,0,118)(255,0,117)(255,0,116)(255,0,115)(255,0,114)(255,0,113)(255,0,111) 
(255,0,111)(255,0,113)(255,0,114)(255,0,115)(255,0,116)(255,0,117)(255,0,118)(255,0,119)(255,0,120)(255,0,121)(255,0,122)(255,0,123)(255,0,124)(255,0,125)(255,0,126)(255,0,127) 
(255,0,95)(255,0,101)(255,0,104)(255,0,107)(255,0,110)(255,0,113)(255,0,116)(255,0,119)(255,0,122)(255,0,125)(255,0,128)(255,0,131)(255,0,134)(255,0,137)(255,0,140)(255,0,143) 
(255,0,79)(255,0,89)(255,0,94)(255,0,99)(255,0,104)(255,0,109)(255,0,114)(255,0,119)(255,0,124)(255,0,129)(255,0,134)(255,0,139)(255,0,144)(255,0,149)(255,0,154)(255,0,159) 
(255,0,63)(255,0,77)(255,0,84)(255,0,91)(255,0,98)(255,0,105)(255,0,112)(255,0,119)(255,0,126)(255,0,133)(255,0,140)(255,0,147)(255,0,154)(255,0,161)(255,0,168)(255,0,175) 
(255,0,47)(255,0,58)(255,0,67)(255,0,76)(255,0,85)(255,0,94)(255,0,103)(250,0,112)(250,0,121)(255,0,137)(255,0,146)(255,0,155)(255,0,164)(255,0,173)(255,0,182)(255,0,191) 
(255,0,31)(255,0,53)(255,0,64)(255,0,75)(255,0,86)(255,0,97)(255,0,108)(255,0,119)(255,0,130)(255,0,141)(255,0,152)(255,0,163)(255,0,174)(255,0,185)(255,0,196)(255,0,207) 
(255,0,15)(255,0,41)(255,0,54)(255,0,67)(255,0,80)(255,0,93)(255,0,106)(255,0,119)(255,0,132)(255,0,145)(255,0,158)(255,0,171)(255,0,184)(255,0,197)(255,0,210)(255,0,223) 
(255,0,0)(255,0,15)(255,0,31)(255,0,47)(255,0,63)(255,0,79)(255,0,95)(255,0,111)(255,0,127)(255,0,143)(255,0,159)(255,0,175)(255,0,191)(255,0,207)(255,0,223)(255,0,239)

我希望将该图像转换到离散频域（DFT）。1D DFT公式为：X[k] = Σ_{n=0}^{N-1} x[n] * exp(-j * 2 * π * k * n / N)。对于图像处理中的2D DFT，我的理解是先对每行进行DFT，再对每列进行DFT，然后将结果相加，例如第1行DFT结果为1，第1列DFT结果为2，则2D DFT中(1,1)位置的值为1+2（假设从第0行第0列开始计数）。我编写了如下Python代码，但实现结果出错，请求帮助排查错误所在：

import cmath
filename = input("Enter a filename:\n")
file = open(filename,'r')
redArray = []
greenArray = []
blueArray = []
redArrayFourier = []
greenArrayFourier = []
blueArrayFourier = []
fourierRowsRed = []
fourierColumnsRed = []
fourierRowsBlue = []
fourierColumnsBlue = []
fourierRowsGreen = []
fourierColumnsGreen = []
rows, cols = 16, 16
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    redArray.append(row)
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    blueArray.append(row)
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    greenArray.append(row)
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    redArrayFourier.append(row)
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    blueArrayFourier.append(row)
for _ in range(rows):
    row = []
    for _ in range(cols):
        row.append(0)
    greenArrayFourier.append(row)
for i in range(rows):
    fourierRowsRed.append(0)
for i in range(rows):
    fourierRowsBlue.append(0)
for i in range(rows):
    fourierRowsGreen.append(0)
for i in range(cols):
    fourierColumnsRed.append(0)
for i in range(cols):
    fourier...

首先得说，你对2D DFT的理解有个核心误区：2D DFT绝对不是行DFT结果加列DFT结果！正确的流程是：

1. 先对图像的每一行单独做1D DFT，得到一个中间结果矩阵
2. 再对这个中间矩阵的每一列单独做1D DFT，最终得到的就是完整的2D DFT结果
   （反过来先做列DFT再做行DFT，结果是完全一致的，但顺序不影响，核心是两次一维变换的级联，不是相加）

再看代码里的具体问题：

1. 图像读取解析逻辑完全缺失 #

代码只打开了文件，但没有读取任何内容，也没有解析像素的逻辑——redArray、greenArray、blueArray始终是全0矩阵，后续计算自然全错。你需要用正则表达式提取每行的RGB值，示例代码：

import re
# 改用with语句自动管理文件，避免资源泄漏
with open(filename, 'r') as file:
    content = file.read().splitlines()
    for i, line in enumerate(content):
        # 提取所有括号内的RGB数值对
        pixels = re.findall(r'\((\d+),(\d+),(\d+)\)', line)
        for j, (r, g, b) in enumerate(pixels):
            redArray[i][j] = int(r)
            greenArray[i][j] = int(g)
            blueArray[i][j] = int(b)

2. DFT计算逻辑完全缺失 #

代码只初始化了一堆数组，但没有实现行DFT和列DFT的计算逻辑。先写一个通用的1D DFT函数，再分别处理行和列：

def dft_1d(signal):
    N = len(signal)
    result = [0 + 0j] * N
    for k in range(N):
        for n in range(N):
            result[k] += signal[n] * cmath.exp(-1j * 2 * cmath.pi * k * n / N)
    return result

# 第一步：对每行做DFT
for i in range(rows):
    red_row_dft = dft_1d(redArray[i])
    green_row_dft = dft_1d(greenArray[i])
    blue_row_dft = dft_1d(blueArray[i])
    for j in range(cols):
        redArrayFourier[i][j] = red_row_dft[j]
        greenArrayFourier[i][j] = green_row_dft[j]
        blueArrayFourier[i][j] = blue_row_dft[j]

# 第二步：对每列做DFT
for j in range(cols):
    # 提取当前列的所有元素
    red_col = [redArrayFourier[i][j] for i in range(rows)]
    green_col = [greenArrayFourier[i][j] for i in range(rows)]
    blue_col = [blueArrayFourier[i][j] for i in range(rows)]
    # 计算列DFT
    red_col_dft = dft_1d(red_col)
    green_col_dft = dft_1d(green_col)
    blue_col_dft = dft_1d(blue_col)
    # 将结果写回傅里叶矩阵
    for i in range(rows):
        redArrayFourier[i][j] = red_col_dft[i]
        greenArrayFourier[i][j] = green_col_dft[i]
        blueArrayFourier[i][j] = blue_col_dft[i]

3. 冗余变量过多 #

代码里的fourierRowsRed、fourierColumnsRed