从Matlab/Octave fft2()函数输出中读出特定点 [英] Reading out specific points off the Matlab / Octave fft2() function output

问题描述

我已经熟悉Octave和功能fft2().在这个玩具示例中，我的目标是产生以下256 x 256 png图像的2D DFT:

I am getting familiarized with Octave and the function fft2(). In this toy example, I am aiming at producing the 2D DFT of the following 256 x 256 png image:

为了易于理解输出，我尝试将此图像转换为256 x 256图像，消除了颜色信息:

To be able to understand the output easily, I try to convert this image into a 256 x 256 image, eliminating color information:

Im = imread('circ.png');
pkg load image
Im = rgb2gray(Im);
figure, imshow(Im)

在完成簿记准备之后，我运行:

After this bookkeeping preliminaries I run:

A = fft2(double(Im));

好.现在，我拍摄相同的图像，并使用ImageJ分析它，检查点(157, 96)处的输出，如下所示:

OK. Now I take the same image, and analyze it with ImageJ, checking the output at point (157, 96), like this:

所以幅度将是sqrt(7.448^2 + 10.458^2) = 12.83

相位arctan(-10.458 / 7.448) = 54.54 degrees.

问题是，如何从fft2()输出中获取这些值?

The question is, How can I get these values out of the fft2() output?

如果有所作为，这就是我绘制八度输出2D DFT的方式:

In case it makes a difference, this is how I plotted the Octave output 2D DFT:

subplot(132);
h = imshow(abs(fftshift(A)),[24 100000]);
h2 = get(h,'Parent');
set(h2,'YDir','Normal');
axis equal tight;
title("2D FFT Magnitude");

subplot(133);
h = imshow(angle(fftshift(A)),[-pi pi]);
h2 = get(h,'Parent');
set(h2,'YDir','Normal');
axis equal tight;
title("2D FFT Phase");

---

这是ImageJ中的过程:

---

and this is the process in ImageJ:

推荐答案

以下是一些观察结果，它们可以澄清所使用的缩放比例:

Here are a few observations which should clarify the scaling used:

• ImageJ的X和Y位置基于0，而Matlab的索引基于1.
• 增加ImageJ的X位置对应于增加Matlab中的列索引，增加ImageJ的Y位置对应于增加Matlab中的行索引，因此在ImageJ中的(X,Y)坐标对将在Matlab的索引(Y+1,X+1)处找到.
• ImageJ在图像的中间显示频率分量为0的图像，因此有效地进行了等效于fftshift(Im)
的测量
• ImageJ将0-255的灰度值缩放为0.0-1.0范围内的浮点值(即除以255缩放所有值)

• ImageJ's X and Y positions are 0-based whereas Matlab's indexing is 1-based.
• Incrementing ImageJ's X position corresponds to incrementing the column index in Matlab, and incrementing ImageJ's Y position corresponds to incrementing the row index in Matlab, so that an (X,Y) coordinate pair in ImageJ will be found at index (Y+1,X+1) in Matlab.
• ImageJ displays the image with the 0 frequency component in the middle of the image, so measurements are effectively made on the equivalent of fftshift(Im)
• ImageJ scales the 0-255 grayscale values to floating point values in the 0.0-1.0 range (i.e. scaling all values by dividing by 255)

因此，请记住:

>> Ashifted = fftshift(A);
>> Ashifted(97,158)/255
ans =   7.4484 - 10.4582i
>> Ashifted(93,165)/255
ans =  12.1928 -  4.9850i

分别与您在位置(X,Y) = (157,96)和(X,Y) = (164,92)处对实部和虚部的测量值完全对应.

which correspond exactly to your illustrated measurements of the real and imaginary parts at positions (X,Y) = (157,96) and (X,Y) = (164,92) respectively.

请注意，通过FFT的线性属性，您还可以对输入进行除法并获得相同的结果:

Note that by the linearity property of the FFT, you could also divide the input and get the same results:

A = fft2(double(Im)/255.0);
>> Ashifted = fftshift(A);
>> Ashifted(97,158)
ans =   7.4484 - 10.4582i
>> Ashifted(93,165)
ans =  12.1928 -  4.9850i

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