Tuesday, May 5, 2015

2D FFT: Effects of Location & Thickness of Diagonal Lines on 2D Magnitude & Phase: Part 05

Problem

This post continues our series of posts on a programmatic investigation of the effects of the location and thickness of horizontal, vertical, and diagonal lines on the magnitude and phase components of 2D FFT. In this post, we will continue our investigation of diagonal lines. Let us thicken the top-left-to-bottom-right diagonal line, move it up, and compute its magnitude and thickness. The MATLAB source of our programmatic investigation is in the last section of The previous post on diagonal lines is here.

Diagonal Line from Top Left to Bottom Right

Figure 1 shows a thickened diagonal line from top left to bottom right.

 Figure 1. dig_tl_br_line_01e.jpg

figure;
imshow(ImgDigTlBrLine_01e);
title('Dig TL-BR LINE 01e');

Now we compute the 2D FFT of the image in Figure 1. The image is converted into a 2D double matrix. The fftshift command positions the 0,0 origin to the center of the image. We compute the absolute magnitude of each value in the 2D matrix and take the log of each value. 1 is added to each value before taking the log, because log(0) is not defined.

%% ff2 magnitude
fftDigTlBrLine01e = fft2(double(ImgDigTlBrLine_01e));
fftDigTlBrLine01e = fftshift(fftDigTlBrLine01e);
fftDigTlBrLine01e = abs(fftDigTlBrLine01e);
fftDigTlBrLine01e = log(fftDigTlBrLine01e+1);
fftDigTlBrLine01e = mat2gray(fftDigTlBrLine01e);

The magnitude image is shown in Figure 2. As can be seen in Figure 2, the FFT magnitude shows a diagonal line in the center orthogonal to the diagonal line in Figure 1. Note that there are other lines orthogonal to the diagonal line in Figure 1 but less pronounced than the other lines. This indicates that there are other frequencies detected in the image.

figure;
imshow(fftDigTlBrLine01e);
colormap(gray);
title('Dig TL-BR LINE 01e FFT2 Magnitude');

 Figure 2. 2D FFT Magnitude of Figure 1

The phase component can be computed and displayed by the following MATLAB code. Figure 3 shows the 2D FFT phase graph of Figure 1.

%%% fft2 phase
fftDigTlBrLine01e_PHASE = fft2(double(ImgDigTlBrLine_01e));
fftDigTlBrLine01e_PHASE = fftshift(fftDigTlBrLine01e_PHASE);
fftDigTlBrLine01e_PHASE = angle(fftDigTlBrLine01e_PHASE);

figure;
imshow(fftDigTlBrLine01e_PHASE, [-pi, pi]);
colormap(gray);

title('Dig TL-BR LINE 01e FFT2 Phase');

 Figure 3. 2D FFT Phase of Figure 1

2D FFT: Effects of Location & Thickness of Diagonal Lines on 2D Magnitude & Phase: Part 04

Problem

This post continues our series of posts on a programmatic investigation of the effects of the location and thickness of horizontal, vertical, and diagonal lines on the magnitude and phase components of 2D FFT. In this post, we will continue our investigation of diagonal lines. Let us thicken the top-left-to-bottom-right diagonal line and compute its magnitude and thickness. The MATLAB source of our programmatic investigation is in the last section of The previous post on diagonal lines is here.

Diagonal Line from Top Left to Bottom Right

Figure 1 shows a thickened diagonal line from top left to bottom right.

 Figure 1. dig_tl_br_line_01d.jpg

figure;
imshow(ImgDigTlBrLine_01d);
title('Dig TL-BR LINE 01d');

Now we compute the 2D FFT of the image in Figure 1. The image is converted into a 2D double matrix. The fftshift command positions the 0,0 origin to the center of the image. We compute the absolute magnitude of each value in the 2D matrix and take the log of each value. 1 is added to each value before taking the log, because log(0) is not defined.

%% ff2 magnitude
fftDigTlBrLine01d = fft2(double(ImgDigTlBrLine_01d));
fftDigTlBrLine01d = fftshift(fftDigTlBrLine01d);
fftDigTlBrLine01d = abs(fftDigTlBrLine01d);
fftDigTlBrLine01d = log(fftDigTlBrLine01d+1);
fftDigTlBrLine01d = mat2gray(fftDigTlBrLine01d);

The magnitude image is shown in Figure 2. As can be seen in Figure 2, the FFT magnitude shows a diagonal line in the center orthogonal to the diagonal line in Figure 1. Note that there are other lines orthogonal to the diagonal line in Figure 1 but less pronounced than the other lines. This indicates that there are other frequencies detected in the image.

figure;
imshow(fftDigTlBrLine01d);
colormap(gray);
title('Dig TL-BR LINE 01d FFT2 Magnitude');

 Figure 2. 2D FFT Magnitude of Figure 1

The phase component can be computed and displayed by the following MATLAB code. Figure 3 shows the 2D FFT phase graph of Figure 1.

%%% fft2 phase
fftDigTlBrLine01d_PHASE = fft2(double(ImgDigTlBrLine_01d));
fftDigTlBrLine01d_PHASE = fftshift(fftDigTlBrLine01d_PHASE);
fftDigTlBrLine01d_PHASE = angle(fftDigTlBrLine01d_PHASE);

figure;
imshow(fftDigTlBrLine01d_PHASE, [-pi, pi]);
colormap(gray);

title('Dig TL-BR LINE 01d FFT2 Phase');

 Figure 3. 2D FFT Phase of Figure 1

2D FFT: Effects of Location & Thickness of Diagonal Lines on 2D Magnitude & Phase: Part 03

Problem

This post continues our series of posts on a programmatic investigation of the effects of the location and thickness of horizontal, vertical, and diagonal lines on the magnitude and phase components of 2D FFT. In this post, we will continue our investigation of diagonal lines. Let us move the top-left-to-bottom-right diagonal line down and compute its magnitude and thickness. The MATLAB source of our programmatic investigation is in the last section of The previous post on diagonal lines is here.

Diagonal Line from Top Left to Bottom Right

Figure 1 shows a diagonal line from top left to bottom right moved diagonally down.

 Figure 1. dig_tl_br_line_01c.jpg

figure;
imshow(ImgDigTlBrLine_01c);
title('Dig TL-BR LINE 01c');

Now we compute the 2D FFT of the image in Figure 1. The image is converted into a 2D double matrix. The fftshift command positions the 0,0 origin to the center of the image. We compute the absolute magnitude of each value in the 2D matrix and take the log of each value. 1 is added to each value before taking the log, because log(0) is not defined.

%% ff2 magnitude
fftDigTlBrLine01c = fft2(double(ImgDigTlBrLine_01c));
fftDigTlBrLine01c = fftshift(fftDigTlBrLine01c);
fftDigTlBrLine01c = abs(fftDigTlBrLine01c);
fftDigTlBrLine01c = log(fftDigTlBrLine01c+1);
fftDigTlBrLine01c = mat2gray(fftDigTlBrLine01c);

The magnitude image is shown in Figure 2. As can be seen in Figure 2, the FFT magnitude shows a diagonal line in the center orthogonal to the diagonal line in Figure 1. Note that there are other lines orthogonal to the diagonal line in Figure 1 but less pronounced than the other lines. This indicates that there are other frequencies detected in the image.

figure;
imshow(fftDigTlBrLine01c);
colormap(gray);
title('Dig TL-BR LINE 01c FFT2 Magnitude');

 Figure 2. 2D FFT Magnitude of Figure 1

The phase component can be computed and displayed by the following MATLAB code. Figure 3 shows the 2D FFT phase graph of Figure 1.

%%% fft2 phase
fftDigTlBrLine01c_PHASE = fft2(double(ImgDigTlBrLine_01c));
fftDigTlBrLine01c_PHASE = fftshift(fftDigTlBrLine01c_PHASE);
fftDigTlBrLine01c_PHASE = angle(fftDigTlBrLine01c_PHASE);

figure;
imshow(fftDigTlBrLine01c_PHASE, [-pi, pi]);
colormap(gray);

title('Dig TL-BR LINE 01c FFT2 Phase');

 Figure 3. 2D FFT Phase of Figure 1

2D FFT: Effects of Location & Thickness of Diagonal Lines on 2D Magnitude & Phase: Part 02

Problem

This post continues our series of posts on a programmatic investigation of the effects of the location and thickness of horizontal, vertical, and diagonal lines on the magnitude and phase components of 2D FFT. In this post, we will continue our investigation of diagonal lines. Let us move the top-left-to-bottom-right diagonal line  up and compute its magnitude and thickness. The MATLAB source of our programmatic investigation is in the last section of The previous post on diagonal lines is here.

Diagonal Line from Top Left to Bottom Right

Figure 1 shows a diagonal line from top left to bottom right moved diagonally up.
 Figure 1. Image dig_tl_br_line_01b.jpg

figure;
imshow(ImgDigTlBrLine_01b);
title('Dig TL-BR LINE 01b');

Now we compute the 2D FFT of the image in Figure 1. The image is converted into a 2D double matrix. The fftshift command positions the 0,0 origin to the center of the image. We compute the absolute magnitude of each value in the 2D matrix and take the log of each value. 1 is added to each value before taking the log, because log(0) is not defined.

%% ff2 magnitude
fftDigTlBrLine01b = fft2(double(ImgDigTlBrLine_01b));
fftDigTlBrLine01b = fftshift(fftDigTlBrLine01b);
fftDigTlBrLine01b = abs(fftDigTlBrLine01b);
fftDigTlBrLine01b = log(fftDigTlBrLine01b+1);
fftDigTlBrLine01b = mat2gray(fftDigTlBrLine01b);

The magnitude image is shown in Figure 2. As can be seen in Figure 2, the FFT magnitude shows a diagonal line in the center orthogonal to the diagonal line in Figure 1. Note that there are other lines orthogonal to the diagonal line in Figure 1 but less pronounced than the other lines. This indicates that there are other frequencies detected in the image.

figure;
imshow(fftDigTlBrLine01b);
colormap(gray);
title('Dig TL-BR LINE 01b FFT2 Magnitude');

 Figure 2. 2D FFT Phase of Figure 1
The phase component can be computed and displayed by the following MATLAB code. Figure 3 shows the 2D FFT phase graph of Figure 1.

%%% fft2 phase
fftDigTlBrLine01b_PHASE = fft2(double(ImgDigTlBrLine_01b));
fftDigTlBrLine01b_PHASE = fftshift(fftDigTlBrLine01b_PHASE);
fftDigTlBrLine01b_PHASE = angle(fftDigTlBrLine01b_PHASE);

figure;
imshow(fftDigTlBrLine01b_PHASE, [-pi, pi]);
colormap(gray);

title('Dig TL-BR LINE 01b FFT2 Phase');

 Figure 3. 2D FFT phase of Figure 1