How to use fourier transform pair on image using opencv correctly?

Question

Firstly, I utilize putText function to create a zero-filled image:

std::string text("Mengranlin");
int rows = 222;
int cols = 112;
double textSize = 1.5;
int textWidth = 2;
int num = 255;
cv::Mat zero_filled_img = cv::Mat::zeros(cols, rows, CV_32F);
putText(zero_filled_img, text, 
cv::Point(zero_filled_img.cols * 0.5, 
zero_filled_img.rows * 0.3),
cv::FONT_HERSHEY_PLAIN, textSize, cv::Scalar(num, num, num), textWidth);
cv::Mat zero_filled_img2;
flip(zero_filled_img, zero_filled_img2, -1);
zero_filled_img += zero_filled_img2;
transpose(zero_filled_img, zero_filled_img);
flip(zero_filled_img, zero_filled_img, 1);

Here is the image:

Secondly, I utilize inverse Fourier transform to the image:

int m = getOptimalDFTSize(rows);
int n = getOptimalDFTSize(cols);
cv::Mat dst;
copyMakeBorder(zero_filled_img, dst, 0, m - rows, 0, n - cols, BORDER_CONSTANT, Scalar::all(0));
cv::Mat planes[] = { cv::Mat_<float>(dst), 
cv::Mat::zeros(dst.size(), CV_32F) };
cv::Mat complex;
cv::merge(planes,2, complex);
idft(complex, complex);
split(complex, planes);
magnitude(planes[0], planes[1], planes[0]);

Thirdly, I utilize Fourier transform to the result of inverse Fourier transform:

cv::merge(planes2, 2, complex);
dft(complex, complex);
split(complex, planes2);
magnitude(planes2[0], planes2[1], planes2[0]);
cv::Mat result = planes2[0];

Finally, I save the image:

result += 1;
log(result, result);
result = result(cv::Rect(0, 0, cols, rows));
int cx = result.cols / 2;
int cy = result.rows / 2;
cv::Mat temp;
cv::Mat q0(result, cv::Rect(0, 0, cx, cy));
cv::Mat q1(result, cv::Rect(cx, 0, cx, cy));
cv::Mat q2(result, cv::Rect(0, cy, cx, cy));
cv::Mat q3(result, cv::Rect(cx, cy, cx, cy));
q0.copyTo(temp);
q3.copyTo(q0);
temp.copyTo(q3);
q1.copyTo(temp);
q2.copyTo(q1);
temp.copyTo(q2);
imwrite("./image/log_result.jpg", result);

Here is the image:

Although the "Mengnalin" can be found from the image, that is very weak. And then, I save the normalization of the result, but I found nothing:

normalize(result, result);
imwrite("./image/normalize_result.jpg", result);
result *= 255;
imwrite("./image/normalize_result255.jpg", result);

Here is the normalization image:

Here is the normalization image x 255:

The experiment is successful when using Matlab. I want to know where the error is?

Below is the complete code that I ran:

std::string text("Mengranlin");
int rows = 222;
int cols = 112;
double textSize = 1.5;
int textWidth = 2;
int num = 255;
cv::Mat zero_filled_img = cv::Mat::zeros(cols, rows, CV_32F);
putText(zero_filled_img, text, cv::Point(zero_filled_img.cols * 0.5, zero_filled_img.rows * 0.3),
    cv::FONT_HERSHEY_PLAIN, textSize, cv::Scalar(num, num, num), textWidth);
cv::Mat zero_filled_img2;
flip(zero_filled_img, zero_filled_img2, -1);
zero_filled_img += zero_filled_img2;
transpose(zero_filled_img, zero_filled_img);
flip(zero_filled_img, zero_filled_img, 1);
cv::Mat de = cv::Mat_<uchar>(zero_filled_img);
cv::imwrite("./image/zero_filled_img.jpg", zero_filled_img);

//idft
int m = getOptimalDFTSize(rows);
int n = getOptimalDFTSize(cols);
cv::Mat dst;
copyMakeBorder(zero_filled_img, dst, 0, m - rows, 0, n - cols, BORDER_CONSTANT, Scalar::all(0));
cv::Mat planes[] = { cv::Mat_<float>(dst), cv::Mat::zeros(dst.size(), CV_32F) };
cv::Mat complex;
cv::merge(planes,2, complex);
idft(complex, complex);
split(complex, planes);
magnitude(planes[0], planes[1], planes[0]);
cv::Mat freq = planes[0];
freq = freq(cv::Rect(0, 0, cols, rows));
normalize(freq, freq, 0, 1, CV_MINMAX);

//dft
cv::Mat planes2[] = {planes[0], planes[1]};
cv::merge(planes2, 2, complex);
dft(complex, complex);
split(complex, planes2);
magnitude(planes2[0], planes2[1], planes2[0]);
cv::Mat result = planes2[0];
//float min_v, max_v; min_max(result, min_v, max_v);
imwrite("./image/img.jpg", result);
result += 1;
imwrite("./image/img_plus_zero.jpg", result);
log(result, result);
result = result(cv::Rect(0, 0, cols, rows));
//float min_v1, max_v1; min_max(result, min_v1, max_v1);
imwrite("./image/log_img.jpg", result);
int cx = result.cols / 2;
int cy = result.rows / 2;
cv::Mat temp;
cv::Mat q0(result, cv::Rect(0, 0, cx, cy));
cv::Mat q1(result, cv::Rect(cx, 0, cx, cy));
cv::Mat q2(result, cv::Rect(0, cy, cx, cy));
cv::Mat q3(result, cv::Rect(cx, cy, cx, cy));
q0.copyTo(temp);
q3.copyTo(q0);
temp.copyTo(q3);
q1.copyTo(temp);
q2.copyTo(q1);
temp.copyTo(q2);
normalize(result, result);
imwrite("./image/normalize_img.jpg", result);
result *= 255;
imwrite("./image/normalize_img255.jpg", result);

Answer 1

Your code splits the output of idft into planes[0] (real component) and planes[1] (imaginary component), then computes the magnitude and writes it to planes[0] :

idft(complex, complex);
split(complex, planes);
magnitude(planes[0], planes[1], planes[0]);

Next, you merge planes[0] and planes[1] as the real and imaginary parts of a complex-valued image, and compute the dft :

cv::Mat planes2[] = {planes[0], planes[1]};
cv::merge(planes2, 2, complex);
dft(complex, complex);

But because planes[0] doesn't contain the real part of the output of idft any more, but its magnitude, dft will not perform the inverse calculation that idft did.

You can fix this easily. Instead of:

magnitude(planes[0], planes[1], planes[0]);
cv::Mat freq = planes[0];

Do:

cv::Mat freq;
magnitude(planes[0], planes[1], freq);

---

You can significantly simplify your code. Try the following code ( zero_filled_img is the input image computed earlier):

// DFT
cv::Mat complex;
dft(zero_filled_img, complex, DFT_COMPLEX_OUTPUT);

// IDFT
cv::Mat result;
idft(complex, result, DFT_REAL_OUTPUT);
imwrite("./image/img.jpg", result);

result should be equal to zero_filled_img within numerical accuracy.

The DFT_COMPLEX_OUTPUT flag forces the creation of a full, complex-valued DFT, even though the input array is real-valued. Likewise, DFT_REAL_OUTPUT causes any imaginary output components to be dropped, this is equivalent to computing the complex IDFT and then taking the real part only.

I have reversed the DFT and IDFT to be conceptually correct (though it is perfectly fine to reverse these two operations). DFT_COMPLEX_OUTPUT only works with the forward transform and DFT_REAL_OUTPUT only works with the inverse transform, so the code above will not work (I believe) if you use these two operations in the order you attempted in your own code.

The code above also doesn't bother with padding to a favourable size. Doing so might reduce computation time, but for such a small image it will not matter at all.

---

Note also that taking the magnitude of the output of the inverse transform (the second transform you apply) is OK in your case, but not in general. This second transform is expected to produce a real-valued output (since the input to the first one was real-valued). Any imaginary component should be 0 within numerical precision. Thus, the real component of the complex output should be kept. If you take the magnitude, you obtain the absolute value of the real component, meaning that any negative values in the original input will become positive values in the final output. In the case of the example images, all pixels are non-negative, but this is not necessarily true. Do the correct thing and take the real component rather than the magnitude.