学习OpenCV：频域及傅里叶变换

关键公式，《数字图像处理》第三版，P147：

1、频谱图具有周期性，归一化为图像大小；

2、频谱图具有对称性，以复数形式表示图像二维，幅值为实部与虚部平方和；

https://blog.csdn.net/daduzimama/article/details/80109139

https://www.zhihu.com/question/67234546/answer/403292397

https://www.cnblogs.com/Imageshop/archive/2019/10/08/11625895.html

示例一：参考HDevelop中的remove_texture_fft.hdev，采用OpenCV实现去除图像中固定纹理，实际上是构建了一个带阻滤波器；这里只是演示，只去除了部分纹理频率，没有完全去除干净，效果肯定没有HDevelop中的好：

Mat ConvertToFreq(Mat imgInput)
{
	int w = getOptimalDFTSize(imgInput.cols);
	int h = getOptimalDFTSize(imgInput.rows);
	Mat padded;
	copyMakeBorder(imgInput, padded, 0, h - imgInput.rows, 0, w - imgInput.cols, BORDER_CONSTANT, Scalar::all(0));
	padded.convertTo(padded, CV_32FC1);
	for (int i = 0; i < padded.rows; i++)//中心化操作，其余操作和上一篇博客的介绍一样
	{
		float* ptr = padded.ptr<float>(i);
		for (int j = 0; j < padded.cols; j++)	ptr[j] *= pow(-1, i + j);
	}
	Mat plane[] = { padded,Mat::zeros(padded.size(),CV_32F) };
	Mat imgFreqComplex;
	merge(plane, 2, imgFreqComplex);
	dft(imgFreqComplex, imgFreqComplex);
	return imgFreqComplex;
}

Mat ConvertFromFreq(Size size, Mat imgInput)
{
	Mat imgTemp;
	idft(imgInput, imgTemp);
	vector<Mat> plane;
	split(imgTemp, plane);
	magnitude(plane[0], plane[1], plane[0]);
	Mat imgResult;
	normalize(plane[0], plane[0], 0, 255, NORM_MINMAX, CV_8UC1);
	Rect rect(0, 0, size.width, size.height);
	try
	{
		imgResult = plane[0](rect);
	}
	catch (...)
	{
		imgResult = plane[0].clone();
	}
	return imgResult;
}

Mat ConvertVisualization(Mat imgInput)
{
	vector<Mat> plane;
	split(imgInput, plane);
	magnitude(plane[0], plane[1], plane[0]);
	plane[0] += Scalar::all(1);
	log(plane[0], plane[0]);
	normalize(plane[0], plane[0], 1, 0, NORM_MINMAX);
	return plane[0];
}

void testFrequency1()
{
	Mat imgInput = imread("plan_01.png", 0);
	Mat imgFreqComplex = ConvertToFreq(imgInput);//中心化后复数形式的频谱图
	Mat imgFreqVisual = ConvertVisualization(imgFreqComplex);//取对数后的幅值频谱图
	Mat imgFreq8U;//归一化8位1通道的幅值频谱图
	normalize(imgFreqVisual, imgFreq8U, 0, 255, NORM_MINMAX, CV_8UC1);
	Mat imgFreqFilter;
	blur(imgFreq8U, imgFreqFilter, Size(5, 5));
	Mat imgFreqThresh;
	threshold(imgFreqFilter, imgFreqThresh, 140, 255, THRESH_BINARY); //经验值
	Mat imgMorph;
	morphologyEx(imgFreqThresh, imgMorph, MORPH_DILATE, getStructuringElement(MORPH_ELLIPSE, Size(5, 5)));
	vector<contour> vecContFind;
	findContours(imgMorph, vecContFind, RETR_LIST, CHAIN_APPROX_NONE);
	vector<contour> vecContSelect;//筛选排除掉中心区域大量的低频分量
	for (size_t j = 0; j < vecContFind.size(); j++)
	{
		contour contTemp = vecContFind[j];
		float fContArea = contourArea(contTemp);
		if (fContArea < 300)//经验值
			vecContSelect.push_back(contTemp);
	}
	Mat imgFreqMask = 255 - Contour2Mask(imgMorph.size(), vecContSelect);
	Mat imgFreqBitwise;//去除纹理相关频率的频谱图
	bitwise_and(imgFreqComplex, imgFreqComplex, imgFreqBitwise, imgFreqMask);
	Mat imgOutput = ConvertFromFreq(imgInput.size(), imgFreqBitwise);//从频域转换回时域，已经归一化为CV_8UC1
}

效果图如下，分别对应原图、去除纹理后、原图对应的复数形式频谱图、去除纹理后对应的复数形式频谱图、取对数后的幅值频谱图、二值化后的幅值频谱图、膨胀后的幅值频谱图、排除掉中心区域的低频分量后的幅值频谱图。

          

          

示例二：在频域中实现高斯滤波，实现了高通低通高斯滤波，对应图像的锐化和模糊；

Mat GeneGaussFilter(Size size, float fSigma, FilterType type)
{
	Mat imgGauss(size, CV_32FC2);
	if (HIGHT_PASS == type)
	{
		for (int i = 0; i < imgGauss.rows; i++)
		{
			float* q = imgGauss.ptr<float>(i);
			for (int j = 0; j < imgGauss.cols; j++)
			{
				float d = pow(i - imgGauss.rows / 2, 2) + pow(j - imgGauss.cols / 2, 2);
				q[2 * j] = 1 - expf(-d / fSigma);
				q[2 * j + 1] = 1 - expf(-d / fSigma);
			}
		}
	}
	else
	{
		for (int i = 0; i < imgGauss.rows; i++)
		{
			float* p = imgGauss.ptr<float>(i);
			for (int j = 0; j < imgGauss.cols; j++)
			{
				float d = pow(i - imgGauss.rows / 2, 2) + pow(j - imgGauss.cols / 2, 2);
				p[2 * j] = expf(-d / fSigma);
				p[2 * j + 1] = expf(-d / fSigma);

			}
		}
	}
	return imgGauss;
}

void testFrequency2()
{
	Mat imgInput = imread("lena.jpg", 0);
	Mat imgFreqComplex = ConvertToFreq(imgInput);
	Mat imgFreqGauss = GeneGaussFilter(imgFreqComplex.size(), 2 * 10 * 10, HIGHT_PASS);
	Mat imgFreqBlur;
	multiply(imgFreqComplex, imgFreqGauss, imgFreqBlur);//矩阵元素对应相乘法，注意，和矩阵相乘区分
	Mat imgOutput = ConvertFromFreq(imgInput.size(), imgFreqBlur);
	Mat imgVisual0 = ConvertVisualization(imgFreqComplex);
	Mat imgVisual1 = ConvertVisualization(imgFreqGauss);
	Mat imgVisual2 = ConvertVisualization(imgFreqBlur);
}

效果图如下，分别对应原图、滤波后、原图的频谱、高斯核的频谱。