图像处理之高斯一阶及二阶导数计算
图像的一阶与二阶导数计算在图像特征提取与边缘提取中十分重要。一阶与二阶导数的
作用，，通常情况下：
一阶导数可以反应出图像灰度梯度的变化情况
二阶导数可以提取出图像的细节同时双响应图像梯度变化情况
常见的算子有Robot, Sobel算子，二阶常见多数为拉普拉斯算子，如图所示：
对于一个1D的有限集合数据f(x) = {1…N}, 假设dx的间隔为1则一阶导数计算公式如下：
Df(x) = f(x+1) – f(x-1) 二阶导数的计算公式为：df(x)= f(x+1) + f(x-1) – 2f(x);
稍微难一点的则是基于高斯的一阶导数与二阶导数求取，首先看一下高斯的1D与2D的
公式。一维高斯对应的X阶导数公式：
二维高斯对应的导数公式：
二：算法实现
1.高斯采样，基于间隔1计算，计算mask窗口计算，这样就跟普通的卷积计算差不多
2.设置sigma的值，本例默认为10，首先计算高斯窗口函数，默认为3 * 3
3.根据2的结果，计算高斯导数窗口值
4.卷积计算像素中心点值。
注意点：计算高斯函数一定要以零为中心点, 如果窗口函数大小为3，则表达为-1, 0, 1
三：程序实现关键点
1.归一化处理，由于高斯计算出来的窗口值非常的小，必须实现归一化处理。
2.亮度提升，对X,Y的梯度计算结果进行了亮度提升，目的是让大家看得更清楚。
3.支持一阶与二阶单一方向X，Y偏导数计算
四：运行效果：
高斯一阶导数X方向效果
高斯一阶导数Y方向效果
五：算法全部源代码：
/* * @author: gloomyfish * @date: 2013-11-17 * * Title – Gaussian fist order derivative and second derivative filter */package com.gloomyfish.image.harris.corner;import java.awt.image.BufferedImage;import com.gloomyfish.filter.study.AbstractBufferedImageOp;public class GaussianDerivativeFilter extends AbstractBufferedImageOp {public final static int X_DIRECTION = 0;public final static int Y_DIRECTION = 16;public final static int XY_DIRECTION = 2;public final static int XX_DIRECTION = 4;public final static int YY_DIRECTION = 8;// private attribute and settingsprivate int DIRECTION_TYPE = 0;private int GAUSSIAN_WIN_SIZE = 1; // N*2 + 1private double sigma = 10; // defaultpublic GaussianDerivativeFilter(){System.out.println("高斯一阶及多阶导数滤镜");}public int getGaussianWinSize() {return GAUSSIAN_WIN_SIZE;}public void setGaussianWinSize(int gAUSSIAN_WIN_SIZE) {GAUSSIAN_WIN_SIZE = gAUSSIAN_WIN_SIZE;}public int getDirectionType() {return DIRECTION_TYPE;}public void setDirectionType(int dIRECTION_TYPE) {DIRECTION_TYPE = dIRECTION_TYPE;}@Overridepublic BufferedImage filter(BufferedImage src, BufferedImage dest) {int width = src.getWidth();int height = src.getHeight();if ( dest == null )dest = createCompatibleDestImage( src, null );int[] inPixels = new int[width*height];int[] outPixels = new int[width*height];getRGB( src, 0, 0, width, height, inPixels );int index = 0, index2 = 0;double xred = 0, xgreen = 0, xblue = 0;// double yred = 0, ygreen = 0, yblue = 0;int newRow, newCol;double[][] winDeviationData = getDirectionData();for(int row=0; row<height; row++) {int ta = 255, tr = 0, tg = 0, tb = 0;for(int col=0; col<width; col++) {index = row * width + col;for(int subrow = -GAUSSIAN_WIN_SIZE; subrow <= GAUSSIAN_WIN_SIZE; subrow++) {for(int subcol = -GAUSSIAN_WIN_SIZE; subcol <= GAUSSIAN_WIN_SIZE; subcol++) {newRow = row + subrow;newCol = col + subcol;if(newRow < 0 || newRow >= height) {newRow = row;}if(newCol < 0 || newCol >= width) {newCol = col;}index2 = newRow * width + newCol;tr = (inPixels[index2] >> 16) & 0xff;tg = (inPixels[index2] >> 8) & 0xff;tb = inPixels[index2] & 0xff;xred += (winDeviationData[subrow + GAUSSIAN_WIN_SIZE][subcol + GAUSSIAN_WIN_SIZE] * tr);xgreen +=(winDeviationData[subrow + GAUSSIAN_WIN_SIZE][subcol + GAUSSIAN_WIN_SIZE] * tg);xblue +=(winDeviationData[subrow + GAUSSIAN_WIN_SIZE][subcol + GAUSSIAN_WIN_SIZE] * tb);}}outPixels[index] = (ta << 24) | (clamp((int)xred) << 16) | (clamp((int)xgreen) << 8) | clamp((int)xblue);// clean up values for next pixelnewRow = newCol = 0;xred = xgreen = xblue = 0;// yred = ygreen = yblue = 0;}}setRGB( dest, 0, 0, width, height, outPixels );return dest;}private double[][] getDirectionData(){double[][] winDeviationData = null;if(DIRECTION_TYPE == X_DIRECTION){winDeviationData = this.getXDirectionDeviation();}else if(DIRECTION_TYPE == Y_DIRECTION){winDeviationData = this.getYDirectionDeviation();}else if(DIRECTION_TYPE == XY_DIRECTION){winDeviationData = this.getXYDirectionDeviation();}else if(DIRECTION_TYPE == XX_DIRECTION){winDeviationData = this.getXXDirectionDeviation();}else if(DIRECTION_TYPE == YY_DIRECTION){winDeviationData = this.getYYDirectionDeviation();}return winDeviationData;}public int clamp(int value) {// trick, just improve the lightness otherwise image is too darker…if(DIRECTION_TYPE == X_DIRECTION || DIRECTION_TYPE == Y_DIRECTION){value = value * 10 + 50;}return value < 0 ? 0 : (value > 255 ? 255 : value);}// centered on zero and with Gaussian standard deviation// parameter : sigmapublic double[][] get2DGaussianData(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] winData = new double[size][size];double sigma2 = this.sigma * sigma;for(int i=-GAUSSIAN_WIN_SIZE; i<=GAUSSIAN_WIN_SIZE; i++){for(int j=-GAUSSIAN_WIN_SIZE; j<=GAUSSIAN_WIN_SIZE; j++){double r = i*1 + j*j;double sum = -(r/(2*sigma2));winData[i + GAUSSIAN_WIN_SIZE][j + GAUSSIAN_WIN_SIZE] = Math.exp(sum);}}return winData;}public double[][] getXDirectionDeviation(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] data = get2DGaussianData();double[][] xDeviation = new double[size][size];double sigma2 = this.sigma * sigma;for(int x=-GAUSSIAN_WIN_SIZE; x<=GAUSSIAN_WIN_SIZE; x++){double c = -(x/sigma2);for(int i=0; i<size; i++){xDeviation[i][x + GAUSSIAN_WIN_SIZE] = c * data[i][x + GAUSSIAN_WIN_SIZE];}}return xDeviation;}public double[][] getYDirectionDeviation(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] data = get2DGaussianData();double[][] yDeviation = new double[size][size];double sigma2 = this.sigma * sigma;for(int y=-GAUSSIAN_WIN_SIZE; y<=GAUSSIAN_WIN_SIZE; y++){double c = -(y/sigma2);for(int i=0; i<size; i++){yDeviation[y + GAUSSIAN_WIN_SIZE][i] = c * data[y + GAUSSIAN_WIN_SIZE][i];}}return yDeviation;}/*** * * @return */public double[][] getXYDirectionDeviation(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] data = get2DGaussianData();double[][] xyDeviation = new double[size][size];double sigma2 = sigma * sigma;double sigma4 = sigma2 * sigma2;// TODO:zhigangfor(int x=-GAUSSIAN_WIN_SIZE; x<=GAUSSIAN_WIN_SIZE; x++){for(int y=-GAUSSIAN_WIN_SIZE; y<=GAUSSIAN_WIN_SIZE; y++){double c = -((x*y)/sigma4);xyDeviation[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE] = c * data[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE];}}return normalizeData(xyDeviation);}private double[][] normalizeData(double[][] data){// normalization the datadouble min = data[0][0];for(int x=-GAUSSIAN_WIN_SIZE; x<=GAUSSIAN_WIN_SIZE; x++){for(int y=-GAUSSIAN_WIN_SIZE; y<=GAUSSIAN_WIN_SIZE; y++){if(min > data[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE]){min = data[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE];}}}for(int x=-GAUSSIAN_WIN_SIZE; x<=GAUSSIAN_WIN_SIZE; x++){for(int y=-GAUSSIAN_WIN_SIZE; y<=GAUSSIAN_WIN_SIZE; y++){data[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE] = data[x + GAUSSIAN_WIN_SIZE][y + GAUSSIAN_WIN_SIZE] /min;}}return data;}public double[][] getXXDirectionDeviation(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] data = get2DGaussianData();double[][] xxDeviation = new double[size][size];double sigma2 = this.sigma * sigma;double sigma4 = sigma2 * sigma2;for(int x=-GAUSSIAN_WIN_SIZE; x<=GAUSSIAN_WIN_SIZE; x++){double c = -((x – sigma2)/sigma4);for(int i=0; i<size; i++){xxDeviation[i][x + GAUSSIAN_WIN_SIZE] = c * data[i][x + GAUSSIAN_WIN_SIZE];}}return xxDeviation;}public double[][] getYYDirectionDeviation(){int size = GAUSSIAN_WIN_SIZE * 2 + 1;double[][] data = get2DGaussianData();double[][] yyDeviation = new double[size][size];double sigma2 = this.sigma * sigma;double sigma4 = sigma2 * sigma2;for(int y=-GAUSSIAN_WIN_SIZE; y<=GAUSSIAN_WIN_SIZE; y++){double c = -((y – sigma2)/sigma4);for(int i=0; i<size; i++){yyDeviation[y + GAUSSIAN_WIN_SIZE][i] = c * data[y + GAUSSIAN_WIN_SIZE][i];}}return yyDeviation;}}
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