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 settings
	private int DIRECTION_TYPE = 0;
	private int GAUSSIAN_WIN_SIZE = 1; // N*2 + 1
	private double sigma = 10; // default

	public 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;
	}

	@Override
	public 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 pixel
                newRow = 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 : sigma
	public 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:zhigang
		for(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 data
		double 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;
	}

}

国足都战胜亚洲强队印尼了，我还有什么理由不坚持写下去！

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