图像处理------高斯一阶及二阶导数计算 分类： 视频图像处理 2015-07-24 15:07 72人阅读 评论(0) 收藏

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;
    }

}

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