图像处理之一阶微分应用

一：数学背景

首先看一下一维的微分公式Δf = f(x+1) – f(x), 对于一幅二维的数字图像f(x,y)而言，需要完

成XY两个方向上的微分，所以有如下的公式：

分别对X,Y两个方向上求出它们的偏微分，最终得到梯度Delta F.

对于离散的图像来说,一阶微分的数学表达相当于两个相邻像素的差值，根据选择的梯度算

子不同，效果可能有所不同，但是基本原理不会变化。最常见的算子为Roberts算子，其它

常见还有Sobel，Prewitt等算子。以Roberts算子为例的X,Y的梯度计算演示如下图：

二：图像微分应用

图像微分(梯度计算)是图像边缘提取的重要的中间步骤，根据X,Y方向的梯度向量值，可以

得到如下两个重要参数振幅magnitude, 角度theta，计算公式如下：

Theta = tan^-1(yGradient/xGradient)

magnitude表示边缘强度信息

theta预言边缘的方向走势。

假如对一幅数字图像，求出magnitude之后与原来每个像素点对应值相加，则图像边缘将被

大大加强，轮廓更加明显，是一个很典型的sharp filter的效果。

 

三：程序效果

X, Y梯度效果，及magnitude效果

图像微分的Sharp效果：

四：程序源代码

package com.process.blur.study;import java.awt.image.BufferedImage;// roberts operator// X direction 1, 0//             0,-1// Y direction 0, 1//  -1, 0public class ImageGradientFilter extends AbstractBufferedImageOp {public final static int X_DIRECTION = 0;public final static int Y_DIRECTION = 2;public final static int XY_DIRECTION = 4;private boolean sharp;private int direction;public ImageGradientFilter() {direction = XY_DIRECTION; // default;sharp = false;}public boolean isSharp() {return sharp;}public void setSharp(boolean sharp) {this.sharp = sharp;}public int getDirection() {return direction;}public void setDirection(int direction) {this.direction = direction;}@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;        double mred, mgreen, mblue;        int newX, newY;        int index1, index2, index3;        for(int row=0; row<height; row++) {        int ta = 0, tr = 0, tg = 0, tb = 0;        for(int col=0; col<width; col++) {        index = row * width + col;                // base on roberts operator        newX = col + 1;        newY = row + 1;        if(newX > 0 && newX < width) {        newX = col + 1;        } else {        newX = 0;        }                if(newY > 0 && newY < height) {        newY = row + 1;        } else {        newY = 0;        }        index1 = newY * width + newX;        index2 = row * width + newX;        index3 = newY * width + col;        ta = (inPixels[index] >> 24) & 0xff;                tr = (inPixels[index] >> 16) & 0xff;                tg = (inPixels[index] >> 8) & 0xff;                tb = inPixels[index] & 0xff;                        int ta1 = (inPixels[index1] >> 24) & 0xff;                int tr1 = (inPixels[index1] >> 16) & 0xff;                int tg1 = (inPixels[index1] >> 8) & 0xff;                int tb1 = inPixels[index1] & 0xff;                                int xgred = tr -tr1;                int xggreen = tg - tg1;                int xgblue = tb - tb1;                        int ta2 = (inPixels[index2] >> 24) & 0xff;                int tr2 = (inPixels[index2] >> 16) & 0xff;                int tg2 = (inPixels[index2] >> 8) & 0xff;                int tb2 = inPixels[index2] & 0xff;                        int ta3 = (inPixels[index3] >> 24) & 0xff;                int tr3 = (inPixels[index3] >> 16) & 0xff;                int tg3 = (inPixels[index3] >> 8) & 0xff;                int tb3 = inPixels[index3] & 0xff;                                int ygred = tr2 - tr3;                int yggreen = tg2 - tg3;                int ygblue = tb2 - tb3;                                mred = Math.sqrt(xgred * xgred + ygred * ygred);                mgreen = Math.sqrt(xggreen * xggreen + yggreen * yggreen);                mblue = Math.sqrt(xgblue * xgblue + ygblue * ygblue);                if(sharp) {                tr = (int)(tr + mred);                tg = (int)(tg + mgreen);                tb = (int)(tb + mblue);                outPixels[index] = (ta << 24) | (clamp(tr) << 16) | (clamp(tg) << 8) | clamp(tb);                } else {                    outPixels[index] = (ta << 24) | (clamp((int)mred) << 16) | (clamp((int)mgreen) << 8) | clamp((int)mblue);                    // outPixels[index] = (ta << 24) | (clamp((int)ygred) << 16) | (clamp((int)yggreen) << 8) | clamp((int)ygblue);                    // outPixels[index] = (ta << 24) | (clamp((int)xgred) << 16) | (clamp((int)xggreen) << 8) | clamp((int)xgblue);                }                                        }        }        setRGB(dest, 0, 0, width, height, outPixels );        return dest;}public static int clamp(int c) {if (c < 0)return 0;if (c > 255)return 255;return c;}}

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