图像处理之一阶微分应用

- created by gloomyfish

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, 0    public 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;      }        @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;          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;      }  }