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图像放缩之双立方插值

一：数学原理

如果已知一个函数f(x)以及它在x=0,x=1处的导数，那么函数可以在[0,1]之间插值，当函数

表达为三次多项式时我们称之谓立方插值。一个三次多项式及其导数：

f(x) =ax^3 +bx^2 + cx + d

f’(x)=3ax^2 + 2bx +c

多项式在x=0, x=1处值及其导数值为：

f(0)= d;

f(1)= a + b + c + d;

f’(0)=c

f’(1)=3a + 2b + c

上述的四个等式可以等价的变换为:

a= 2f(0) – 2f(1) + f’(0) + f’(1)

b= -3f(0) + 3f(1) – 2f’(0) – f’(1)

c= f’(0)

d= f’(1)

假设你有四个点值p0, p1, p2, p3分别在x=-1, x=0, x=1, x=2, 把值分别指定到f(0), f(1), f’(0),

f’(1)中为:

f(0)= p1

f(1)= p2

f’(0)= (p2 – p0)/2

f’(1)= (p3-p1)/2

这个我们的立方插值公式变成:

f(p0,p1,p2,p3, x) = (-1/2p0 + 3/2p1 -3/2p2+ 1/2p3)x^3 + (p0-5/2p1 + 2p2 -1/2d)x^2 + (-1/2p0 +

1/2p2)x + p1

双立方插值是立方插值在二维空间的表达, 插值公式可以表述为:

G(x, y) = f (f (p00, p01, p02, p03, y), f(p10,p11, p12, p13, y), f(p20, p21, p22, p23, y), f(p30, p31, p32, p33, y), x)

解出其中的16个参数，即可得带G(x, y)目标插值点的值。

二：双立方插值优缺点

双立方插值在图像放大过程可以保留更多的图像细节，放大以后的图像带有反锯齿的功能，

同时图像和源图像相比效果更加真实, 缺点是计算量比较大，是常见的三种图像放大算法中

计算量最大的一种，据说Photoshop的图像放大就是基本双立方插值的优化算法

三：程序运行效果如下：

四：关键代码解析

不想解释太多，最重要的是代入计算的是浮点数坐标的小数部分，即 x, y的取值范围均在[0,1]之间

五：基于Java的程序完全源代码

public class BiCubicInterpolationScale implements ImageScale {
 
  private double a00, a01, a02, a03;
  private double a10, a11, a12, a13;
  private double a20, a21, a22, a23;
  private double a30, a31, a32, a33;
  private int srcWidth;
  private int srcHeight;
  
  @Override
  public int[] imgScale(int[] inPixelsData, int srcW, int srcH, int destW, int destH) {
    double[][][] input3DData = processOneToThreeDeminsion(inPixelsData, srcH, srcW);
    int[][][] outputThreeDeminsionData = new int[destH][destW][4];
    double[][] tempPixels = new double[4][4];
    float rowRatio = ((float)srcH)/((float)destH);
    float colRatio = ((float)srcW)/((float)destW);
    this.srcWidth = srcW;
    this.srcHeight = srcH;
    for(int row=0; row<destH; row++) {
      // convert to three dimension data
      double srcRow = ((float)row)*rowRatio;
      double j = Math.floor(srcRow);
      double t = srcRow - j;
      for(int col=0; col<destW; col++) {
        double srcCol = ((float)col)*colRatio;
        double k = Math.floor(srcCol);
        double u = srcCol - k;
        for(int i=0; i<4; i++) {
          tempPixels[0][0] = getRGBValue(input3DData,j-1, k-1,i);
          tempPixels[0][1] = getRGBValue(input3DData,j-1, k, i);
          tempPixels[0][2] = getRGBValue(input3DData, j-1,k+1, i);
          tempPixels[0][3] = getRGBValue(input3DData, j-1, k+2,i);
          
          tempPixels[1][0] = getRGBValue(input3DData, j, k-1, i);
          tempPixels[1][1] = getRGBValue(input3DData, j, k, i);
          tempPixels[1][2] = getRGBValue(input3DData, j, k+1, i);
          tempPixels[1][3] = getRGBValue(input3DData, j, k+2, i);
          
          tempPixels[2][0] = getRGBValue(input3DData, j+1,k-1,i);
          tempPixels[2][1] = getRGBValue(input3DData, j+1, k, i);
          tempPixels[2][2] = getRGBValue(input3DData, j+1, k+1, i);
          tempPixels[2][3] = getRGBValue(input3DData, j+1, k+2, i);
          
          tempPixels[3][0] = getRGBValue(input3DData, j+2, k-1, i);
          tempPixels[3][1] = getRGBValue(input3DData, j+2, k, i);
          tempPixels[3][2] = getRGBValue(input3DData, j+2, k+1, i);
          tempPixels[3][3] = getRGBValue(input3DData, j+2, k+2, i);
          
          // update coefficients
          updateCoefficients(tempPixels);
          outputThreeDeminsionData[row][col][i] = getPixelValue(getValue(t, u));
        }
 
      }
    }
    
    return convertToOneDim(outputThreeDeminsionData, destW, destH);
  }
  
  public double getRGBValue(double[][][] input3DData, double row, double col, int index) {
    if(col >= srcWidth) {
      col = srcWidth - 1;
    }
    
    if(col < 0) {
      col = 0;
    }
    
    if(row >= srcHeight) {
      row = srcHeight - 1;
    }
    
    if(row < 0) {
      row = 0;
    }
    return input3DData[(int)row][(int)col][index];
  }
  
  public int getPixelValue(double pixelValue) {
    return pixelValue < 0 ? 0: pixelValue >255.0d ?255:(int)pixelValue;
  }
  
  public void updateCoefficients (double[][] p) {
    a00 = p[1][1];
    a01 = -.5*p[1][0] + .5*p[1][2];
    a02 = p[1][0] - 2.5*p[1][1] + 2*p[1][2] - .5*p[1][3];
    a03 = -.5*p[1][0] + 1.5*p[1][1] - 1.5*p[1][2] + .5*p[1][3];
    a10 = -.5*p[0][1] + .5*p[2][1];
    a11 = .25*p[0][0] - .25*p[0][2] - .25*p[2][0] + .25*p[2][2];
    a12 = -.5*p[0][0] + 1.25*p[0][1] - p[0][2] + .25*p[0][3] + .5*p[2][0] - 1.25*p[2][1] + p[2][2] - .25*p[2][3];
    a13 = .25*p[0][0] - .75*p[0][1] + .75*p[0][2] - .25*p[0][3] - .25*p[2][0] + .75*p[2][1] - .75*p[2][2] + .25*p[2][3];
    a20 = p[0][1] - 2.5*p[1][1] + 2*p[2][1] - .5*p[3][1];
    a21 = -.5*p[0][0] + .5*p[0][2] + 1.25*p[1][0] - 1.25*p[1][2] - p[2][0] + p[2][2] + .25*p[3][0] - .25*p[3][2];
    a22 = p[0][0] - 2.5*p[0][1] + 2*p[0][2] - .5*p[0][3] - 2.5*p[1][0] + 6.25*p[1][1] - 5*p[1][2] + 1.25*p[1][3] + 2*p[2][0] - 5*p[2][1] + 4*p[2][2] - p[2][3] - .5*p[3][0] + 1.25*p[3][1] - p[3][2] + .25*p[3][3];
    a23 = -.5*p[0][0] + 1.5*p[0][1] - 1.5*p[0][2] + .5*p[0][3] + 1.25*p[1][0] - 3.75*p[1][1] + 3.75*p[1][2] - 1.25*p[1][3] - p[2][0] + 3*p[2][1] - 3*p[2][2] + p[2][3] + .25*p[3][0] - .75*p[3][1] + .75*p[3][2] - .25*p[3][3];
    a30 = -.5*p[0][1] + 1.5*p[1][1] - 1.5*p[2][1] + .5*p[3][1];
    a31 = .25*p[0][0] - .25*p[0][2] - .75*p[1][0] + .75*p[1][2] + .75*p[2][0] - .75*p[2][2] - .25*p[3][0] + .25*p[3][2];
    a32 = -.5*p[0][0] + 1.25*p[0][1] - p[0][2] + .25*p[0][3] + 1.5*p[1][0] - 3.75*p[1][1] + 3*p[1][2] - .75*p[1][3] - 1.5*p[2][0] + 3.75*p[2][1] - 3*p[2][2] + .75*p[2][3] + .5*p[3][0] - 1.25*p[3][1] + p[3][2] - .25*p[3][3];
    a33 = .25*p[0][0] - .75*p[0][1] + .75*p[0][2] - .25*p[0][3] - .75*p[1][0] + 2.25*p[1][1] - 2.25*p[1][2] + .75*p[1][3] + .75*p[2][0] - 2.25*p[2][1] + 2.25*p[2][2] - .75*p[2][3] - .25*p[3][0] + .75*p[3][1] - .75*p[3][2] + .25*p[3][3];
  }
  
  public double getValue (double x, double y) {
    double x2 = x * x;
    double x3 = x2 * x;
    double y2 = y * y;
    double y3 = y2 * y;
 
    return (a00 + a01 * y + a02 * y2 + a03 * y3) +
           (a10 + a11 * y + a12 * y2 + a13 * y3) * x +
           (a20 + a21 * y + a22 * y2 + a23 * y3) * x2 +
           (a30 + a31 * y + a32 * y2 + a33 * y3) * x3;
  }
 
  /* <p> The purpose of this method is to convert the data in the 3D array of ints back into </p>
   * <p> the 1d array of type int. </p>
   * 
   */
  public int[] convertToOneDim(int[][][] data, int imgCols, int imgRows) {
    // Create the 1D array of type int to be populated with pixel data
    int[] oneDPix = new int[imgCols * imgRows * 4];
 
    // Move the data into the 1D array. Note the
    // use of the bitwise OR operator and the
    // bitwise left-shift operators to put the
    // four 8-bit bytes into each int.
    for (int row = 0, cnt = 0; row < imgRows; row++) {
      for (int col = 0; col < imgCols; col++) {
        oneDPix[cnt] = ((data[row][col][0] << 24) & 0xFF000000)
            | ((data[row][col][1] << 16) & 0x00FF0000)
            | ((data[row][col][2] << 8) & 0x0000FF00)
            | ((data[row][col][3]) & 0x000000FF);
        cnt++;
      }// end for loop on col
 
    }// end for loop on row
 
    return oneDPix;
  }// end convertToOneDim
  
  private double [][][] processOneToThreeDeminsion(int[] oneDPix2, int imgRows, int imgCols) {
    double[][][] tempData = new double[imgRows][imgCols][4];
    for(int row=0; row<imgRows; row++) {
      
      // per row processing
      int[] aRow = new int[imgCols];
      for (int col = 0; col < imgCols; col++) {
        int element = row * imgCols + col;
        aRow[col] = oneDPix2[element];
      }
      
      // convert to three dimension data
      for(int col=0; col<imgCols; col++) {
        tempData[row][col][0] = (aRow[col] >> 24) & 0xFF; // alpha
        tempData[row][col][1] = (aRow[col] >> 16) & 0xFF; // red
        tempData[row][col][2] = (aRow[col] >> 8) & 0xFF;  // green
        tempData[row][col][3] = (aRow[col]) & 0xFF;       // blue
      }
    }
    return tempData;
  }
}

图像放缩之双立方插值

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