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; } }
