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;
}
}
六：尾声
这三篇文章，是讨论关于图像放缩的基本原理，方法，及实现，如果想要得到全部的源代码
可以联系我。如果文章有任何错误，欢迎指出。
