Nonrigid image deformation using moving regularized least quares

MLS算法是基于最小二乘法的移动网格变形算法，该算法依赖于网格大小的选择，而本文介绍的这篇论文，就是对网格变形的改进，使他不依赖于网格的选择。

为了方便更多的人看懂，本文主要讲算法实现的流程，不侧重于数学公式的推导之类。所以这里直接给出算法实现过程：

其中：

上面的算法流程经作者实践，比较耗时，于是作者做了优化，对公式8做了改进，如下：

论文效果图如下：

本人代码实现这个算法如下：

// 对矩阵求逆，结果保存在a中
int f_GaussInverseMatrix(double A[], double B[], int nColumns)
{  memcpy(B, A, sizeof(double) * nColumns * nColumns);int *is,*js,i,j,k,l,u,v;  double d,p;  is = new int[nColumns];  js = new int[nColumns];  for (k=0; k<=nColumns-1; k++)  {  d=0.0;  for (i=k; i<=nColumns-1; i++)  for (j=k; j<=nColumns-1; j++)  {  l=i*nColumns+j; p=fabs(B[l]);  if (p>d) { d=p; is[k]=i; js[k]=j;}  }  if (d+1.0==1.0)  {  free(is); free(js); printf("err**not inv\n");  return(1);  }  if (is[k]!=k)  for (j=0; j<=nColumns-1; j++)  {  u=k*nColumns+j; v=is[k]*nColumns+j;  p=B[u]; B[u]=B[v]; B[v]=p;  }  if (js[k]!=k)  for (i=0; i<=nColumns-1; i++)  {  u=i*nColumns+k; v=i*nColumns+js[k];  p=B[u]; B[u]=B[v]; B[v]=p;  }  l=k*nColumns+k;  B[l]=1.0f/B[l];  for (j=0; j<=nColumns-1; j++)  if (j!=k)  { u=k*nColumns+j; B[u]=B[u]*B[l];}  for (i=0; i<=nColumns-1; i++)  if (i!=k)  for (j=0; j<=nColumns-1; j++)  if (j!=k)  {  u=i*nColumns+j;  B[u] -= B[i*nColumns+k]*B[k*nColumns+j];  }  for (i=0; i<=nColumns-1; i++)  if (i!=k)  {  u=i*nColumns+k;  B[u] = -B[u]*B[l];  }  }  for (k=nColumns-1; k>=0; k--)  {  if (js[k]!=k)  for (j=0; j<=nColumns-1; j++)  {  u=k*nColumns+j; v=js[k]*nColumns+j;  p=B[u]; B[u]=B[v]; B[v]=p;  }  if (is[k]!=k)  for (i=0; i<=nColumns-1; i++)  {   u=i*nColumns+k; v=i*nColumns+is[k];  p=B[u]; B[u]=B[v]; B[v]=p;  }  }  free(is); free(js);  return 0;
}  // 求矩阵的乘积 C = A.*B
void f_matrixMul(double a[], double b[], int m, int n, int k, double c[])
{  int i, j, l, u;  for (i = 0; i < m; i++)  {for (j = 0; j < k; j++)  {  u = i * k + j; c[u] = 0.0;  for (l = 0; l < n; l++)  {c[u] += a[i * n + l] * b[l * k + j];}}  }
}

#define MIN2(a, b) ((a) < (b) ? (a) : (b))
#define MAX2(a, b) ((a) > (b) ? (a) : (b))
#define CLIP3(x, a, b) MIN2(MAX2(a,x), b)
//------------------------------------------------------------------
//Function: MRLS
//Params:
//return: 0 or false
//Reference: Nonrigid image deformation using moving regularized least squares. Jiayi Ma,2013
//-------------------------------------------------------------------
int f_MLSR(unsigned char *srcData, int width, int height, int stride, int srcPoint[], int dstPoint[], int pointNum)
{int ret = 0;unsigned char* tempData = (unsigned char*)malloc(sizeof(unsigned char)* height * stride);memcpy(tempData, srcData, sizeof(unsigned char) * height * stride);//Processdouble *srcPx = (double*)malloc(sizeof(double) * pointNum);double *srcPy = (double*)malloc(sizeof(double) * pointNum);double *dstPx = (double*)malloc(sizeof(double) * pointNum);double *dstPy = (double*)malloc(sizeof(double) * pointNum);double* GammaIJ= (double*)malloc(sizeof(double) * pointNum * pointNum); double* GammaIJT= (double*)malloc(sizeof(double) * pointNum * pointNum); double* GammaP = (double*)malloc(sizeof(double) * pointNum); double* S      = (double*)malloc(sizeof(double) * pointNum); double* W      = (double*)malloc(sizeof(double) * pointNum * pointNum); double* WInv   = (double*)malloc(sizeof(double) * pointNum * pointNum); double* tempIJ = (double*)malloc(sizeof(double) * pointNum * pointNum); double belta = 2.0;double alpha = 2.0;double lambda = 1.0;double fbelta = 1.0 / (belta * belta);for(int i = 0; i < pointNum; i++){srcPx[i] = (double)srcPoint[2 * i] / width;srcPy[i] = (double)srcPoint[2 * i + 1] / height;dstPx[i] = (double)dstPoint[2 * i] / width;dstPy[i] = (double)dstPoint[2 * i + 1] / height;}int w = pointNum;for(int j = 0; j < pointNum; j++){for(int i = 0; i < pointNum; i++){GammaIJ[i + j * w] = exp(-((double)(srcPx[i] - srcPx[j]) * (srcPx[i] - srcPx[j]) + (srcPy[i] - srcPy[j]) * (srcPy[i] - srcPy[j])) * fbelta);}}unsigned char* pSrc = srcData;int aa, bb, cc, dd, pos, pos1;double r1, r2, r3;unsigned char *pSrcL1;unsigned char *pSrcL2;unsigned char* p = tempData;for(int j = 0; j < height; j++){for(int i = 0; i < width; i++){double px = (double)i / width, py = (double)j / height;for(int n = 0; n < pointNum; n++){for(int m = 0; m < pointNum; m++){if(n == m)W[m + n * w] = pow(((double)(px - srcPx[m]) * (px - srcPx[m]) + (py - srcPy[m]) * (py - srcPy[m])), -alpha);elseW[m + n * w] = 0;}}//compute inverse matrix of Wf_GaussInverseMatrix(W, WInv, pointNum);//compute Gamma + lambda * WInvfor(int n = 0; n < pointNum; n++){for(int m = 0; m < pointNum; m++){//W is a diagonal matrix with the i-th entry Wi(p),other's 0GammaIJT[m + n * w] = GammaIJ[m + n * w] + lambda * WInv[m + n * w];}}//compute inverse matrix of (Gamma + lambda * WInv)f_GaussInverseMatrix(GammaIJT, tempIJ, pointNum);//compute GammaP   exp(-(p-xi) * (p - xi) / (belta * belta))for(int m = 0; m < pointNum; m++)GammaP[m] = exp((-((double)(px - srcPx[m]) * (px - srcPx[m]) + (py - srcPy[m]) * (py - srcPy[m])) * fbelta));//compute Sf_matrixMul(GammaP, tempIJ, 1, pointNum, pointNum, S);//compute fp   T + SYdouble sumx = 0, sumy = 0;for(int m = 0; m < pointNum; m++){sumx += S[m] * srcPx[m];sumy += S[m] * srcPy[m];}px = px - sumx;py = py - sumy;sumx = 0, sumy = 0;for(int m = 0; m < pointNum; m++){sumx += S[m] * dstPx[m];sumy += S[m] * dstPy[m];}px = px + sumx;py = py + sumy;double x_in = CLIP3(px * width, 0, width - 1);double y_in = CLIP3(py * height, 0, height - 1);//please change interpolation to get better effects.int xx = (int)x_in;int yy = (int)y_in;     pSrcL1 = p + yy * stride;pSrcL2 = pSrcL1 + stride;pos = (xx << 2);aa = pSrcL1[pos];bb = pSrcL1[pos + 4];cc = pSrcL2[pos];dd = pSrcL2[pos + 4];r1 = aa + (bb - aa) * (x_in - xx);r2 = cc + (dd - cc) * (x_in - xx);r3 = r1 + (r2 - r1) * (y_in - yy);pSrc[0]=(unsigned char)(CLIP3(r3, 0, 255));//B分量aa = pSrcL1[pos + 1];bb = pSrcL1[pos + 4 +1];cc = pSrcL2[pos + 1];dd = pSrcL2[pos + 4 + 1];r1 = aa + (bb - aa) * (x_in - xx);r2 = cc + (dd - cc) * (x_in - xx);r3 = r1 + (r2 - r1) * (y_in - yy);pSrc[1]=(unsigned char)(CLIP3(r3, 0, 255));//G分量aa = pSrcL1[pos + 2];bb = pSrcL1[pos + 4 + 2];cc = pSrcL2[pos + 2];dd = pSrcL2[pos + 4 + 2];r1 = aa + (bb - aa) * (x_in - xx);r2 = cc + (dd - cc) * (x_in - xx);r3 = r1 + (r2 - r1) * (y_in - yy);pSrc[2]=(unsigned char)(CLIP3(r3, 0, 255));//R分量aa = pSrcL1[pos + 3];bb = pSrcL1[pos + 4 + 3];cc = pSrcL2[pos + 3];dd = pSrcL2[pos + 4 + 3];r1=aa + (bb - aa) * (x_in - xx);r2=cc + (dd - cc) * (x_in - xx);r3=r1 + (r2 - r1) * (y_in - yy);pSrc[3]=(unsigned char)(CLIP3(r3, 0, 255));//A分量pSrc += 4;}}free(tempData);free(srcPx);free(srcPy);free(dstPx);free(dstPy);free(GammaIJ);free(GammaP);free(S     );free(W     );free(WInv  );free(tempIJ);free(GammaIJT);return ret;
};

具体效果，本人与上一篇博文保持一致仍然以瘦脸效果为例，原图和效果图如下：

原图(蓝色点为原始点，红色点为拖拽点)

瘦脸效果图

图片来自网络，如有侵权敬请告知

该算法优缺点：

算法变形效果非常好，但是在10个点位个数以上，速度会随点位增加而变慢，对于全脸100+个特征点变形而言，速度太慢；算法适用于10个点位左右的人脸五官变形，比如瘦脸，胖脸，小脸，嘴巴变形等等，当然也可以做其他变形功能需求；

本人QQ1358009172

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