本文主要总结递归高斯算法的基本原理,分析,实现,指令优化实现等
原始高斯滤波算法计算量很大,网上资料很多,不赘述了,自行search
二维高斯滤波在图像处理中非常非常重要,最常见的是高斯模糊,做模糊滤镜,其本质是空域的低通滤波器,在本人另外的两篇博客同态滤波和MSRCR中有提及。对于同态滤波,需要对对数变换后的结果进行高通滤波,常规是变换到频域进行处理,也可以不进行频域变换,直接进行空域高斯滤波实现低通滤波,然后再用减法实现高通滤波。对于MSRCR算法,每个颜色通道需要进行三个强度的高斯滤波,复杂度可想而知。而且常规的高斯滤波需要考虑滤波窗大小,可以自己参考opencv里面的高斯滤波,可以自己搜一下复杂度分析,对于视频等实时处理,原始算法显然没什么实用性。
后面有人提出先进行水平一维滤波,然后再进行竖直方向一维滤波,对于一个点的滤波,计算数据量由NxN减少为2*N,但还是和窗口大小N有关
有看到一种递归减半滤波平面大小的递归实现,原理不太清楚,可参考如下资料,文中fastFilter函数的定义:
http://blog.csdn.net/smallstones/article/details/41787079
然后是本文主要介绍的IIR递归高斯滤波,这个滤波器能近似接近高斯滤波器,也是分成两次一维滤波,关键是不需要设定滤波窗口大小,复杂度跟窗口大小无关,对于一维滤波先进行一次正向滤波,然后进行一次逆向滤波,所以对于大小为NXN大小的二维矩阵,计算复杂度为O(NXN)
计算公式参考:
http://www.cnblogs.com/ImageVision/archive/2012/06/11/2545555.html
实现代码:
GIMP中有代码:http://gimp.sourcearchive.com/documentation/2.6.1/contrast-retinex_8c-source.html
typedef struct { gint scale; gint nscales; gint scales_mode; gfloat cvar; } RetinexParams; /* * Calculate the coefficients for the recursive filter algorithm * Fast Computation of gaussian blurring. */ static void compute_coefs3 (gauss3_coefs *c, gfloat sigma) { /* * Papers: "Recursive Implementation of the gaussian filter.", * Ian T. Young , Lucas J. Van Vliet, Signal Processing 44, Elsevier 1995. * formula: 11b computation of q * 8c computation of b0..b1 * 10 alpha is normalization constant B */ gfloat q, q2, q3; q = 0; if (sigma >= 2.5) { q = 0.98711 * sigma - 0.96330; } else if ((sigma >= 0.5) && (sigma < 2.5)) { q = 3.97156 - 4.14554 * (gfloat) sqrt ((double) 1 - 0.26891 * sigma); } else { q = 0.1147705018520355224609375; } q2 = q * q; q3 = q * q2; c->b[0] = (1.57825+(2.44413*q)+(1.4281 *q2)+(0.422205*q3)); c->b[1] = ( (2.44413*q)+(2.85619*q2)+(1.26661 *q3)); c->b[2] = ( -((1.4281*q2)+(1.26661 *q3))); c->b[3] = ( (0.422205*q3)); c->B = 1.0-((c->b[1]+c->b[2]+c->b[3])/c->b[0]); c->sigma = sigma; c->N = 3; /* g_printerr ("q %f\n", q); g_printerr ("q2 %f\n", q2); g_printerr ("q3 %f\n", q3); g_printerr ("c->b[0] %f\n", c->b[0]); g_printerr ("c->b[1] %f\n", c->b[1]); g_printerr ("c->b[2] %f\n", c->b[2]); g_printerr ("c->b[3] %f\n", c->b[3]); g_printerr ("c->B %f\n", c->B); g_printerr ("c->sigma %f\n", c->sigma); g_printerr ("c->N %d\n", c->N); */ } static void gausssmooth (gfloat *in, gfloat *out, gint size, gint rowstride, gauss3_coefs *c) { /* * Papers: "Recursive Implementation of the gaussian filter.", * Ian T. Young , Lucas J. Van Vliet, Signal Processing 44, Elsevier 1995. * formula: 9a forward filter * 9b backward filter * fig7 algorithm */ gint i,n, bufsize; gfloat *w1,*w2; /* forward pass */ bufsize = size+3; size -= 1; w1 = (gfloat *) g_try_malloc (bufsize * sizeof (gfloat)); w2 = (gfloat *) g_try_malloc (bufsize * sizeof (gfloat)); w1[0] = in[0]; w1[1] = in[0]; w1[2] = in[0]; for ( i = 0 , n=3; i <= size ; i++, n++) { w1[n] = (gfloat)(c->B*in[i*rowstride] + ((c->b[1]*w1[n-1] + c->b[2]*w1[n-2] + c->b[3]*w1[n-3] ) / c->b[0])); } /* backward pass */ w2[size+1]= w1[size+3]; w2[size+2]= w1[size+3]; w2[size+3]= w1[size+3]; for (i = size, n = i; i >= 0; i--, n--) { w2[n]= out[i * rowstride] = (gfloat)(c->B*w1[n] + ((c->b[1]*w2[n+1] + c->b[2]*w2[n+2] + c->b[3]*w2[n+3] ) / c->b[0])); } g_free (w1); g_free (w2); }
/* @function: 1channel 2D gauss filter */ void gausssmooth1ch(float *src, float *dst, int width, int height, gauss3_coefs *c) { for (int y=0; y<height; y++) { gausssmooth(src+y*width, dst+y*width, width, 1, c); } for (int x=0; x<width; x++) { gausssmooth(dst+x, dst+x, height, width, c); } }
加速实现:
intel有详细的介绍和SSE实现:https://software.intel.com/en-us/articles/iir-gaussian-blur-filter-implementation-using-intel-advanced-vector-extensions
总而言之,这个算法非常非常实用