SIFT算法相关的几个常见疑问

﹏ヽ暗。殇╰゛Y 2022-04-08 08:10 238阅读 0赞

参考博客：

SIFT算法的Matlab实现  
[https://www.sun11.me/blog/2016/sift-implementation-in-matlab/][https_www.sun11.me_blog_2016_sift-implementation-in-matlab]

SIFT特征详解  
[https://www.cnblogs.com/wangguchangqing/p/4853263.html][https_www.cnblogs.com_wangguchangqing_p_4853263.html]

下面我列一下博客中讲到，但涉及源码时又不够透彻的地方（其实是否透彻还是取决于个人的经验和理解）

### 用到的平滑叠加原理 ###

对图像做一次![\\sigma\_0][sigma_0]的高斯平滑，再对图像做一次![\\sigma\_1][sigma_1]的高斯平滑，等效于对图像做一次![\\sigma =\\sqrt\{ \\sigma\_\{0\}^2+\\sigma\_\{1\}^2\}][sigma _sqrt_ _sigma_0_2_sigma_1_2]的高斯平滑。

###  (1) σ的初始化 ###

为了图像反走样的需要，通常假设输入图像是经过高斯平滑处理的，其值为![\\sigma\_n][sigma_n]=0.5，即半个像元。意思就是说我们采集到的图像![I(x,y)][I_x_y]，已经被σ=![\\sigma\_n][sigma_n]=0.5的高斯滤波器平滑过了(事实上有没有滤波过和滤波的大小取决于照相机的算法，我们并不知道)。所以我们不能直接对![I(x,y)][I_x_y]用![\\sigma\_0][sigma_0]的高斯滤波器平滑，而应该用![\\sigma = \\sqrt\{\\sigma\_0^2 - \\sigma\_n^2\}][sigma _ _sqrt_sigma_0_2 - _sigma_n_2]的高斯滤波器去平滑![I(x,y)][I_x_y]，即

![FirstLayer(x,y) = I(x,y)\\otimes G(x,y,\\sqrt\{\\sigma\_0^2 - \\sigma\_n^2\})][FirstLayer_x_y_ _ I_x_y_otimes G_x_y_sqrt_sigma_0_2 - _sigma_n_2]

进一步， 已经知到了![I(x,y)][I_x_y]看成是已经被![\\sigma\_n][sigma_n]=0.5模糊过的图像，那么第１层将![I(x,y)][I_x_y]放大2倍后得到![I\_s(x,y)][I_s_x_y]则可以看为是被  
2![\\sigma\_n][sigma_n]=1的高斯核模糊过的图像。所以，源码中体现的该组(octaves)中第一张图像的滤波就是这样的，

![FirstLayer(x,y) = I\_s(x,y)\\otimes G(x,y,\\sqrt\{\\sigma\_0^2 - (2\\sigma\_n)^2\})][FirstLayer_x_y_ _ I_s_x_y_otimes G_x_y_sqrt_sigma_0_2 - _2_sigma_n_2]

再进一步，根据前面的原理，考虑同一组中，任一前后两张图片之间的σ的差别是

![\\sigma\_\{(diff,i)\} = \\sqrt\{(\\sigma\_0k^\{i+1\})^2 - (\\sigma\_0 k^i)^2\} = \\sigma\_0 k^\{i\} \* \\sqrt\{k^2 - 1^2\}][sigma_diff_i_ _ _sqrt_sigma_0k_i_1_2 - _sigma_0 k_i_2_ _ _sigma_0 k_i_ _ _sqrt_k_2 - 1_2]

所以，为了构建连续的尺度空间，源代码中初始的σ取值是这样的

![\\sigma= \\sigma\_0 \* \\sqrt\{k^2 - 1^2\}][sigma_ _sigma_0 _ _sqrt_k_2 - 1_2]

### (2) 图像数目为什么是S+3 ? ###

要在S个差分图中寻找极值点，计算时就需要S+3张图像！因为S+3张图像可以得到S+2个差分图，进一步在判断（寻找）极值的时候，最前面那个和最后那个差分图是起辅助作用的，因为差分图中需要前后都有差分图才能在其中找极值（要有３张差分图，才能构成尺度空间找极值），这个关系就是

S+3张图像　－－>>　S+2张差分图　　－－>>　可在其中的S张差分图中找极值

这也就是下图的理解由来

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Rhbm14MjE5_size_16_color_FFFFFF_t_70][]

### （３）抛物线插值 ###

这个各博客中都没提，我顺便推导了一下

根据抛物线插值公式，对三个点![(-1,l), (0,c),(1,r)][-1_l_ _0_c_1_r]进行插值计算是这样的

![y = \\frac\{l(x - 0)(x - 1)\}\{2\} - c(x + 1)(x - 1) + \\frac\{r(x + 1)(x - 0)\}\{2\}][y _ _frac_l_x - 0_x - 1_2_ - c_x _ 1_x - 1_ _ _frac_r_x _ 1_x - 0_2]

对极值点，有

![\\frac\{dy\}\{dx\} = \\frac\{1\}\{2\} (-4 c x+l (2 x-1)+2 r x+r) = 0][frac_dy_dx_ _ _frac_1_2_ _-4 c x_l _2 x-1_2 r x_r_ _ 0]

求解即得

![x=\\frac\{r-l\}\{2 (l-2 c+r)\}][x_frac_r-l_2 _l-2 c_r]

### （４）为什么使用DOG代替LOG? ###

SIFT的计算量无疑是很大的，使用近似的DOG代替LOG就是为了节省计算。 如下所示，DoG的系数是常数，在巨量的计算中，计算起来要快得多。

DOG  
![LoG\\triangleq \\bigtriangleup G\_\{\\sigma\}(x,y) = \\frac\{\\partial^2\}\{\\partial x^2\} G\_\{\\sigma\}(x,y) + \\frac\{\\partial^2\}\{\\partial y^2\} G\_\{\\sigma\}(x,y) = \\frac\{x^2+y^2-2\\sigma^2\}\{\\sigma^4\}e^\{-(x^2+y^2)/(2\\sigma^2)\}][LoG_triangleq _bigtriangleup G_sigma_x_y_ _ _frac_partial_2_partial x_2_ G_sigma_x_y_ _ _frac_partial_2_partial y_2_ G_sigma_x_y_ _ _frac_x_2_y_2-2_sigma_2_sigma_4_e_-_x_2_y_2_2_sigma_2]

LOG  
![DoG\\triangleq =G\_\{\\sigma\_1\}(x,y) - G\_\{\\sigma\_2\}(x,y) =\\frac\{1\}\{\\sqrt\{2\\pi\}\} \\left( \\frac\{1\}\{\\sigma\_1\}e^\{-(x^2+y^2)/(2\\sigma\_1^2)\} - \\frac\{1\}\{\\sigma\_2\}e^\{-(x^2+y^2)/(2\\sigma\_2^2)\} \\right)][DoG_triangleq _G_sigma_1_x_y_ - G_sigma_2_x_y_ _frac_1_sqrt_2_pi_ _left_ _frac_1_sigma_1_e_-_x_2_y_2_2_sigma_1_2_ - _frac_1_sigma_2_e_-_x_2_y_2_2_sigma_2_2_ _right]

我没有仔细考核，也没找到相关的具体指标，但找到了这个帖子，供参考

〈【计算机视觉】SIFT中LoG和DoG比较：[https://blog.csdn.net/xiaowei\_cqu/article/details/27692123][https_blog.csdn.net_xiaowei_cqu_article_details_27692123]〉

这里要注意：DoG需要在每层金字塔要多做一次高斯操作（即需要S+3张高斯模糊图来得到得到S+2张DoG图，而LoG只需要直接计算S+2张图像），但整体上，DoG通过减法取代LoG核大大减少了运算次数，节省了运算时间。

[https_www.sun11.me_blog_2016_sift-implementation-in-matlab]: https://www.sun11.me/blog/2016/sift-implementation-in-matlab/
[https_www.cnblogs.com_wangguchangqing_p_4853263.html]: https://www.cnblogs.com/wangguchangqing/p/4853263.html
[sigma_0]: https://private.codecogs.com/gif.latex?%5Csigma_0
[sigma_1]: https://private.codecogs.com/gif.latex?%5Csigma_1
[sigma _sqrt_ _sigma_0_2_sigma_1_2]: https://private.codecogs.com/gif.latex?%5Csigma%20%3D%5Csqrt%7B%20%5Csigma_%7B0%7D%5E2&plus;%5Csigma_%7B1%7D%5E2%7D
[sigma_n]: https://private.codecogs.com/gif.latex?%5Csigma_n
[I_x_y]: https://private.codecogs.com/gif.latex?I%28x%2Cy%29
[sigma _ _sqrt_sigma_0_2 - _sigma_n_2]: https://private.codecogs.com/gif.latex?%5Csigma%20%3D%20%5Csqrt%7B%5Csigma_0%5E2%20-%20%5Csigma_n%5E2%7D
[FirstLayer_x_y_ _ I_x_y_otimes G_x_y_sqrt_sigma_0_2 - _sigma_n_2]: https://private.codecogs.com/gif.latex?FirstLayer%28x%2Cy%29%20%3D%20I%28x%2Cy%29%5Cotimes%20G%28x%2Cy%2C%5Csqrt%7B%5Csigma_0%5E2%20-%20%5Csigma_n%5E2%7D%29
[I_s_x_y]: https://private.codecogs.com/gif.latex?I_s%28x%2Cy%29
[FirstLayer_x_y_ _ I_s_x_y_otimes G_x_y_sqrt_sigma_0_2 - _2_sigma_n_2]: https://private.codecogs.com/gif.latex?FirstLayer%28x%2Cy%29%20%3D%20I_s%28x%2Cy%29%5Cotimes%20G%28x%2Cy%2C%5Csqrt%7B%5Csigma_0%5E2%20-%20%282%5Csigma_n%29%5E2%7D%29
[sigma_diff_i_ _ _sqrt_sigma_0k_i_1_2 - _sigma_0 k_i_2_ _ _sigma_0 k_i_ _ _sqrt_k_2 - 1_2]: https://private.codecogs.com/gif.latex?%5Csigma_%7B%28diff%2Ci%29%7D%20%3D%20%5Csqrt%7B%28%5Csigma_0k%5E%7Bi&plus;1%7D%29%5E2%20-%20%28%5Csigma_0%20k%5Ei%29%5E2%7D%20%3D%20%5Csigma_0%20k%5E%7Bi%7D%20*%20%5Csqrt%7Bk%5E2%20-%201%5E2%7D
[sigma_ _sigma_0 _ _sqrt_k_2 - 1_2]: https://private.codecogs.com/gif.latex?%5Csigma%3D%20%5Csigma_0%20*%20%5Csqrt%7Bk%5E2%20-%201%5E2%7D
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3Rhbm14MjE5_size_16_color_FFFFFF_t_70]: /images/20220407/a2edf6ec257d4dcca449e99da0727499.png
[-1_l_ _0_c_1_r]: https://private.codecogs.com/gif.latex?%28-1%2Cl%29%2C%20%280%2Cc%29%2C%281%2Cr%29
[y _ _frac_l_x - 0_x - 1_2_ - c_x _ 1_x - 1_ _ _frac_r_x _ 1_x - 0_2]: https://private.codecogs.com/gif.latex?y%20%3D%20%5Cfrac%7Bl%28x%20-%200%29%28x%20-%201%29%7D%7B2%7D%20-%20c%28x%20&plus;%201%29%28x%20-%201%29%20&plus;%20%5Cfrac%7Br%28x%20&plus;%201%29%28x%20-%200%29%7D%7B2%7D
[frac_dy_dx_ _ _frac_1_2_ _-4 c x_l _2 x-1_2 r x_r_ _ 0]: https://private.codecogs.com/gif.latex?%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%28-4%20c%20x&plus;l%20%282%20x-1%29&plus;2%20r%20x&plus;r%29%20%3D%200
[x_frac_r-l_2 _l-2 c_r]: https://private.codecogs.com/gif.latex?x%3D%5Cfrac%7Br-l%7D%7B2%20%28l-2%20c&plus;r%29%7D
[LoG_triangleq _bigtriangleup G_sigma_x_y_ _ _frac_partial_2_partial x_2_ G_sigma_x_y_ _ _frac_partial_2_partial y_2_ G_sigma_x_y_ _ _frac_x_2_y_2-2_sigma_2_sigma_4_e_-_x_2_y_2_2_sigma_2]: https://private.codecogs.com/gif.latex?LoG%5Ctriangleq%20%5Cbigtriangleup%20G_%7B%5Csigma%7D%28x%2Cy%29%20%3D%20%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20x%5E2%7D%20G_%7B%5Csigma%7D%28x%2Cy%29%20&plus;%20%5Cfrac%7B%5Cpartial%5E2%7D%7B%5Cpartial%20y%5E2%7D%20G_%7B%5Csigma%7D%28x%2Cy%29%20%3D%20%5Cfrac%7Bx%5E2&plus;y%5E2-2%5Csigma%5E2%7D%7B%5Csigma%5E4%7De%5E%7B-%28x%5E2&plus;y%5E2%29/%282%5Csigma%5E2%29%7D
[DoG_triangleq _G_sigma_1_x_y_ - G_sigma_2_x_y_ _frac_1_sqrt_2_pi_ _left_ _frac_1_sigma_1_e_-_x_2_y_2_2_sigma_1_2_ - _frac_1_sigma_2_e_-_x_2_y_2_2_sigma_2_2_ _right]: https://private.codecogs.com/gif.latex?DoG%5Ctriangleq%20%3DG_%7B%5Csigma_1%7D%28x%2Cy%29%20-%20G_%7B%5Csigma_2%7D%28x%2Cy%29%20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7D%20%5Cleft%28%20%5Cfrac%7B1%7D%7B%5Csigma_1%7De%5E%7B-%28x%5E2&plus;y%5E2%29/%282%5Csigma_1%5E2%29%7D%20-%20%5Cfrac%7B1%7D%7B%5Csigma_2%7De%5E%7B-%28x%5E2&plus;y%5E2%29/%282%5Csigma_2%5E2%29%7D%20%5Cright%29
[https_blog.csdn.net_xiaowei_cqu_article_details_27692123]: https://blog.csdn.net/xiaowei_cqu/article/details/27692123