opencv3之SVD

cv::SVD::compute

static void cv::SVD::compute	(	InputArray	src,
		OutputArray	w,
		OutputArray	u,
		OutputArray	vt,
		int	flags = 0
	)

參數：

//A=u*w*vt
src	decomposed matrix
w	calculated singular values
u	calculated left singular vectors
vt	transposed matrix of right singular values
flags	operation flags

flags可以選的參數：

Enumerator	flag
SVD::MODIFY_A	1	allow the algorithm to modify the decomposed matrix; it can save space and speed up processing. currently ignored.
SVD::NO_UV	2	indicates that only a vector of singular values w is to be processed, while u and vt will be set to empty matrices
SVD::FULL_UV	4	when the matrix is not square, by default the algorithm produces u and vt matrices of sufficiently large size for the further A reconstruction; if, however, FULL_UV flag is specified, u and vt will be full-size square orthogonal matrices.

//使用方法
Mat A, w, u, vt;
cv::SVD::compute(A, w, u, vt,SVD::FULL_UV);
cv::SVDecomp(A,w,u,vt,cv::SVD::MODIFY_A | cv::SVD::FULL_UV);

cv::SVDecomp

也是調用的上面函數

void cv::SVDecomp(InputArray src, OutputArray w, OutputArray u, OutputArray vt, int flags)
{
    CV_INSTRUMENT_REGION()

    SVD::compute(src, w, u, vt, flags);
}

SVD計算方法

奇異值分解定義：

有一個m×n的實數矩陣A，我們想要把它分解成如下的形式：
$A=U\Sigma V^T$
其中U和V均爲單位正交陣

具體計算可以參考：https://byjiang.com/2017/11/18/SVD/

或者：奇異值分解與應用

主要利用如下性質：
$AA^T=U\Sigma V^TV\Sigma^TU^T=U\Sigma\Sigma^TU^T\\ A^TA=V\Sigma^T U^TU\Sigma V^T=V\Sigma^T\Sigma V^T\\$

$\Sigma\Sigma^T= \left[ \begin{matrix} \sigma_1^2 & 0 & \cdots & 0 \\ 0 & \sigma_2^2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma_m^2 \\ \end{matrix} \right], \Sigma^T\Sigma= \left[ \begin{matrix} \sigma_1^2 & 0 & \cdots & 0 \\ 0 & \sigma_2^2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma_n^2 \\ \end{matrix} \right]$

參考

https://docs.opencv.org/3.1.0/d2/de8/group__core__array.html#gab477b5b7b39b370bb03e75b19d2d5109

SVD(奇異值分解)算法及其評估

https://wenku.baidu.com/view/38693ef2e109581b6bd97f19227916888486b916.html?rec_flag=default&sxts=1563780499924