什么是张量

  张量是多维数组的泛概念。一维数组我们通常称之为向量，二维数组我们通常称之为矩阵，但其实这些都是张量的一种。以此类推，我们也会有三维张量、四维张量以及五维张量。那么零维张量是什么呢？其实零维张量就是一个数。

张量的基本操作

两个张量的内积

$<\chi,y>=\sum_{i_1=1}^{I_1} \sum_{i_2=1}^{I_2} ... \sum_{i_N=1}^{I_N} x_{i_1 i_2 ... i_N} y_{i_1 i_2 ... i_N}$<χ,y>=i1​=1∑I1​​i2​=1∑I2​​...iN​=1∑IN​​xi1​i2​...iN​​yi1​i2​...iN​​

介绍

$observation[I] = image[I_{clean}]+noise[\eta]$observation[I]=image[Iclean​]+noise[η]

算法描述

张量简介

HOSVD分解

基于HOSVD的去噪算法

算法流程

Algorithm 1:HOSVD denoising
input :$I_{noisy}$Inoisy​：original imag，$\sigma$σ
output:$I_{denoised}$Idenoised​：Denoised image
Set kHard = 8
Precomputes Block matching with threshold：$3 \times chnls \times \sigma^2 \times kHard^2$3×chnls×σ2×kHard2
foreach pixel position (i,j) do
Define matrix unfold1(kHard,kHard \times nSx_r \times chnls)
Define matrix unfold1(kHard,kHard \times nSx_r \times chnls)
Define matrix unfold2(kHard,kHard \times nSx_r \times chnls)
Define matrix unfold3(chnls,kHard^2 \times nSx_r)
Define matrix unfold4(nSx_r,kHard^2 \times chnls)
Fill unfold1，unfold1，unfold1，unfold1 from $I_{noisy}$Inoisy​
Compute the SVD $svd_i$svdi​ of $unfold_i$unfoldi​
Define matrix U42
if chnls==1 then
$\llcorner$└U42=$svd_4$svd4​.matrixU()$\otimes$⊗$svd_2$svd2​.matrixU()

if chnls==3 then
$\llcorner$└U42=$svd_4$svd4​.matrixU()$\otimes$⊗$svd_3$svd3​.matrixU()$\otimes$⊗$svd_2$svd2​.matrixU()

Define matrix core=$svd_1.matrixU()^T \times unfold_1 \times U42$svd1​.matrixU()T×unfold1​×U42
Threshold core using $\sqrt{2 \times \log(kHard^2 \times nSx_r \times chnls)}$2×log(kHard2×nSxr​×chnls)​
Store denoised tensor fromCore = $svd_i.matrixU() \times core \times U42^T$svdi​.matrixU()×core×U42T

Perform averaging of all ovelapping pathes to produce $I_{denoised}$Idenoised​