Statistical Texture Modeling for Medical Volume Using Linear Tensor Coding

Algorithm 2

Iteration algorithm of LTC.

IN: a series of 𝑁th order tensors, 𝒜_𝑖∈ℝ^{𝐼1×𝐼2×⋯×𝐼𝑁},  𝑖=1,2,…,𝑀. Define 𝑁 Matrices

𝐔^(𝑛)_opt∈ℝ^{𝐼𝑛×𝐽𝑛} (𝐽_𝑛=1,  𝑛=1,2,…,𝑁) with orthogonal column vectors.

OUT: 𝑆 Rank-1 basis tensor ℬ_𝑠,  𝑠≤𝑆𝑆 depends on convergence.

Iterate for 𝑠 until convergence

 (1) Initial values: 𝒜_𝑖=𝒜_𝑖−𝒜^Rec𝑠_𝑖, define 𝒜^Rec𝑠_𝑖=0.

 (2) (a) Initial values: 𝑘=0 and 𝐔₀^(𝑛) whose columns are determined as the first 𝐽_𝑛 leading

      eigenvectors of the matrices ∑^𝑀_𝑖=1(𝐀_𝑖(𝑛)⋅𝐀^𝑇_𝑖(𝑛)).

   (b) Iterate for 𝑘 until convergence

    (i) Maximize 𝑆^′=∑^𝑀_𝑖=1𝒞_𝑖×₁𝐔^𝑇(1)2,  𝒞_𝑖=𝒜_𝑖×₂𝐔_𝑘^𝑇(2)×⋯×_𝑁𝐔_𝑘^𝑇(𝑁)

      Solution: 𝐔⁽¹⁾ whose columns are determined as the first 𝐽₁ leading eigenvectors

      of ∑^𝑀_𝑖=1(𝐂_𝑖(1)⋅𝐂^𝑇_𝑖(1))

      Set 𝐔⁽¹⁾_𝑘+1=𝐔⁽¹⁾.

   (ii) Maximize 𝑆^′=∑^𝑀_𝑖=1𝒞_𝑖×₂𝐔^𝑇(2)2,  𝒞_𝑖=𝒜_𝑖×₁𝐔^𝑇(1)_𝑘+1×₃𝐔_𝑘^𝑇(3)×⋯×_𝑁𝐔_𝑘^𝑇(𝑁)

       Solution: 𝐔⁽²⁾ whose columns are determined as the first 𝐽₂ leading eigenvectors

       of ∑^𝑀_𝑖=1(𝐂_𝑖(2)⋅𝐂^𝑇_𝑖(2))

       Set 𝐔⁽²⁾_𝑘+1=𝐔⁽²⁾.

       …

   (iii) Maximize 𝑆^′=∑^𝑀_𝑖=1𝒞_𝑖×_𝑛𝐔^𝑇(𝑛)2,

       𝒞_𝑖=𝒜_𝑖×₁𝐔^𝑇(1)_𝑘+1×⋯×_𝑛−1𝐔^{𝑇(𝑛−1)}_𝑘+1×_𝑛+1𝐔_𝑘^𝑇(𝑛+1)×⋯×_𝑁𝐔_𝑘^𝑇(𝑁)

       Solution: 𝐔^(𝑛) whose columns are determined as the first 𝐽_𝑛 leading eigenvectors

       of ∑^𝑀_𝑖=1(𝐂_𝑖(𝑛)⋅𝐂^𝑇_𝑖(𝑛))

       Set 𝐔^(𝑛)_𝑘+1=𝐔^(𝑛).

       …

   (iv) Maximize 𝑆^′=∑^𝑀_𝑖=1𝒞_𝑖×_𝑁𝐔^𝑇(𝑁)2,  𝒞_𝑖=𝒜_𝑖×₁𝐔^𝑇(1)_𝑘+1×⋯×_𝑁−1𝐔^{𝑇(𝑁−1)}_𝑘+1

       Solution: 𝐔^(𝑁) whose columns are determined as the first 𝐽_𝑁 leading eigenvectors

       of ∑^𝑀_𝑖=1(𝐂_𝑖(𝑁)⋅𝐂^𝑇_𝑖(𝑁))

       Set 𝐔^(𝑁)_𝑘+1=𝐔^(𝑁).

   𝑘=𝑘+1

  (c) Set 𝐔⁽¹⁾_opt=𝐔_𝑘⁽¹⁾,  𝐔⁽²⁾_opt=𝐔_𝑘⁽²⁾,…,𝐔^(𝑁)_opt=𝐔_𝑘^(𝑁).

 (3) Size of 𝐔^(𝑖)_opt is 𝐼_𝑖×1, each basis ℬ_𝑠=𝐔⁽¹⁾_opt⊗𝐔⁽²⁾_opt⊗⋯⊗𝐔^(𝑁)_opt.

 (4) For each data, coefficient on this basis 𝑐_𝑖,𝑠=𝒜_𝑖×₁𝐔_𝑘^𝑇(1)×₂𝐔_𝑘^𝑇(2)×⋯×_𝑁𝐔_𝑘^𝑇(𝑁).

 (5) For each data 𝒜^Rec𝑠_𝑖=𝑐_𝑖,𝑠⋅ℬ_𝑠.