Statistically Matched Wavelet Based Texture Synthesis in a Compressive Sensing Framework

Algorithm 2

Texture synthesis in a CS framework.

Inputs: Initial Reference Point, Observation Vector, Optimization Parameter

Initialize:   𝑠_𝑎=𝑁×1 vector(guess point), 𝑦=𝑘×1 measurement vector,

𝑇 (tolerance for primal-dual algorithm) = 5, 𝐼 (Maximum primal dual iterations) = 20,

𝑇_𝑔 (Tolerance for conjugate gradients) = 1𝑒−8, 𝐼_𝑔 (Maximum conjugate gradients) = 300

If (Valid Starting Reference, 𝑠_𝑎)

   Minimize 𝑓(𝑥) subject to Θ𝑥=𝑠_𝑎 (Where, 𝑓-convex, rankΘ=𝑀<𝑁)

    (Use Primal Dual Interior Point Methods with Equality Constraint)

   While (Surrogate duality gap <𝑇 OR Iterations >𝐼) do

 (1)  Optimality Condition: ∇𝑓(𝑥^∗)+Θ^𝑇𝑉^∗=0, Θ𝑥^∗=𝑠_𝑎

 (2)  Compute the Newton step and decrement Δ𝑥_𝑛𝑡, 𝜆(𝑥)

 (3)  Solve 𝐾×𝐾 Positive definite system of equations from Newton step

    (3a)  Use Conjugate Gradients Method, MATLAB command, cgsolve

    (3b)  Stop Conjugate Gradient algorithm, If (norm(Θ_𝑥−𝑠_𝑎)/norm(𝑠_𝑎)<𝑇_𝑔)

 (4)  Do line Search. Choose step size 𝐭 by backtracking line search

 (5)  Update. 𝑥=𝑥+𝑡Δ𝑥_𝑛𝑡

 (6)  Update Central and Dual residuals

   end while

end If

Output: Sparse representation Wavelet Coefficients 𝐬̂