View of An algorithm for wavelet thresholding based image denoising by representing images in hexagonal lattice

Journal of Applied Research and Technology

www.jart.ccadet.unam.mx

Journal of Applied Research and Technology 16 (2018) 103-114

Jeevan K. M.^*, S. Krishnakumar

Received dd mm aaaa; accepted dd mm aaaa Available online dd mm aaaa

𝐼𝑁(𝑥, 𝑦) = 𝐼(𝑥, 𝑦) + 𝑁(𝑥, 𝑦) (1)

𝑝(𝑧) = 1 𝜎√2𝜋𝑒⁻

(𝑧−𝜇)²

2𝜎² (2)

μ σ

ψ

ψ ω

∫ 𝜓(𝑥)𝑑𝑥 = 0 (3)

𝛼

−𝛼

∫ |𝜓(𝑥)|²𝑑𝑥 < 𝛼 (4)

𝛼

−𝛼

ψ ψ

𝑇_{𝑣𝑖𝑠𝑢}= 𝜎_𝑛√2𝑙𝑜𝑔𝐿 (7) 𝜎_𝑛²

𝑇_{𝑠𝑢𝑟𝑒}= min(𝑡, 𝜎√2𝑙𝑜𝑔𝐿) (8)

σ

𝑇_𝐵= (𝜎𝑛2

𝜎_𝐹) (9) 𝜎_𝑛²

𝜎_𝐹

𝜎_𝑛=𝑀𝑒𝑑𝑖𝑎𝑛⟦{𝑌𝑖𝑗}⟧

0.6745 , 𝑌_𝑖𝑗 ∈ 𝑆𝑢𝑏𝑏𝑎𝑛𝑑 (10) 𝑥 = 𝑇ℎ(𝑦, 𝑇) = {𝑦, 𝑖𝑓 |𝑦| ≥ 𝑇

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5)

𝑥 = 𝑇_𝑠(𝑦, 𝑇) = {

𝑦 − 𝑇, 𝑖𝑓 𝑦 ≥ 𝑇 𝑦 + 𝑇, 𝑖𝑓 𝑦 ≤ −𝑇

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(6)

𝜎_𝑛²

𝜎𝐹

𝜎_𝑛=𝑀𝑒𝑑𝑖𝑎𝑛⟦{𝑌𝑖𝑗}⟧

0.6745 , 𝑌_𝑖𝑗 ∈ 𝑆𝑢𝑏𝑏𝑎𝑛𝑑 (10)

𝛽𝑖𝑗 = (1 −𝑇_𝑢²

𝑆_𝑖𝑗²) (11)

𝑆_𝑖𝑗²

𝑆_𝑖𝑗² 𝑇_𝑢²

𝐶_𝑖𝑗 = 𝐶_𝑖𝑗𝛽_𝑖𝑗 (12)

𝑔_{𝜆𝜃𝜎ᵧᵩ}(𝑥, 𝑦) = exp (−𝑥^′2+ ᵧ²𝑦^′2

2𝜎² ) cos (2𝜋𝑥^′

𝜆 + ᵩ) (13) 𝑔_{𝜆𝜃𝜎ᵧᵩ}(𝑥, 𝑦) = exp (−𝑥^′2+ ᵧ²𝑦^′2

2𝜎² ) sin (2𝜋𝑥^′

𝜆+ ᵩ) (14)

x^′= xcosθ + ysinθ xy^′= −xsinθ + ycosθ

σ γ

λ ɵ φ

σ

𝑇_{𝑠𝑢𝑟𝑒}= min(𝑡, 𝜎√2𝑙𝑜𝑔𝐿)

σ

𝜕_𝑖,𝑗 = (1 −𝑇𝑠𝑢𝑟𝑒2

𝑆_𝑖𝑗² ) (15) 𝑆_𝑖,𝑗²

𝑇_{𝑠𝑢𝑟𝑒}²

𝜕_𝑖,𝑗 𝜕_𝑖,𝑗

σ γ

λ ɵ φ

σ γ

γ λ

λ

σ λ

𝑆_𝑖,𝑗² = 𝐶_{𝑖−1,𝑗−1}² + 𝐶_{𝑖−1,𝑗}² + 𝐶_{𝑖−1,𝑗+1}² + 𝐶_{𝑖,𝑗−1}² + 𝐶_𝑖,𝑗² + 𝐶𝑖.𝑗+12 + 𝐶𝑖+1,𝑗−12 + 𝐶𝑖+1,𝑗2 + 𝐶𝑖+1,𝑗+12

𝑆𝑖,𝑗 2 𝑇𝑠𝑢𝑟𝑒2

𝐶_𝑖𝑗 = 𝐶_𝑖𝑗(1 −^{𝑇𝑆𝑢𝑟𝑒2}

𝑆_𝑖𝑗² )

𝑆_𝑖,𝑗² 𝑇_𝑢²

𝑆_𝑖,𝑗² 𝑇𝑠𝑢𝑟𝑒2

𝑀𝑆𝐸 = 1

𝑁∑ ∑(𝑄_𝑖𝑗− 𝑃_𝑖𝑗)² (16)

𝑗 𝑖

𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔₁₀(²⁵⁵²

𝑀𝑆𝐸) (17)

σ

σ σ

σ.

σ



σ

σ

σ

σ

σ