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SSIM Mathematical Foundation

This document describes the mathematical formulas used in @blazediff/ssim for structural similarity image quality assessment.

SSIM (Structural Similarity Index)

SSIM measures image quality by comparing luminance, contrast, and structure between two images. Higher values indicate better quality (1 = identical, 0 = completely different).

Local Statistics

For each window (typically 11×11 Gaussian):

μₓ = mean(x)     // Mean of window x
μᵧ = mean(y)     // Mean of window y
σₓ² = var(x)     // Variance of window x
σᵧ² = var(y)     // Variance of window y
σₓᵧ = cov(x,y)   // Covariance between windows x and y

SSIM Formula

SSIM(x,y) = l(x,y) × c(x,y) × s(x,y)

Where:

l(x,y) = (2μₓμᵧ + C₁) / (μₓ² + μᵧ² + C₁)           // Luminance
c(x,y) = (2σₓσᵧ + C₂) / (σₓ² + σᵧ² + C₂)           // Contrast
s(x,y) = (σₓᵧ + C₂/2) / (σₓσᵧ + C₂/2)              // Structure

Simplified form:

SSIM(x,y) = ((2μₓμᵧ + C₁)(2σₓᵧ + C₂)) / ((μₓ² + μᵧ² + C₁)(σₓ² + σᵧ² + C₂))

Constants

C₁ = (K₁ × L)²
C₂ = (K₂ × L)²

Where:

  • K₁ = 0.01 (default)
  • K₂ = 0.03 (default)
  • L = 255 (dynamic range for 8-bit images)

Gaussian Window

Default window is 11×11 Gaussian with σ = 1.5:

w = fspecial('gaussian', 11, 1.5)

Mean SSIM

MSSIM = mean(SSIM_map)

Average SSIM across all windows.

MS-SSIM (Multi-Scale SSIM)

MS-SSIM extends SSIM by evaluating quality at multiple scales through iterative downsampling.

Algorithm

  1. Compute contrast and structure at each scale
  2. Compute luminance only at the coarsest scale
  3. Combine using weighted geometric mean

Multi-Scale Formula

MS-SSIM = [l₅(x,y)]^α₅ × ∏(i=1 to 5) [cᵢ(x,y) × sᵢ(x,y)]^βᵢ

Where:

  • Scale 1: Original resolution
  • Scale 2-5: Iteratively downsampled by 2×
  • Default weights: [0.0448, 0.2856, 0.3001, 0.2363, 0.1333]

Downsampling

Between scales, apply 2× downsampling:

img_down = imresize(img, 0.5, 'bicubic')

Typical ranges:

  • 1.00: Perfect match (identical images)
  • 0.95-1.00: Excellent quality
  • 0.85-0.95: Good quality
  • 0.70-0.85: Fair quality
  • < 0.70: Poor quality

References

  • Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. (2004). "Image quality assessment: From error visibility to structural similarity." IEEE Transactions on Image Processing, 13(4), 600-612.
  • Wang, Z., Simoncelli, E. P., & Bovik, A. C. (2003). "Multi-scale structural similarity for image quality assessment." IEEE Asilomar Conference on Signals, Systems and Computers.
  • Reference implementation: https://www.cns.nyu.edu/~lcv/ssim/

Testing

See TESTING.md for instructions on running MATLAB/Octave validation tests.