Math

Fourier in two dimensions: inside JPEG

Decompose images into orthonormal 8×8 cosine blocks, remove or quantize coefficients and measure reconstruction error.

Actual application image

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A grayscale image is split into 8×8 cosine blocks. Keep low-frequency zigzag coefficients and optionally quantize them with a JPEG-style luminance table. Click a source block to inspect its coefficients. This demonstrates the transform stage, not file encoding.

Equations used in the module

Equations
Cuv = ¼ αuαv ΣxΣy (Ixy−128) cos((2x+1)uπ/16) cos((2y+1)vπ/16)
α0=1/√2; αk=1 otherwise
Quantized coefficient = Q · round(C/Q)
PSNR = 10 log₁₀(255²/MSE)

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