8.12.3. MINFFT-03 — 2D DCT and JPEG Compression

JPEG and MPEG intra frames split an image into 8x8 blocks and code each block with a 2D DCT, quantization, and the inverse. This tutorial walks one 8x8 block through that pipeline; for a full image-in / image-out demo with a PSNR sweep, see examples/minfft/dct_jpeg.das.

8.12.3.1. 2D plans

make_dct_plan_2d(rows, cols) builds a plan for rows x cols blocks. The data is a row-major array of rows*cols floats. idct(dct(x)) scales by 2N per axis, so an 8x8 block round-trips at 256 = (2*8)*(2*8):

var plan = make_dct_plan_2d(8, 8)
dct(blk, coeff, plan)        // forward
idct(coeff, back, plan)      // inverse; back[i] / 256.0f recovers blk[i]

8.12.3.2. Orthonormal scaling

The standard JPEG quantization table is designed for the orthonormal DCT. minfft’s DCT-II is unnormalized, so each coefficient is rescaled by a per-axis factor (s[0] = 1/(4*sqrt2), else 1/4) before quantizing, and unscaled afterwards. The scale cancels on a clean round-trip; it only changes how the quant table weights each frequency.

8.12.3.3. The pipeline

Forward DCT, quantize against the JPEG luminance table, dequantize, inverse DCT:

let level = round(ortho[i] / JPEG_LUMA[i])   // quantize (nearest integer)
deq[i] = (level * JPEG_LUMA[i]) / (s[u] * s[v])      // dequantize, back to minfft scale

Most coefficients quantize to zero, yet the block reconstructs closely:

-> 4 of 64 coefficients survive quantization
max per-pixel error: 3.1

Those zeros are what an entropy (Huffman) coder stores cheaply — the only stage this tutorial omits from baseline JPEG.