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Discrete cosine transform

The concept

How a JPEG photo fits in a fraction of the space of its raw pixels

The discrete cosine transform (DCT) rewrites a row or block of pixels as a mix of cosine waves, from completely flat to finely striped. Nothing is lost: the original pixels can be rebuilt exactly from the wave weights.

Take two neighbouring pixels with brightness 10 and 10. The DCT says: lots of the flat pattern, none of the striped one. Smooth areas of a photo turn into a few big numbers and many near-zero ones, and near-zero numbers are cheap to store.

What it’s forTurns N pixel values into N weights, one per cosine pattern.

Xₖ = Σₙ xₙ · cos(π/N · (n + ½) · k)

xₙ: brightness of pixel n in the row
N: number of pixels in the row; JPEG uses 8
k: which pattern: 0 is flat, higher k means finer stripes
Xₖ: how much of pattern k the row contains

What it givesA list of pattern weights where smooth images put almost everything in the first few.

Pixels [10, 10], N = 2: X₀ = 10 + 10 = 20
X₁ = 10·cos(π/4) + 10·cos(3π/4) = 7.07 − 7.07 = 0
Pixels [10, 6]: X₀ = 16, X₁ = 4 × 0.707 ≈ 2.83

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Discrete cosine transform

Real-life use case

Squeeze a JPEG too hard and you see 8×8 blocks

JPEG cuts each image into blocks of 8×8 pixels and runs a two-dimensional DCT on each, giving 64 weights per block. Save a photo at very low quality and those blocks become visible as little squares.

The squares appear because each block is simplified on its own, so neighbouring blocks no longer line up at their edges. Common video formats such as H.264 use the same idea with an integer version of the transform.

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Discrete cosine transform

How it helps

Fine stripes your eye barely sees get rounded to zero

The DCT alone saves nothing; it moves the picture's detail into a form that is easy to throw away. In real photos most of each block's energy sits in the low-frequency weights, because neighbouring pixels are usually similar.

JPEG then divides each weight by a step size and rounds, called quantization. Fine-stripe patterns get bigger steps because eyes notice them least, so most round to zero; Huffman coding then packs those zeros tightly.

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Topics in News

How a JPEG photo fits in a fraction of the space of its raw pixels

Squeeze a JPEG too hard and you see 8×8 blocks

Fine stripes your eye barely sees get rounded to zero

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