How JPEG Shrinks an Image

A row of pixels is a function of position. A cosine series rewrites it as an average plus waves of increasing frequency. Keep the big low-frequency waves, round away the tiny high ones, and the picture barely changes while the file shrinks.

How it works

Fourier cosine series

In one sentence

The even $2L$ -periodic extension of $f$ on $[0, L]$ has the Fourier cosine series $\frac{a_0}{2} + \sum a_n\cos(n\pi x/L)$ with $a_n = \frac2L\int_0^Lf(x)\cos(n\pi x/L)\,dx$ .

General formula

f(x) = \frac{{\htmlData{sym=1}{a_0}}}{2} + \sum_{{\htmlData{sym=3}{n}}=1}^{\infty}{\htmlData{sym=0}{a_n}}\cos\frac{{\htmlData{sym=3}{n}}\pi x}{\htmlData{sym=2}{{L}}}, \qquad {\htmlData{sym=0}{a_n}} = \frac{2}{\htmlData{sym=2}{{L}}}\int_0^{\htmlData{sym=2}{L}} f(x)\cos\frac{{\htmlData{sym=3}{n}}\pi x}{\htmlData{sym=2}{{L}}}\,dx

Each cosine is a brightness wave across the row. The coefficients measure how much of each wave the row contains; the first is just the average.

What the symbols mean

$a_n$
Cosine coefficient
How much of the nth brightness wave the row contains.
$a_0$
Average
Twice the row's mean brightness.
$L$
Row length
1 in the graph.
$n$
Frequency
How many half-waves fit across the row.

A row of pixels is a function of position; the cosine series writes it as its average plus cosines of increasing frequency.
Cosines suit images because their even extension doesn't jump at the edges, so coefficients shrink fast for smooth rows.
JPEG keeps the large low-frequency coefficients and rounds away small high-frequency ones: fewer numbers, nearly the same picture.
Sharp edges need many terms; cutting them off leaves ripples, the ringing you see around text in a low-quality JPEG.

Learn the math

Fourier Cosine SeriesDiff Eq · Lesson 45, Boundary Value Problems and Fourier Series The Fourier Series ToolkitPDE · Lesson 3, Foundations