DCT

Concept

Cosine transform

• fourier transform but only with the cosine component
• even signals can be represented with a linear combination of cosine fuctions
• used in:
  • JPEG
  • MPEG

similarly, the DFT of an even signal real, but the DFT of an odd signal is imaginary

Discrete Cosine Transform(DCT)

• like the DFT’s periodic assumption, but requiring symmetric extensions
• different variations of tiling

usually DCT-2 is used for compression

Even symmetry

• symmetry removes the need for phase

DCT-2

• tile by flipping

JPEG compression

• lossy compression
• 2D DCT and filtering out the high frequency components

Steps

1. color space transform, RGB to YCbCr (Y = luminance, Cb = blue diff, Cr = red diff)
2. image is divided into 8x8 chunks
3. apply 2D DCT to each chunk
   • which yields an 8x8 set of coefficients for the 8x8 basis functions
     

note the high frequency allow for the most detail, filtering them out causes a loss in detail

4. the coefficients are divided by the values in the quantization table and rounded
   • usually lowers the high frequencies until they are mostly 0
5. RLE and huffman coding used to further compress the data
   • having many consecutive 0s is good for RLE

Frequency values

• pixels(samples) can be represented by and transformed into independent basis functions(frequencies)