Image Compression Roger Cheng (JPEG slides courtesy of Brian Bailey) Spring 2007 Slideshow and powerpoint viewer: Ubiquitous use of digital images Goal Store image data as efficiently as po

Goal Store image data as efficiently as possible Ideally, want to – – Maximize image quality Minimize storage space and processing resources Can’t have best of both worlds What are some good compromises?

Digital vs. Analog Modern image processing is done in digital domain If we have an analog source image, convert it to digital format before doing any processing

Two main schools of image compression Lossless – – – – Stored image data can reproduce original image exactly Takes more storage space Uses entropy coding only (or none at all) Examples: BMP, TIFF, GIF Lossy – – – – Stored image data can reproduce something that looks “close” to the original image Uses both quantization and entropy coding Usually involves transform into frequency or other domain Examples: JPEG, JPEG2000

BMP (Bitmap) Use 3 bytes per pixel, one each for R, G, and B Can represent up to 224 = 16.7 million colors No entropy coding File size in bytes = 3*length*height, which can be very large Can use fewer than 8 bits per color, but you need to store the color palette Performs well with ZIP, RAR, etc.

GIF (Graphics Interchange Format) Can use up to 256 colors from 24-bit RGB color space – If source image contains more than 256 colors, need to reprocess image to fewer colors Suitable for simpler images such as logos and textual graphics, not so much for photographs Uses LZW lossless data compression

JPEG (Joint Photographic Experts Group) Most dominant image format today Typical file size is about 10% of that of BMP (can vary depending on quality settings) Unlike GIF, JPEG is suitable for photographs, not so much for logos and textual graphics

Preprocess Shift – – values [0, 2P - 1] to [-2P-1, 2P-1 - 1] e.g. if (P=8), shift [0, 255] to [-127, 127] DCT requires range be centered around 0 Segment each component into 8x8 blocks Interleave – components (or not) may be sampled at different rates

Interleaving Non-interleaved: scan from left to right, top to bottom for each color component Interleaved: compute one “unit” from each color component, then repeat – – full color pixels after each step of decoding but components may have different resolution

Color Transformation (optional) Down-sample – – compress without loss of quality (color space) e.g., YUV 4:2:2 or 4:1:1 Example: – – – chrominance components 640 x 480 RGB to YUV 4:1:1 Y is 640x480 U is 160x120 V is 160x120

Interleaving ith color component has dimension (xi, yi) – – maximum dimension value is 216 [X, Y] where X=max(xi) and Y=max(yi) Sampling among components must be integral – Hi and Vi; must be within range [1, 4] – [Hmax, Vmax] where Hmax=max(Hi) and Vmax=max(Vi) xi = X * Hi / Hmax yi = Y * Vi / Vmax

Forward DCT Convert – – from spatial to frequency domain convert intensity function into weighted sum of periodic basis (cosine) functions identify bands of spectral information that can be thrown away without loss of quality Intensity values in each color plane often change slowly

Understanding DCT For example, in R3, we can write (5, 2, 9) as the sum of a set of basis vectors – we know that [(1,0,0), (0,1,0), (0,0,1)] provides one set of basis functions in R3 (5,2,9) = 5*(1,0,0) + 2*(0,1,0) + 9*(0,0,1) DCT is same process in function domain

1D Forward DCT Given a list of n intensity values I(x), where x = 0, …, N-1 Compute the N DCT coefficients: 2 F (u ) n where n 1 C (u ) I ( x ) cos x 0 1 C (u ) 2 1 for (2 x 1) , u 0...n 1 2n u 0, otherwise

1D Inverse DCT Given a list of n DCT coefficients F(u), where u = 0, …, n-1 Compute the n intensity values: 2 I ( x) n where n 1 F (u )C (u ) cos u 0 1 C (u ) 2 1 for ( 2 x 1) , x 0...n 1 2n u 0, otherwise

Extend DCT from 1D to 2D Perform 1D DCT on each row of the block Again for each column of Y 1D coefficients – alternatively, transpose the matrix and perform DCT on the rows X

Equations for 2D DCT Forward DCT: m 1 n 1 2 (2 x 1)u F (u, v) C (u )C (v) I ( x, y ) * cos 2n nm y 0 x 0 (2 y 1)v * cos 2 m Inverse DCT: 2 m 1 n 1 (2 x 1)u (2 y 1)v I ( y, x) F (v, u )C (u )C (v) cos * cos 2n 2m nm v 0 u 0

Quantization Divide each coefficient by integer [1, 255] – – In the decoding process, multiply the quantized coefficients by the inverse of the table – – comes in the form of a table, same size as a block multiply the block of coefficients by the table, round result to nearest integer get back a number close to original error is less than 1/2 of the quantization number Larger quantization numbers cause more loss

Entropy Encoding Compress sequence of quantized DC and AC coefficients from quantization step – further increase compression, without loss Separate – DC from AC components DC components change slowly, thus will be encoded using difference encoding

DC Encoding DC – – represents average intensity of a block encode using difference encoding scheme use 3x3 pattern of blocks Because difference tends to be near zero, can use less bits in the encoding – – categorize difference into difference classes send the index of the difference class, followed by bits representing the difference

AC Encoding Use – – zig-zag ordering of coefficients orders frequency components from low->high produce maximal series of 0s at the end Apply RLE to ordering

Huffman Encoding Sequence of DC difference indices and values along with RLE of AC coefficients Apply Huffman encoding to sequence – Exploits sequence’s statistics by assigning frequently used symbols fewer bits than rare symbols Attach appropriate headers Finally have the JPEG image!

JPEG Decoding Steps Basically reverse steps performed in encoding process – – – – Parse compressed image data and perform Huffman decoding to get RLE symbols Undo RLE to get DCT coefficient matrix Multiply by the quantization matrix Take 2-D inverse DCT of this matrix to get reconstructed image data

Reconstruction Error Resulting image is “close” to original image Usually measure “closeness” with MSE (Mean Squared Error) and PSNR (Peak Signal to Noise Ratio) – Want low MSE and high PSNR

Analysis Near perfect image at 2.76M, so-so image at 88K Sharpness decreases as file size decreases False contours visible at 144K and 88K – Can be fixed by dithering image before quantization Which file size is the best? – – No correct answer to this question Answer depends upon how strict we are about image quality, what purpose image is to be used for, and the resources available

Conclusion Image compression is important Image compression has come a long way Image compression is nearly mature, but there is always room for improvement