Trevor McCasland Arch Kelley. Goal: reduce the size of stored files and data while retaining all necessary perceptual information Used to create an.

Trevor McCasland Arch Kelley

Goal: reduce the size of stored files and data while retaining all necessary perceptual information Used to create an encoded copy of the original data with a (much) smaller size Compression Ratio = Uncompressed Size/Compressed Size Typical Compression Ratios GIF JPEG(low) JPEG(mid) JPEG(high) PNG 4:1- 10:1- 30:1- 60:1- 10-30% 10:1 20:1 50:1 100:1 smaller than GIF

The following JPEGs are compressed with different ratios 1:1(low) 1:10(low) 1:30(mid) 1:30 with 5Xzoom

Lossy: Original data->Lossy Compressor ->Compressed Data Reverse: Compressed Data->Decompressor ->Altered Data (not the same as original!) Lossless: Original data->Lossy Compressor ->Compressed Data Reverse: Compressed Data->Decompressor ->Original Data (exactly the same!) Lossy is best used for data that we can afford to irreversibly alter (images, audio, video)

Common techniques used in lossy compression methods include: Color spacing/chroma downsampling (images) Quantization Discrete cosine transform (DCT) Zigzag ordering and run-length encoding Entropy coding (Huffman coding)

1) Image contents change slowly across the image, i.e., it is unusual for intensity values to vary widely several times in a small area, for example, within an 8x8 image block 2) Humans are much less likely to notice the loss of very high spatial frequency components than the loss of lower frequency components High frequencies can be thrown out without noticeable change to the image

3) Visual accuracy in distinguishing closely spaced lines is much greater for black and white than for color Taken together, these three observations can be used to compress images in a way that reduces loss of visual quality while significantly downsizing the filesize

A way of defining the boundaries in which an image can be represented using color Examples: RGB, YCbCr, YPbPr, YUV RGB is most basic color space

A pixels color is determined by its RGB (red blue green) value Eg. R=30, G=100, B=50 Image formats using lossy compression often convert this data into a format that separates luminance (brightness) and chrominance (hue) Eg. Y = (R + G + B) / 3, Cb = B - Y Cr = R - Y Data of original image (top) is Separated into luminance data (left) and chrominance data (right) *Operation takes O(1) time for each 8x8 block and is done on n blocks => Running time O(n)

Used to reduce the possible values that can be used to represent the chrominance of a pixel Specific color spacing allows for chroma downsampling Throw out portions of the chrominance (color) data in a group of pixels to reduce the total space used Use the chroma from one part of the group to display the other part of the group Source of data loss (lossy method)

Different patterns exist for disposing of chrominance (UV values in figures) for portions or pixel groups Most common is 4:2:2 4:4:4 is pointless because no data is being disposed (no downsampling occurs) 4:2:2 sampling 4:1:1 sampling 4:2:1 sampling 4:4:4 sampling

DCT main purpose is to remove redundancy of neighboring pixels to provide compression The transform maps the correlated data to transformed uncorrelated coefficients High compaction of information

1D DCT-II x[n] = original value X[k]= transformed value N= number of columns k= transform coefficient 1D-DCT-II 1D-IDCT-II

2D DCT XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX XXXX 1D-DCT Create an intermediate Sequence by computing 1D-DCT on rows Compute 1D-DCT on the columns

The DCT coefficient matrix is U The input coefficient matrix is A Where

High information compaction at (0,0)

Each basis function is multiplied by its coefficient and then added to the transformed image

complexity O(n ), running it 2n times to build a 2D DCT with complexity O(n ). We can do better again by replacing the O(n ) DCT algorithm with one factored similarly to a Fast Fourier Transform which would have O(nlogn)complexity. O(2n*nlogn) = O(n logn)

Reduce the range of values used to represent image/audio data Similar to chroma downsampling, but applied to a full array of values Achieved by multiplying a constant quantization matrix with the DCT coefficient matrix Quantization matrix can be user-defined Can adjust quantization level (throw out more or less data) by altering matrix

X Q =round(X n,m /Q n,m ) X=Q= XQ=XQ= Ex: Time complexity: O(n 2 ) where n=#columns and rows (n 2 O(1) operations)

Main source of data loss in lossy compression algorithms Only works on data represented using frequencies Encoder uses different quantization levels for different frequencies based on human vision preferences Usually results in a much smaller filesize (typical JPEG compression ratio is 30:1)

Reorder quantized matrix to put non-zero elements in a sortable sequence -26 -3 0 -3 -3 -6 2 4 1 -4 1 1 5 1 2 -1 1 -1 2 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [-26, -3 0, -3, , -1, 0, 0] --Do not store zeroes after final element in zigzag row with a non-zero element

Simple method of storing similar data using a single value and a run length Ex: WWWWWWWWWWWWBWWWWWWWWWWWWBBBWWWWWW WWWWWWWWWWWWWWWWWWBWWWWWWWWWWWWWW becomes 12W1B12W3B24W1B14W Time complexity: O(n) where n=# of characters in string

RLE string is entropy coded to add an extra layer of compression/further reduce filesize Entropy coding is a lossless method Common algorithm used is Huffman Coding Makes trees that look like: *Not important to understand lossy compression

JPEG Lossy image filetype that follows the process exactly MPEG-2 Uses chroma downsampling and different quantization values to adjust level of compression Streaming video lets users adjust quality MP3 Quantization removes frequencies that humans cant hear by rounding them to zero Many, many more

Questions? -Where does the loss of data actually occur? -Why do highly compressed images look blocky? -What flaws appear in highly compressed audio? -How long does it take to learn all of this? O(n n ) time

Trevor McCasland Arch Kelley. Goal: reduce the size of stored files and data while retaining all necessary perceptual information Used to create an.

Documents

luminance data

chrominance color data

correlated data

altered data

image slide

compressed datadecompressor

sampling slide

chrominance data right