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: % 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 pixel’s 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, …, -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 can’t 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