Digital Image Processing Image Compression Mohamed N Ahmed

Digital Image Processing Image Compression Mohamed N. Ahmed, Ph. D. Software Research

Digital Image Processing Image Compression • Everyday an enormous amount of information is stored, processed, and transmitted • • • Financial data Reports Inventory Cable TV Online Ordering and tracking Software Research

Digital Image Processing Image Compression • Because much of this information is graphical or pictorial in nature, the storage and communications requirements are immense. • Image compression addresses the problem of reducing the amount of data requirements to represent a digital image. • Image Compression is becoming an enabling technology: HDTV. • Also it plays an important role in Video Conferencing, remote sensing, satellite TV, FAX, document and medical imaging. Software Research

Digital Image Processing Image Compression • We want to remove redundancy from the data • Mathematically 2 D array Of pixels Transformation Software Research Statistically Uncorrelated data

Digital Image Processing Day 4 : Image Compression Outline: 1. Fundamentals Coding Redundancy Interpixel Redundancy Psychovisual Redundancy Fidelity Criteria 2. Error-Free Compression Variable-length Coding LZW Coding Predictive Coding 3. Lossy Compression Transform Coding Wavelet Coding 4. Image Compression Standards Software Research

Digital Image Processing Fundamentals • The term data compression refers to the process of reducing the amount of data required to represent a given quantity of information • Data Information • Various amount of data can be used to represent the same information • Data might contain elements that provide no relevant information : data redundancy • Data redundancy is a central issue in image compression. It is not an abstract concept but mathematically quantifiable entity Software Research Some Images are adopted from R. C. Gonzalez & R. E. Woods

Digital Image Processing Data Redundancy • Let n 1 and n 2 denote the number of information carrying units in two data sets that represent the same information • The relative redundancy RD is define as : where CR, commonly called the compression ratio, is Software Research

Digital Image Processing Data Redundancy • If n 1 = n 2 , CR=1 and RD=0 • If n 1 >> n 2 , CR and RD • If n 1 << n 2 , CR and RD no redundancy high redundancy undesirable • A compression ration of 10 (10: 1) means that the first data set has 10 information carrying units (say, bits) for every 1 unit in the second (compressed) data set. • In Image compression , 3 basic redundancy can be identified » Coding Redundancy » Interpixel Redundancy » Psychovisual Redundancy Software Research

Digital Image Processing Coding Redundancy • Recall from the histogram calculations where p(rk) is the probability of a pixel to have a certain value rk If the number of bits used to represent rk is l(rk), then Software Research

Digital Image Processing Coding Redundancy • Example: Software Research

Digital Image Processing Coding Redundancy Variable-Length Coding Software Research

Digital Image Processing Inter-pixel Redundancy Here the two pictures have Approximately the same Histogram. We must exploit Pixel Dependencies. Each pixel can be estimated From its neighbors. Software Research

Digital Image Processing Run-Length Encoding Example of Inter-pixel Redundancy removal Software Research

Digital Image Processing Psycho-visual Redundancy The human visual system is more sensitive to edges Middle Picture: Uniform quantization from 256 to 16 gray levels CR= 2 Right picture: Improved gray level quantization (IGS) CR= 2 Software Research

Digital Image Processing Fidelity Criteria The error between two functions is given by: So, the total error between the two images is The root-mean-square error averaged over the whole image is Software Research

Digital Image Processing Fidelity Criteria • A closely related objective fidelity criterion is the mean square signal to noise ratio of the compresseddecompressed image Software Research

Digital Image Processing Fidelity Criteria Software Research

Digital Image Processing Compression Model The source encoder is responsible for removing redundancy (coding, inter-pixel, psycho-visual) The channel encoder ensures robustness against channel noise. Software Research

Digital Image Processing Compression Types Compression Error-Free Compression (Loss-less) Software Research Lossy Compression

Digital Image Processing Error-Free Compression • Some applications require no error in compression (medical, business documents, etc. . ) • CR=2 to 10 can be expected. • Make use of coding redundancy and inter-pixel redundancy. • Ex: Huffman codes, LZW, Arithmetic coding, 1 D and 2 D run-length encoding, Loss-less Predictive Coding, and Bit-Plane Coding. Software Research

Digital Image Processing Huffman Coding • The most popular technique for removing coding redundancy is due to Huffman (1952) • Huffman Coding yields the smallest number of code symbols per source symbol • The resulting code is optimal Software Research

Digital Image Processing Huffman Codes Software Research

Digital Image Processing Huffman Codes Software Research

Digital Image Processing Workshop Obtain the Huffman codes for the following sequence: – 555588427772222447777222222444 – What is the average code length with and without compression ? Software Research

Digital Image Processing Fixed Length: LZW Coding • Error Free Compression Technique • Remove Inter-pixel redundancy • Requires no priori knowledge of probability distribution of pixels • Assigns fixed length code words to variable length sequences • Patented Algorithm US 4, 558, 302 • Included in GIF and TIFF and PDF file formats Software Research

Digital Image Processing LZW Coding • Coding Technique – A codebook or a dictionary has to be constructed – For an 8 -bit monochrome image, the first 256 entries are assigned to the gray levels 0, 1, 2, . . , 255. – As the encoder examines image pixels, gray level sequences that are not in the dictionary are assigned to a new entry. – For instance sequence 255 -255 can be assigned to entry 256, the address following the locations reserved for gray levels 0 to 255. Software Research

Digital Image Processing LZW Coding • Example Consider the following 4 x 4 8 bit image 39 39 126 126 Dictionary Location Entry 0 1. 255 256 0 1. 255 - 511 Initial Dictionary Software Research

Digital Image Processing LZW Coding 39 39 126 126 Dictionary Location 126 126 • Is 39 in the dictionary……. . Yes • What about 39 -39…………. No • Then add 39 -39 in entry 256 • And output the last recognized symbol… 39 Entry 0 1. 255 256 0 1. 255 39 -39 - 511 Software Research

Digital Image Processing Workshop • Code the following image using LZW codes 39 39 126 126 * How can we decode the compressed sequence to obtain the original image ? Software Research

Digital Image Processing LZW Coding Software Research

Digital Image Processing Bit-Plane Coding • An effective technique to reduce inter pixel redundancy is to process each bit plane individually • The image is decomposed into a series of binary images. • Each binary image is compressed using one of well known binary compression techniques. Software Research

Digital Image Processing Bit-Plane Decomposition Software Research

Digital Image Processing Bit-Plane Encoding Constant Area Coding One Dimensional Run Length coding 1 b 2 w 1 b 3 w Two Dimensional Run Length coding 12 w Software Research 4 b 1 w

Digital Image Processing Loss-less Predictive Encoding Software Research

Digital Image Processing Loss-less Predictive Encoding Software Research

Digital Image Processing Lossy Compression Quantizer Software Research

Digital Image Processing Lossy Compression Software Research

Digital Image Processing DPCM Software Research

Digital Image Processing DPCM Software Research

Digital Image Processing Transform Coding • A reversible linear transform (such as Fourier Transform) is used to map the image into a set of transform coefficients • These coefficients are then quantized and coded. • The goal of transform coding is to decorrelate pixels and pack as much information into small number of transform coefficients. • Compression is achieved during quantization not during the transform step Software Research

Digital Image Processing Transform Coding Software Research

Digital Image Processing 2 D Transforms • Energy packing – 2 D transforms pack most of the energy into small number of coefficients located at the upper left corner of the 2 D array Energy Packing Software Research

Digital Image Processing 2 D Transforms • Consider an image f(x, y) of size N x N Forward transform g(x, y, u, v) is the forward transformation kernel or basis functions Software Research

Digital Image Processing 2 D Transforms • Inverse transform h(x, y, u, v) is the inverse transformation kernel or basis functions Software Research

Digital Image Processing Discrete Cosine Transform • One of the most frequently used transformations for image compression is the DCT. for u=0 for u=1, 2, …, N-1 Software Research

Digital Image Processing Discrete Cosine Transform Software Research

Digital Image Processing 2 D Transforms Software Research

Digital Image Processing Effect of Window Size Software Research

Digital Image Processing Quantization Quantizer Software Research

Digital Image Processing Quantization • Each transformed coefficient is quantized Software Research

Digital Image Processing Quantization Software Research

Digital Image Processing Bit allocation and Zig Zag Ordering Software Research

Digital Image Processing DCT and Quantization Right Column Software Research

Digital Image Processing Wavelet Coding Software Research

Digital Image Processing Wavelet Transform 1 2 3 4 Put a pixel in each quadrant- No size change Software Research

Digital Image Processing Wavelet Transform a b c d • Now let » » a = (x 1+x 2+x 3+x 4)/4 b =(x 1+x 2 -x 3 -x 4)/4 c =(x 1+x 3 -x 2 -x 4)/4 d =(x 1+x 4 -x 2 -x 3)/4 Software Research

Digital Image Processing Wavelet Transform Software Research

Digital Image Processing Wavelet Transform Software Research

Digital Image Processing Wavelet Transform Software Research

Digital Image Processing Wavelet Coding • High Frequency coefficients tend to be very small --- 0 • They can be quantized very effectively without distorting the results Software Research

Digital Image Processing Wavelet Transform DCT Wavelet Software Research

Digital Image Processing Wavelet Transform Software Research

Digital Image Processing Image Compression Standards • Binary Compression Standards • • CCITT G 3 -> 1 D Run Length Encoding CCITT G 4 -> 2 D Run Length encoding JBIG 1 -> Lossless adaptive binary compression JBIG 2 -> Lossy/Lossless adaptive binary compression Software Research

Digital Image Processing JBIG/JBIG 2 Software Research

Digital Image Processing Image Compression Standards • Continuous Tone Still Image Compression Standards • JPEG 2000 • Mixed Raster Content (MRC) Software Research

Digital Image Processing MRC Software Research

Digital Image Processing Video Compression Software Research