JPEG Compression. What is JPEG?

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1 JPEG Compression Michael W. Chou Scott Siegrist EEA Spring April, Professor Ingrid Verbauwhede What is JPEG? JPEG is short for the 'Joint Photographic Experts Group'. The JPEG standard is fairly complex because, rather than defining an image file format, it defines a number of related image compression techniques.

2 Advantages of JPEG Compression The goal is to define a universal lossy compression standard for images. Takes advantage of the fact that the eye has a hard time distinguishing between nearly identical colors. Computationally intensive Good at compressing photographs Other compression techniques are better at compressing text and drawings JPEG Characteristics Always Lossy Compression True 4-bit color (6 million colors) Compression ration of - : Good performance for pictures that are smooth with a lot of colors. Bad performance for pictures with sharp edges.

3 Uncompressed TIFF (4 x x 4bpp - 6KB) JPEG (9 KB 5.8% of original image)

4 JPEG ( KB % of original image) Dataflow of JPEG Compression Algorithm JPEG File Compression Sampling Quantization Discrete Cosine Transform 4

5 Sampling JPEG File Compression Sampling Quantization Discrete Cosine Transform Sampling: RGB Color System Three component representation of the color of a pixel Represents the intensities of the red, green, and blue components 4 bit True Color Each component represented with 8 bits of precision The components each contain roughly the same amount of information 5

6 Human Visual System The human eye has a tendency to notice variations of brightness intensity much more than variations of the color in an image The human eye is not as sensitive to highfrequency chrominance (color) components as it is to luminance (intensity) components We can take advantage of this by transforming the color space of RGB to another representation YUV (YCrCb) Color Space An ideal format for JPEG compression The brightness and color information in an image are separated Concentrates the most important info into one component, allowing for greater compression Y component represents the color intensity of the image (equivalent to a black and white television signal) U and V represent the relative redness and blueness of the image 6

7 YUV Transformation A linear transformation from RGB to YUV and from YUV to RGB Y =.R +.587G +.4B U = -.687R -.G +.5B + 8 V =.5R.487G.8B + 8 R = Y +.4V G = Y.444(U 8).744(V 8) B = Y +.7(U 8) 8 = Sample Prescision/ YUV and Scanning Scan a block of compressed data that contains a single pass through the image for one or more of the color components Each color component is completely encoded in a single scan The compressed pixel information is usually stored in a contiguous region in the file 7

8 YUV and Scanning (cont.) The information in the U and V components is not as important as the information in the Y component In doing the transformation we reduce the number of component blocks from 6 to 4 in a lossy scheme 8x8 blocks of RGB (6 component blocks) We transform these to blocks of Y values (one for each) and block each of U and V The U and V are the averages of the two pixel component values from corresponding locations in the block Perform the DCT on first Y block, then second Y block, followed by DCT on the averaged U and V blocks, written to compressed image in that order Discrete Cosine Transform JPEG File Compression Sampling Quantization Discrete Cosine Transform 8

9 DCT on 8x8 blocks 8 pixels 8 pixels We will break the image into nonoverlapping 8x8 blocks. For each block u(m,n), we will take an 8x8 DCT 64 pixels 64 pixels DCT Discrete Cosine Transform 8x8 block of Pixels are represented using cosine functions of different frequencies. A real unitary Transform Unlike the DFT (FFT) which uses complex basis functions. The result is a transform that will give you real values in the transform domain. v = C u C = C(k,n)= k = N π ( n + ) cos N N n N k N n N 9

10 Understanding the DCT The DCT is best understood in terms of a DFT of a sequence of length N derived from u(n) The DFT implies a periodicity that can result in discontinuities that do not reflect the nature of the signal u(n) Discontinuity is created u(n) = [,,, 4] u(n) n n Principle behind DCT. Make the input symmetric by doubling the length of u(n) u' ( n) = u( n) + u(n n) u(n) n

11 . Delay u (n) by ½ unit u (n) = u (n- ½) u(n) u (n) will have even symmetry Below is one possibly DCT definition V ( k) = N n= πk u( n)cos ( n + N ) ½.5 N-½ n Unitary D-DCT Not surprisingly, it turns out that you can get better compression using the DCT if you take into account the horizontal and vertical correlation between pixels simultaneously. Forward DCT V ( k, l) = α( k) α( l) Backward DCT u( m, n) = N N N N α( k) = α( ) = N π (m + ) k π (n + ) l u( m, n)cos cos N m= n= N π (m + ) k π (n + ) l α( k) α( l) V ( k, l)cos cos N k = l = N N

12 DCT / DFT Comparison DFT Fu F = ½ -j - j - - j - j DCT Cu C = D DCT / DFT Comparison U(n) DFT DCT Comment [,,, ] [,,, ] [,,, ] DC transforms to impulse in both. [,,, ] [, -,, -] [.65,.7, -.7, -.65] [.5,.5,.5,.5] [,,, ] [, j,, j] [.5,.65,.5,.7] [,.765,,.848] [,,, ] Impulse transforms to impulse in only DFT In this high frequency sequence, the DCT periodicity is actually not as clean. For signals whose periodic extensions are smooth, the energy localizations with DFT is better than DCT Low Frequency sequence. In general, the N -> N mapping implicit in the DCT gives smoother sequences than the periodic N extension of the DFT. This is why the DCT has better energy compaction for most signals.

13 Example of DCT u(m,n) Low Frequencies Mid Frequencies High Frequencies v(k,l) DCT Other Tranforms used in Image Processing KL Transform Very important theoretically, but not used because no fast algorithm exists and depends on statistics of the image. KL Transform is optimal in producing transform coefficients that are uncorrelated Has best average energy compaction for an ensemble of images Singular Value Decomposition (SVD) Best energy compaction for a given image.

14 Why 8x8 blocks? V(k) can be coded using R bits/coefficient, with Distortion D p = correlation ( p ) R( D) = log ( ) log p D N Increasing N beyond some point does little difference R(D) Typical R approaches asymptote for block sizes of 8x8. Note: As p goes to, the asymptote (/N)log(-p ) gets more negative. ( log D ) p Asymptote gets larger You can do bigger blocks and get better performance. 8x8 Block Size NxN Quantization JPEG File Compression Sampling Quantization Discrete Cosine Transform 4

15 Quantization Quantized Value = Round (coefficient / Quantum Value) v( k, l) v'( k, l) = round q( k, l) Choosing a quantum value as small as would convert over half of the coefficients to zeros. The JPEG standard does not specify the quantization values to be used. This is left up to the application. However it does provide some quantization tables that have been tested empirically and found to generate good results Example of Quantization = zeros v(k,l) v (k,l) Quantization 5

16 Quantization Table Y Component U and V Components Zig-Zag Ordering = DC = zeros The goal is to group all the zeros together, to allow compression

17 Run Length Coding Simple, lossless compression scheme Consecutive pixels with the same value are encoded using a run-length and value pair x x x x x x x x x x ->xa x So in our example on the last page: We will take advantage of the fact that there is a long string of coefficients that are s. We will run-length code these. Compression JPEG File Compression Sampling Quantization Discrete Cosine Transform 7

18 Compression/Source Coding Represent the information produced by a random source (random variable/process) with a different symbol alphabet to reduce the size Entropy description of the amount of uncertainty of a random source. A quantitative measure that describes the number of bits on average required to represent a source Lossless compression schemes used Huffman Coding A variable-length encoding algorithm that is based on the source symbol patterns Optimum for codes that prefix free Algorithm Pair up the two symbols with the lowest probability These two symbols represent a new symbol Repeating pairing with the new symbol in place of the pair Each pairing represents a pair of branches from a node in a tree Assign a or to each branch in the tree Concatenate the each branch symbol to get the coded symbol 8

19 A Simple Example Alphabet = {,,,4,5} Probabilities = {.5,.,.,.,.5} Uncoded = bits per symbol Entropy =.6 bits per symbol Huffman average code length =.5 bits Arithmetic Coding An alternative technique allowed by the JPEG standard Proprietary encoding technique requiring a license Not permitted in baseline JPEG Offers a 5-% better compression Algorithm A point in the [,] interval is the encoded message The segment in the interval chosen symbol by symbol Each symbol has a corresponding segment on the interval After each symbol, the current segment chosen is divided as if it were the original interval with the segments scaled to fit into the current segment 9

20 Huffman Coding in JPEG Used on the reduced coefficients after the DCT and quantization JPEG standard doesn t specify how the codes are created, but tables are provided Two types of DCT coefficients DC first coefficient AC all the other coefficients Different Huffman Tables used Different procedures used to Huffman code that take advantage of the nature of the coefficients AC/DC = DC Coefficients = AC Coefficients

21 Huffman Tables Difference in the nature of the AC and DC coefficients leads to different methods of coding for more optimal compression Different tables for DC and AC coding Also differentiate between Luminance (Y) and Chrominance (U,V) information Baseline however only allows two tables, so the DC code uses same table for Luminance and Chrominance (same for AC) Huffman Coding of the DC Coefficients Use a differential prediction model Idea is that the DC coefficient itself is unpredictable, but we expect to have little change between neighbors (Very high correlation) Increase the probability that the value that we encode will be small by taking the difference between the coefficient and the DC value from the previous block

22 Huffman Coding of the DC Coefficients Range (SSSS) Difference -, -,-,, -7,,-4, 4,,7-5,,-8, 8,,5 -,,-6, 6,, -6,,-,,,6-7,,-64, 64,,7-55,,-8, 8,,55-5,,-56, 56,,5 -,,-5, 5,, -47,,-4, 4,,47 Additional Bits,,,,,,,,,,,,,,,,,,,, Huffman Coding of the DC Coefficients Baseline JPEG only allows for bit difference values for 8 bit input sample SSSS is only a range for absolute values Code is completed when additional bits are appended to the SSSS Huffman code tables for SSSS can be different depending on whether looking at luminance or chrominance Additional bits are always the same whether looking at luminance or chrominance values

23 Huffman Coding of the AC Coefficients The values of the AC coefficients tend towards zero after the quantization step Selection of a symbol is based on the number of zeroes (RRRR) that preceding the current coefficient and the range of the current coefficient The range of the current coefficient is calculated in the same manner as the DC coefficient (the SSSS symbol is selected in the same manner) Huffman Coding of the AC Coefficients The Run Size symbol is selected by cross referencing the RRRR and SSSS values and getting the code from the tables provided in the JPEG standard There are two special cases No non-zero coefficient values in the current array More than 6 zeroes in a sequence AC coefficients are processed from the first AC coefficient to the last (or 64 th ) in zig-zag ordering Processed until the first non-zero coefficient, or 6 zeroes are found

24 JPEG File format (JFIF) JPEG File Compression Sampling Quantization Discrete Cosine Transform JFIF JPEG File Interchange Format No file format in the standard Image orientation is top-down: first encoded image samples are in the upper left hand corner, follows from left to right Defines header fields and sections to provide the necessary information for an image to be decoded Compression parameters include Quantization tables, Huffman tables, etc. 4

25 JPEG Headers/Markers Signature for identifying JPEG files Identified by reading the first bytes of an image First 4 bytes xff xd8, xff xe (SOI and APP markers) Skip next two bytes Next 5 bytes = JFIF\ JPEG Headers/Markers (cont) Compressed Image File Formats by John Miano SOI Start of Image APP Allows for JFIF Extensions (the extensions to JPEG are not used for baseline image formats) and specifies a few parameters i.e. Aspect Ratio: 8x6 4x, 56x56 x 5

26 JFIF File Flow SOI APP DQT Define Quantization Tables SOF Start of Frame DHT Define Huffman Tables SOS Start of Scan (encoded data) EOI References Miano, John. Compression Image File Formats. New York: New York, 9 Lecture Notes from John Villasenor prepared for EEA taught at UCLA Department of Electrical Engineering in Henry Samueli School of Engineering. JPEG File Interchange Format." C-Cube Microsystems. Updated Sept. 9. Retrieved 8 April < JPEG Overview" Retrieved 8 April < 9/section.htm>. 6

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