Chapter 3: Multimedia Systems  Communication Aspects and Services Chapter 4: Multimedia Systems Storage Aspects Chapter 5: Multimedia Usage

 Lydia Washington
 11 months ago
 Views:
Transcription
1 Chapter : Basics Audio Technology Images and Graphics Video and Animation.: Images and Graphics Digital image representation Image formats and color models JPEG, JPEG Image synthesis and graphics systems Image analysis Chapter 3: Multimedia Systems  Communication Aspects and Services Chapter 4: Multimedia Systems Storage Aspects Chapter 5: Multimedia Usage Page
2 Digital Image Representation A digital image is a spatial representation of an object (D, 3D scene or another image  real or virtual) Definition of digital image : Let I, J, K Z be a finite interval. Let G N with G < be the grey scale level / color depth (intensity value of a picture element = a pixel) of the image. () A Dimage is a function f: I J G () A 3Dimage is a function f: I J K G (3) If G = {,}, the function is a binary (or bit) image, otherwise it is a pixel image The Resolution depends on the size of I and J (and K) and describes the number of pixels per row resp. column. Example To display a 55line television picture (NTSC) without noticeable degradation with a Video Graphics Array (VGA) video controller, 64x48 pixels and 56 discrete grey levels give an array of bit numbers and a total of bit. Page
3 Image Representation An Image Capturing Format is specified by: spatial resolution (pixel x pixel) and color encoding (bits per pixel) Example: captured image of a DVD video with 4:3 picture size: spatial resolution: 768 x 576 pixel color encoding: bit (binary image), 8bit (color or grayscale), 4bit (colorrgb) An Image Storing Format is a dimensional array of values representing the image in a bitmap or pixmap, respectively. Also called raster graphics. The data of the fields of a bitmap is a binary digit, data in a pixmap may be a collection of: 3 numbers representing the intensities of red, green, and blue components of the color 3 numbers representing indices to tables of red, green and blue intensities Single numbers as index to a table of color triples Single numbers as index to any other data structures that represents a color / color system Further properties can be assigned with the whole image: width, height, depth, version, etc. Page 3
4 Color Models Lehrstuhl für Informatik 4 Why storing values for red, green, blue? Color perception by the human brain is possible through the additive composition of red, green and blue light (RGB system). The relative intensities of RGB values are transmitted to the monitor where they are reproduced at each point in time. On a computer monitor, each pixel is given as an overlay of those three image tones with different intensities by this, any color can be reproduced. But: another possible color model: CYMK When printing an image, other color components are used cyan, yellow, magenta, kontrast which in all can also reproduce all colors. Thus, many image processing software and also some image storing formats also support this model. Page 4
5 Color Models Lehrstuhl für Informatik 4 Another possibility is to use a different representation of color information by means of the YUV system where Y is the brightness (or luminance) information U and V are color difference signals (chrominance) Y, U and V are functions of R, G and B Why? As the human eye is more sensitive to brightness than to chrominance, separate brightness information from the color information and code the more important luminance with more bit than the chrominance this can save bits in the representation format. Page 5
6 Color Models Lehrstuhl für Informatik 4 Usual scheme: Y =.3 R +.59 G +. B (the color sensitivity of the human eye is considered) U = c (BY); V = c (RY) c, c = constants reflecting perception aspects of the human eye and the human brain! Possible Coding: YUV signal Y =.3 R +.59 G +. B U = (BY).493 = .48 R .9 G B V = (RY).877 =.64 R .57 G .96 B This is a system of 3 equations for determining Y, U, V from R, G, B or for recalculating R, G, B from Y, U, V The resolution of Y is more important than the resolution of U and V Spend more bits for Y than for U and V (Y : U : V = 4 : : ) The weighting factors in the calculation of the Y signal compensate the color perception misbalance of the human eye Page 6
7 Image Formats Lehrstuhl für Informatik 4 Lots of different image formats are in use today, e.g. GIF (Graphics Interchange Format) Compressed with some basic lossless compression techniques to 5% of original picture without loss. Supports 4bit colors. BMP (Bitmap) Devideindependent representation of an image: uses RGB color model, without compression. Color depth up to 4bit, additional option of specifying a color table to use. TIFF (Tagged Image File Format) Supports grey levels, RGB, and CYMK color model. Also supports lots of different compression methods. Additionally contains a descriptive part with properties a display should provide to show the image. PostScript Images are described without reference to special properties as e.g. resolution. Nice feature for printers, but hard to include into documents where you have to know the image size... JPEG (Joint Photographics Expert Group) Lots of possible compressions, mostly with loss! Page 7
8 Why Compression? Highresolution image: e.g pixel, 4 bit color depth = bit Image formats like GIF: Lossless compression (entropy encoding) for reducing data amount while keeping image quality JPEG: Lossy compression remove some image details to achieve a higher compression rate by suppressing higher frequencies Combined with lossless techniques TradeOff between file size and quality JPEG is a joint standard of ISO and ITUT In June 987, an adaptive transformation coding technique based on DCT was adopted for JPEG In 99, JPEG became a ISO international standard Page 8
9 JPEG Implementation Independent of image size Applicable to any image and pixel aspect ratio Color representation JPEG applies to color and greyscaled still images Image content Of any complexity, with any statistical characteristics Properties of JPEG Stateoftheart regarding compression factor and image quality Run on as many available standard processors as possible Compression mechanisms are available as softwareonly packages or together with specific hardware support  use of specialized hardware should speed up image decompression Encoded data stream has a fixed interchange format Fast coding is also used for video sequences: Motion JPEG Page 9
10 How could We compress? Entropy encoding Data stream is considered to be a simple digital sequence without semantics Lossless coding, decompression process regenerates the data completely Used regardless of the media s specific characteristics Examples: Runlength encoding, Huffman encoding, Arithmetic encoding Source encoding Semantics of the data are taken into account Lossy coding (encoded data are not identical with original data) Degree of compression depends on the data contents Example: Discrete Cosine Transformation (DCT) as transformation technique of the spatial domain into the twodimensional frequency domain Hybrid encoding Used by most multimedia systems Combination of entropy and source encoding Examples: JPEG, MPEG, H.6 Page
11 Compression Steps in JPEG Uncompressed Image Compressed Image Image Preparation Pixel Block, MCU Image Processing DCT Predictor Quantization (approximation of real numbers by rational numbers) Entropy Encoding Runlength Huffman Arithmetic MCU: Minimum Coded Unit DCT: Discrete Cosine Transform Page
12 Compression Steps in JPEG Image Preparation Analogtodigital conversion Image division into blocks of N N pixels Suitable structuring and ordering of image information Image Processing  Source Encoding Transformation from time to frequency domain using DCT In principle no compression itself but computation of new coefficients as input for compression process Quantization Mapping of real numbers into rational numbers (approximation) A certain loss of precision will in general be unavoidable Entropy Encoding Lossless compression of a sequential digital data stream Page
13 The Principle Lehrstuhl für Informatik 4 Original get rid of invisible details Transformation Quantization Encode JPEG Picture The opposite Quantization Table Huffman, Run Length Encoding JPEG Decoder Dequantization Retransformation the details cannot be reconstructed Original Without Quantization : Encoding gain would be very poor (or nonexisting) Transformation and Retransformation must be inverse to each other Task of transformation: produce a picture representation which may be encoded with a high gain of reduction Page 3
14 Variants of Image Compression JPEG is not a single format, but it can be chosen from a number of modes: Lossy sequential DCTbased mode (baseline process) Must be supported by every JPEG implementation Block, MCU, FDCT, Runlength, Huffman Expanded lossy DCTbased mode Enhancement to the baseline process by adding progressive encoding Lossless mode Low compression ratio perfect reconstruction of original image No DCT, but differential encoding by prediction Hierarchical mode Accommodates images of different resolutions Selects its algorithms from the three other modes Page 4
15 First Step: Image Preparation General image model Independence from image parameters like size and pixel ratio Description of most of the wellknown picture representations Source picture consists of to 55 components (planes) C i Components may be assigned to RGB or YUV values For example, C may be assigned to red color information Each component C i can have a different number of superpixels X i, Y i (A superpixel is a rectangle of pixels which all have the same value) Y i C C C 3 C N superpixel X i C i Page 5
16 Picture Preparation  Components Resolution of the components may be different: A A X B X B D X 3 D Y Y Y 3 3 A N B M D M X = X = X Y = Y = Y 3 A greyscale image consists (in most cases) of a single component RGB color representation has three components with equal resolution YUV color image processing uses Y = 4 Y = 4 Y 3 and X = 4 X = 4 X 3 Page 6
17 Image Preparation  Dimensions Dimensions of a compressed image are defined by X (maximum of all X i ), Y (maximum of all Y i ), H i and V i (relative horizontal and vertical sampling ratios for each component i) X i Y i with H = and V = i min X j j i miny j j H i and V i must be integers in the range of to 4. This restriction is needed for the interleaving of components Example: Y = 4 pixels, X = 6 pixels Y C C C 3 X = 6, Y = 4 H = V = X X = 6, Y = H = V = X 3 = 3, Y 3 = H 3 = V 3 = Page 7
18 Image Preparation Data Ordering An image is divided into several components which can be processed one by one. But: how to prepare a component for processing? Observation for most parts of an image: not so much difference between the values in a rectangle of N N pixels For further processing: divide each component of an image into blocks of N N pixels Thus, the image is divided into data units (blocks): Lossless mode uses one pixel as one data unit Lossy mode uses blocks of 8 8 pixels (with 8 or bits per pixel) Page 8
19 Image Preparation  Data Ordering Noninterleaved data ordering: The easiest but not the most convenient sequence of data processing Data units are processed component by component For one component, the processing order is lefttoright and toptobottom With the noninterleaved technique, a RGBencoded image is processed by: First the red component only Then the blue component, followed by the green component This is (for speed reasons) less suitable than data unit interleaving Page 9
20 Interleaved Data Ordering Often more suitable: interleave data units Interleaving means: don t process all blocks component by component, but mix data units from all components Interleaved data units of different components: Combination to Minimum Coded Units (MCUs) If all components have the same resolution MCU consists of one data unit for each component If components have different resolutions. For each component, regions of data units are determined; data units in one region are ordered lefttoright and topto bottom. Each component consists of the same number of regions 3. MCU consists of one region in each component Up to 4 components can be encoded in interleaved mode (according to JPEG) Each MCU consists of at most ten data units Page
21 Image Preparation  MCUs MCU example: four components C, C, C 3, C a a a a b b c c d C : H =, V = C : H =, V = C 3 : H 3 =, V 3 = C 4 : H 4 =, V 4 = MCUs: 9 data units per MCU MCU = a a a a b b c c d MCU = a a 3 a a 3 b b 3 c c d MCU 3 = a 4 a 5 a 4 a 5 b 4 b 5 c c d MCU 4 = a a a 3 a 3 b b c c 3 d where a ij : data units of C b ij : data units of C c ij : data units of C 3 d ij : data units of C 4 H V i i xi = min x y i = min y j j Page
22 Compression Steps in JPEG Uncompressed Image Compressed Image Image Preparation Pixel Block, MCU Image Processing DCT Quantization (approximation of real numbers by rational numbers) Entropy Encoding Runlength Huffman Arithmetic MCU: Minimum Coded Unit DCT: Discrete Cosine Transform Result of image preparation: sequence of 8 8 blocks, the order is defined by MCUs The samples are encoded with 8 bit/pixel Next step: image processing by source encoding Page
23 Source Encoding Transformation Encoding by transformation: Data are transformed into another mathematical domain, which is more suitable for compression. The inverse transformation must exist and must be easy to calculate Most widely known example: Fourier transformation uv m n π i i m ux π n vy x = y = xy F f e e = The parameters m and n indicate the granularity Most effective transformation for image compression: Discrete Cosine Transformation (DCT) m n πu(x+ ) πv(y+ ) uv δ nm x= y= xy m n F f cos cos = Fast Fourier Transformation (FFT) Page 3
24 Discrete Cosine Transformation Let f xy be a pixel (x,y) in the original picture. ( x N ; y N ) N N (x + ) u π (y + ) vπ F : = γ c c f cos cos, u,v {,...,N }, N N uv N u v xy x= y= c u u = = u > f xy space domain (i.e. geometric ) F uv frequency domain (indicates how fast the information moves inside the rectangle) F is the lowest frequency in both directions, i.e. a measure of the average pixel value F uv with small total frequency (i.e u+v small) are (in general) larger than F uv with large u+v Page 4
25 Retransformation: Inverse Cosine Transformation N N (x + ) u π (y + ) vπ fxy = δ N cu cv Fuv cos cos u = v = N N Simplest example (just for demonstration): Let f xy = f = constant N F = γ f cos( ) = γ f all other F = N N uv x y f = f = δ c c F cos(...) cos(...) N u v uv u v = δ c c F N N γ N f = δ N! = if δ = γ then γ = N N N N Page 5
26 Example Lehrstuhl für Informatik 4 N=8 (Standard): N=: Fuv = cu c v 4... (x + ) u π (y + ) vπ Fuv = cu cv fxy cos cos x = y = 4 4 uπ vπ uπ 3πv 3π u πv 3π u 3πv = cu cv f cos cos + f cos + f cos + f cos F = [ f + f + f + f ] i.e. f if fxy f Transformed values can be much smaller than original values: π 3π π 3π F = f cos + f cos + f cos + f cos = f f + f f positive + negative terms, i.e. if f xy f F Page 6
27 Baseline Process  Image Processing How can DCT be useful for JPEG?  F uv for larger values of u and v are often very small! First step of image processing: Samples are encoded with 8 bits/pixel; each pixel is an integer in the range [,55] Pixel values are shifted to the range [8, 7] (complement representation) Data units of 8 x 8 pixel values are defined by f xy [8, 7], where x, y are in the range [, 7] Each value is transformed using the Forward DCT (FDCT): 7 7 (x+ )u π (y+ )vπ uv 4 u v xy 6 6 x= y= F = c c f cos cos for u / v = where c u/v = and u,v [,7 ] otherwise Cosine expressions are independent of f xy fast calculation is possible Result: From 64 coefficients f xy we get 64 coefficients F uv in the frequency domain Page 7
28 Meaning of Coefficients Low High Transformation to frequencies Low Low 8 8 block F F + F F F 3 High Low High High... Page 8
29 Baseline Process  Image Processing Coefficient F : DCcoefficient Corresponds to the lowest frequency in both dimensions Determines the fundamental color of the data unit of 64 pixels Normally the values for F are very similar in neighbored blocks Other coefficients (F uv for u+v > ): ACcoefficients Nonzero frequency in one or both dimensions Reconstruction of the image: Inverse DCT (IDCT) If FDCT and IDCT could be calculated with full precision DCT would be lossless In practice: precision is restricted (real numbers!), thus DCT is lossy different implementations of JPEG decoder may produce different images Reason for the transformation: Experience shows that many ACcoefficients have a value of almost zero, i.e. they are zero after quantization entropy encoding may lead to significant data reduction. Page 9
30 Compression Steps in JPEG Uncompressed Image Compressed Image Image Preparation Pixel Block, MCU Image Processing Predictor DCT Quantization (approximation of real numbers by rational numbers) Entropy Encoding Runlength Huffman Arithmetic Result of image processing: 8 8 blocks of DC/AC coefficients MCU: Minimum Coded Unit Till now, no compression is done this task is enabled by DCT: Discrete Cosine Transformquantization Page 3
31 Quantization Lehrstuhl für Informatik 4 Observation: F uv N smaller values N How to enforce that even more values are zero? Answer: by Quantization. Divide F uv by Quantum uv = Q uv and take the nearest integer as the result [ ] F = F / Q Q uv uv uv Dequantization: F Q = F (only an approximation of F uv ) Q * uv uv uv most values are zero Q Example: N=8; quantization step=, Q uv = (u+v)+3 uv = Page 3
32 Baseline Process  Quantization Quantization process: Divide DCTcoefficient value F uv by an integer number Q uv and round the result to the nearest integer Quantization of all DCTcoefficients results in a lossy transformation some image details given by higher frequencies are cut off. JPEG application provides a table with 64 entries, each used for quantization of one DCTcoefficient each coefficient can be adjusted separately A high compression factor is achievable on the expense of image quality large quantization numbers: high data reduction but information loss increases No default values for quantization tables are specified in JPEG Page 3
33 Example Lehrstuhl für Informatik 4 Input values from exemplary greyscale image First: subtract 8 from each element Then: perform FDCT Page 33
34 Example Lehrstuhl für Informatik 4 DC coefficient FDCT Output Values (because of space reasons only the part before the comma ) Quantization Matrix for Quality Level F uv Page 34
35 Example Lehrstuhl für Informatik 4 Effects of Quantization F* uv  reconstruction after dequantization Correct value was instead of Quantized Matrix Indication of quality loss Page 35
36 Example Lehrstuhl für Informatik 4 Reconstructed image after performing the inverse DCT: Error in reconstruction: Page 36
37 Lehrstuhl für Informatik 4 Problem of Quantization Cutting of higher frequencies leads to partly wrong color information the higher the quantization coefficients, the more disturbance is in a 8 8 block Result: edges of blocks can be seen Page 37
38 Compression Steps in JPEG Uncompressed Image Compressed Image Image Preparation Pixel Block, MCU Image Processing Predictor DCT Quantization (approximation of real numbers by rational numbers) Entropy Encoding Runlength Huffman Arithmetic Result of quantization: 8 8 blocks of DC/AC coefficients with lots of zeros MCU: Minimum Coded Unit How to process and encode the data efficiently? DCT: Discrete Cosine Transform Page 38
39 Entropy Encoding Lehrstuhl für Informatik 4 Baseline Process  Entropy Encoding Initial step: map 8 8 block of transformed values F Q uv to a 64 element vector which can be further process by entropy encoding DCcoefficients determine the basic color of the data units in a frame; variation between DCcoefficients of successive frames is typically small The DCcoefficient is encoded as difference between the current coefficient and the previous one ACcoefficients: processing order uses zigzag sequence DCcoefficient ACcoefficients, higher frequencies Coefficients with lower frequencies are encoded first, followed by higher frequencies. Result: sequence of similar data bytes efficient entropy encoding Page 39
40 Example Lehrstuhl für Informatik 4 Zigzag ordering DCcoefficient: code coefficients for one block as difference to the previous one ACcoefficients: consider each block separately, order data using zigzag sequence to achieve long sequences of zerovalues: Entropy encoding: Runlength encoding of zero values of quantized ACcoefficients Huffman encoding on DC and ACcoefficients Page 4
41 Runlength Encoding Runlength encoding is a contentdependent coding technique Sequences of the same bytes are replaced by the number of their occurrences A special flag byte is used which doesn t occur in the byte stream itself Coding procedure: If a byte occurs at least four consecutive times, the number of occurrences 4 (offset = 4) is counted The compression data contain this byte followed by the special flag and the number of occurrences 4 As a consequence: Representation of 4 to 59 bytes with three bytes is possible (with corresponding compression effect) Example with! as special flag: Uncompressed sequence: Runlength coded sequence: ABCCCCCCCCDEFGGG ABC!4DEFGGG Offset of 4, since for smaller blocks there would be no reduction effect; e.g. with offset 3: D! DDD (both strings have same length) Page 4
42 Runlength Encoding Similar it is done in JPEG: The zerovalue is the only one appearing in longer sequences, thus use a more efficient coding by only compressing zero sequences: code nonzero coefficients together with their runlength, i.e. the number of zeros preceding the nonzero value Runlength {,...,5}, i.e. 4 Bit for representing the length of zero sequences Coded sequence: runlength, size, amplitude with runlength number of subsequent zerocoefficients size number of bits used for representing the following coefficient amplitude value of that following coefficient using size bits By adapting the size of representing a coefficient to its value achieves a further compression because most coefficients for higher frequencies have very small values If (runlength, size) = (5, ) then there are more than 5 zeros after each other. (,) = EoB symbol (End of Block) indicates the termination of the actual rectangle (EoB is very frequently used) Page 4
43 Example Lehrstuhl für Informatik 4 Size i Amplitude  3, ,3 37,...,4 4,...,7 45,...,8 8,...,5  i +,..., i i,..., i  3,...,5 5,...,3 is for instance represented by: size = 4, amplitude = The sequence is encoded by = 8 = 9 = = = = 3 = 4 = 5 = 5 = 4 = 3 = = = = 9 = 8 4 bits complement Representation (other representations are possible) 5,, 5,, 5, 7, zeroes 35 zeros at all, followed by a value represented using 7 bit With 7 bit, is In a second step, the string may be still reduced by Huffman encoding principles Page 43
44 Huffman Encoding The Huffman code is an optimal code using the minimum number of bits for a string of data with given probabilities per character Statistical encoding method: For each character, a probability of occurrence is known by encoder and decoder Frequently occurring characters are coded with shorter strings than seldomly occurring characters Successive characters are coded independent of each other Resulting code is prefix free unique decoding is guaranteed A binary tree is constructed to determine the Huffman codewords of the characters: Leaves represent the characters that are to be encoded Nodes contain the occurrence probability of the characters belonging to the subtree Edges of the tree are assigned with and Page 44
45 Huffman Encoding Algorithm for computing the Huffman code:.) List all characters as well as their frequencies.) Select the two list elements with the smallest frequency and remove them from the list 3.) Make them the leafs of a tree, whereby the probabilities for both elements are being added; place the tree into the list 4.) Repeat steps and 3, until the list contains only one element 5.) Mark all edges: Father left son with Father right son with The code words result from the path from the root to the leafs Page 45
46 Huffman Encoding Example Suppose that characters A, B, C, D and E occur with probabilities p(a) =.7, p(b) =.36, p(c) =.6, p(d) =.4, p(e) =.7 p(adceb) =. p(ced) =.37 p(c) =.6 p(ed) =. p(a) =.7 p(e) =.7 p(d) =.4 p(ab) =.63 p(b) =.36 Resulting Code: x w(x) A B C D E Page 46
47 Huffman Encoding in JPEG Coding of runlength ( {,, 5}), size ( {,, }) (i,j): i preceding zeroes ( i 5) in front of a nonzero value coded with j bits The table has 6+ = 6 entries with significantly different occurrence probabilities EoB is relatively frequent ZRL: at least 6 successive zeroes, i.e. ZRL = (5,) Some values such as (5,) are extremely rare: 5 preceding zeros in front of a very large value is practically impossible! The same holds for most of the combinations in the table. Thus: Huffman coding of the table entries will lead to significant further compression! size runlength 3... EoB (impossible) (,3) (impossible).. (i,j).. 4 (impossible) 5 ZRL (5,) Page 47
48 Huffman Encoding in JPEG Different Huffman tables for (runlength, size) are used for different 8x8 blocks, basing on their contents Thus the coding begins with a HTN (Huffmantablenumber) The coding of amplitudes may also change from block to block Amplitude codes are stored in the preceding (runlength, size) coding table A 8 8 block thus is coded as follows: [VLC, DC coefficient, sequence of (runlength, size, amplitude) for the AC coefficients] VLC = variable length code: contains actual HTN + actual VLI (Variable Length Integer), i.e. coding method for next amplitude Page 48
49 Alternative to Huffman: Arithmetic Coding Characteristics: Achieves optimality (coding rate) as the Huffman coding Difference to Huffman: the entire data stream has an assigned probability, which consists of the probabilities of the contained characters. Coding a character takes place with consideration of all previous characters. The data are coded as an interval of real numbers between and. Each value within the interval can be used as code word. The minimum length of the code is determined by the assigned probability. Disadvantage: the data stream can be decoded only as a whole. Page 49
50 Arithmetic Coding: Example Code data ACAB with p A =.5, p B =., p C =.3 p A =.5 p B =. p C = p AA =.5 p AB =. p AC =.5 p BA p BB p BC p CA p CB p CC p ACA =.75 p ACB =.3 p ACC = p ACAA =.375 p ACAB =.5 p ACAC = ACAB can be coded by each binary number from the interval [.3875,.45), rounded up to log (p ACAB ) = 6.6 i.e. 7 bit, e.g.. Page 5
51 Variants of Image Compression JPEG is not a single format, but it can be chosen from a number of modes: Lossy sequential DCTbased mode (baseline process) Presented before, but not the only method Expanded lossy DCTbased mode Enhancement to the baseline process by adding progressive encoding Lossless mode Low compression ratio perfect reconstruction of original image No DCT, but differential encoding Hierarchical mode Accommodates images of different resolutions Selects its algorithms from the three other modes Page 5
52 Variants: Expanded Lossy DCTbased Mode With sequential encoding as in the baseline process the whole image is coded and decoded in a single run. An alternative to sequential encoding is progressive encoding, done in the entropy encoding step. Two alternatives for progressive encoding are possible: Spectral selection At first, coefficients of low frequencies are passed to entropy encoding, coefficients of higher frequencies are processed in successive runs Successive approximation All coefficients are transferred in one run, but mostsignificant bits are encoded prior to lesssignificant bits. possible coding alternatives in the expanded mode: Using sequential encoding, spectral selection, or successive approximation (3 variants) Using Huffman or Arithmetic encoding ( variants) Using 8 or bits for representing the samples ( variants) Most popular mode: sequential display mode with 8 bits/sample and Huffman encoding Page 5
53 Expanded Lossy DCTbased Mode (Example) Sequential encoding: image is coded and decoded in a single run Step Step Step 3 Progressive encoding: image is coded and decoded in refining steps Step Step Step 3 Page 53
54 Variants: Lossless Mode Lossless mode uses differential encoding (Differential encoding is also known as prediction or relative encoding) Sequence of characters whose values are different from zero, but which do not differ much. Calculate only the difference wrt. the previous value (used also for DCcoefficients) Differential encoding for still images: Avoid using DCT/quantization Instead: calculation of differences between nearby pixels or pixel groups Edges are represented by large values Areas with similar luminance and chrominance are represented by small values Homogenous area is represented by a large number of zeros further compression with runlength encoding is possible as for DCT Page 54
55 Variants: Lossless Mode Uses data units of single pixels for image preparation Any precision between and 6 bits/pixel can be used Image processing and quantization use a predictive technique instead of transformation encoding 8 predictors are specified for each pixel X by means of a combination of the already known adjacent samples A, B, and C predictor predicted values X Uncompressed data C A B X The actual predictor should give the best approximation of x by the already known values A,B,C no prediction A B 3 C 4 A+BC 5 A+(BC)/ 6 B+(AC)/ 7 (A+B)/ Predictor Entropy encoder Compressed data The number of the chosen predictor and the difference of the prediction to the actual value are passed to entropy encoding (Huffman or Arithmetic Encoding) Example: (4,): X is exactly given by A+BC (7,): X is (A+B)/+ Page 55
56 Variants: Hierarchical Mode The Hierarchical mode uses either the lossy DCTbased algorithms or the lossless compression technique The idea: encoding of an image at different resolutions Algorithm: Image is initially sampled at a low resolution Subsequently, the resolution is raised and the compressed image is subtracted from the previous result The process is repeated until the full resolution of the image is obtained in a compressed form Disadvantage: Requires substantially more storage capacity Advantage: Compressed image is immediately available at different resolutions scaling becomes cheap Page 56
57 JPEG Lehrstuhl für Informatik 4 Improvement of the original JPEG standard: 6 Bit color depth (up to 8 billion colors) Progressive mode is not only an option, but mandatory Definition of Regions of Interest choose an image region which will be less compressed than the rest of the image. Thus, images can be individually coded, improving the subjective image quality Integration of watermarks invisibly embed additional information which can be recognized by certain programs; watermarks cannot be removed from the image Resync Marker improve fault tolerance and error correction by setting markers; if an transmission error corrupts the data, not all following data are lost like in normal JPEG Most important: better compression, faster coding and decoding process by using Wavelets instead of DCT. Wavelets describe transforming functions, how fast image information are changing. No pixel blocks are used in compression, infinite functions are performed on finite image regions. Thus the wavelets can describe edges and hard changes better than DCT. Page 57
58 Wavelets Lehrstuhl für Informatik 4 Wavelets are mathematical tools which allow to decompose functions in a hierarchical way, i.e. to describe a function by means of: [overall shape, first detail, second detail,...], e.g. by ( most important features, refinements, less important topics,... ) Thus, the most important aspects are available with a relatively small number of bits Application of Wavelets: Image compression Image editing Animation Signal processing... Page 58
59 Haar Transforms The simplest example of wavelet technology is the Haar transform. There are onedimensional Haar wavelet transforms (which allow the compression of the representation of piecewise constant functions) twodimensional Haar wavelet transforms (for image compression,...) higherdimensional Haar wavelet transforms Example (onedimensional): Simple example to show how we can reduce the amount of bits needed for representation by transformation and compression Let a string of pixels be given as follows: Then we do (recursively): calculate the average of successive pairs calculate the distance from the average ( detail ) Page 59
60 D Haar Transforms In the first step of this procedure we get: [average ; average ;...; detail ; detail ;...] Thereafter we get: [average, ; detail, ;...] where average, is the mean value of average and average and detail, is the mean value of detail and detail This procedure may be continued. The detail values become less and less important From the transformed strings we can fully reconstruct the original Very often the details are very small and may be suppressed. In such a case we cannot exactly reconstruct the original but the errors are comparatively small. Page 6
61 Example Lehrstuhl für Informatik Averages Details Global average detail detail coefficient coarsest resolution Thus the sequence [9, 7, 3, 5, 6, 8, 6, 4] has been transformed to [6; ;, ;,,,] detail coefficients finest resolution Page 6
62 Example Lehrstuhl für Informatik 4 Reconstruction: 6; ;, ;, , , 6 = = 68 = = 67 = = 69 = = 83 = 4 + () 5 = 4  () 6 = 7 + () 8 = 7  () 6 = = 5  If we would suppress the finest detail coefficients (i.e.,, , , ) then we would reconstruct to: instead of: Page 6
63 Generalizsation Lehrstuhl für Informatik 4 Generalization to piecewise constant functions (instead of strings): Let V j be vector space of all functions which are piecewise constant in the j equal subintervals of [:] Every onedimensional image with j pixels can be considered as an element of V j (e.g. a twopixel image has two constant parts over the intervals [,.5) and [.5,) The figure shows an element of V 3 Obviously: V 3 is a refinement of V etc. 3 V V V V... A basis for the vector space V j is a set of functions by which all functions of V j may be represented as linear combinations of the basis functions f(x) The piecewise constant sections might be an approximation of the function f(x) according to a given norm x Page 63
64 Generalization Lehrstuhl für Informatik 4 A simple basis of V j (there are many other alternatives for a basis) is: φ j i (x) i i + x ; j j = i i + x ; j j for i =,..., j  Definition:. The inner product of two functions f, g V j is defined as. The L norm is defined by : = < uu > = u( x) u( x) Very often we try to approximate a given function f(x) by a function f(x) according to the L norm. I.e. find f(x) such that f ( x ) f ( x ) = minimum for all f ( x ) 3 φ dx 3 φ 4 f g : = f(x ) g( x ) dx Page 64
65 Generalization Lehrstuhl für Informatik 4 In the following we will use another basis for V j than the very simple basis which consists only of the elementary components shown before This will lead us to the concept of a Haar Wavelet basis Construction procedure of a suitable basis for V j+ from a basis of V j : Suppose that we already have a basis for V j Then define a new vector space W j as orthogonal complement of V j in V j+, i.e. W j := vector space of functions in V j+ which are orthogonal to all functions in V j ( orthogonal means that <u v>= if u V j ; v W j ) W j is the detail of V j+ which cannot be expressed by V j The basis functions of V j together with the basis functions of W j form a basis of V j+ Page 65
66 Generalization Lehrstuhl für Informatik 4 Definition: The elements of a basis of W j j (i.e. linearly independent functions ψ i (x) which span W j ) are called Wavelets. Immediate consequence: j. ψ together with the basis of V j are a basis of V j+ i (x) j. ψ i (x) is orthogonal to j j j k i k φ (x), i.e. ψ (x) φ (x)dx = The detail coefficients are coefficients of the wavelet basis functions. Page 66
67 Generalization Lehrstuhl für Informatik 4 Wavelets for W j (i.e. elements of V j+ which are orthogonal to basis elements of V j ) are denoted as Haar wavelets Haar wavelets of W j (there would be other possibilities for basis functions): ψ j i (x): = if x ; j j for i =,..., j  i i + + if x ; j j i + i + otherwise Page 67
68 Example Lehrstuhl für Informatik 4 Example: V basis functions Haar wavelets for W φ ψ ψ φ φ + ψ + ψ 3 φ Page 68
69 Example Lehrstuhl für Informatik We can (using these functions) redo the string example as follows: S( x ) = string (written as a function of x ) = 9 φ (x) + 7 φ (x) + 3 φ (x) + 5 φ (x) + 6 φ (x) + 8 φ (x) + 6 φ (x) + 4 φ (x) = 8 φ (x) + 4 φ (x) + 7 φ (x) + 5 φ (x) + ψ (x) ψ (x) ψ (x) + ψ (x) 3 3 = 6 φ (x) + 6 φ (x) + ψ (x) + ψ (x) + ψ (x) ψ (x) ψ (x) + ψ (x) 3 = 6 φ (x) + ψ (x) + ψ (x) + ψ (x) + ψ (x) ψ (x) ψ (x) + ψ (x) 3 Page 69
70 Example Lehrstuhl für Informatik 4 Graphical representation: 6x x + x + x + x x x + x = φ ψ ψ ψ ψ ψ ψ ψ 3 Page 7
71 Example Lehrstuhl für Informatik 4 φ Starting with the basis function for V we can refine it by the W detail coefficient, i.e. by to the basis for V ψ This in turn can be refined by two detail coefficients of V : φ, ψ, ψ, ψ ψ, ψ of W to the basis With four more detail coefficients of W we get a basis for V 3 consisting of the eight basis functions φ ; ψ ; ψ, ψ ; ψ, ψ, ψ, ψ With eight more detail coefficients of W 3 we get a basis for V 4 etc. 3 This procedure leads to a better approximation of a given function (or of a D image) Page 7
72 Approximation Of Continuous Functions Approximation of a continuous function f(x) (dotted line) by averages and detail coefficients We start with V. Then: V  appr. + W detail = V appr. V  appr. + W detail = V appr. V  appr. + W detail = V 3 appr. V 3 appr. + W 3 detail = V 4 appr. We can also work backwards, i.e. start with the V 4 approximation and go to the V 3 approximation by suppressing the W 3 detail coefficients etc. V 4 approximation V 3 approximation V approximation average of f(x) V approximation W 3 detail coefficients W detail coefficients W detail coefficients W detail coefficients V approximation Page 7
73 Approximation Of Continuous Functions φ, ψ, ψ, ψ,... All Haar basis functions are orthogonal to each other (of course not to itself); i.e. <f g> = if f g are basis functions Orthogonality to all other basis functions is not necessarily valid for other systems of basis functions In addition to orthogonality we can provide orthonormality, i.e. <f f> = The basis functions φ ψ j* i j* i ( x) : = j ( x) : = j j φ ( x) i j ψ ( x) i are orthonormal If the basis functions are orthonormalized, then the detail coefficients and the j average coefficient have to be multiplied by In our example we then get: S(x) = 6 φ + ψ + ψ + ψ + ψ ψ ψ + ψ 3 = 6 φ + ψ + ψ + ψ + ψ ψ ψ + ψ * * * * * * * * 3 Page 73
74 Decomposition Lehrstuhl für Informatik 4 Decomposition of a sequence of h numbers together with a normalization: each original coefficient with superscript j is multiplied by (j/) proc DecompositionStep(C:array[..h] of reals) for i:= to h/ do C [i]:=(c[i]+c[i])*sqrt() C [h/+i]:=(c[i]c[i])*sqrt() end for C:=C ; end proc proc Decomposition(C:array[..h] of reals) C := C/sqrt(h); // normalize input coefficients while h> do DecompositionStep(C[..h]); h:=h/; end while; end proc Page 74
75 Lehrstuhl für Informatik 4 Usage for Data Compression Haar wavelets for data compression: Suppose that a function f(x) is given by a linear combination of m basis functions, e.g. by a linear combination of m Haar wavelet functions: m f(x) = c u (x) i= i We want to reduce the number of coefficients, i.e. we seek for a function f(x) which:. Is similar to f(x), i.e. i for example: u ( x ) = φ( x ),..., u 8( x ) = ψ3( x ) f(x) f(x) ε for some norm. May be represented by fewer coefficients (possibly with other basis functions) u i( x ) instead of u i( x ) ~ = m ~ f ( x) c~ i u~ i ( x) with m ~ < m i = Finding the best f(x) is a difficult problem if all possible Thus we restrict ourselves to the Haar Wavelet basis u(x) i are taken into account. Page 75
76 Lehrstuhl für Informatik 4 Data Compression If we do not change the basis function but reduce the number of coefficients then it is easy to show that: If the basis functions are orthonormal, then for m < m coefficients and for the L  norm, i.e. f f =< f(x) f(x) f(x) f(x) >, the error is minimized if we keep the coefficients of c,..., c m whose absolute values are the largest ones Proof: Let σ,..., σ m be a permutation of,..., m Let m f(x) = c u i= σ i σ i f ( x)  ~ f ( x) = = σ i i = m~ + m m c m u c σ i i = m~ + j = m~ + c m j c σ i σ j j = m~ + σ < u u σ i σ u J σ j > = m i = m~ + ( c σ i ) since u i are orthonormal c,...,c The error is minimized if for σ the smallest coefficients in absolute m+ σm values are used Page 76
77 Data Compression Approximation of f(x) by V 3, W 3 i.e. 8 averages 8 details φ ( x ),..., φ ( x ) ψ ( x ),..., ψ ( x ) orthogonal functions!! (for simplicity the * has been suppressed) And (further on) by: φ ; ψ ; ψ ; ψ ; ψ,..., ψ ; ψ,..., ψ global average 6 coefficients The sequence of pictures shows the effect (i.e. loss of exactitude) if more and more coefficients are eliminated Page 77
78 D Haar Wavelets We have shown how to use onedimensional Haar wavelets for representation and compression of onedimensional functions Now we generalize to Dimages First we show how to apply the ( averaging + detail ) technique to Dimages a pixel of Dimage First approach: Standard decomposition. Apply the Dapproach to each row. Result: Average + details for each row average x ++++ x ++++ x ++++ x ++++ Details of different resolution (see previous examples). Apply the Dapproach to each column of the results obtained by step I.e.: first the rows, then the columns global average ααββββ XXXX++++ Detail coefficients Page 78
79 D Haar Wavelets Second Approach: Nonstandard decomposition Alternate between rows and columns. Out of the n values per row: calculate n averages + n details. Take the result and do the same for the columns: i.e. calculate m averages + m details for the m values per column Result of steps and : averages step (for left half only) detail coefficients step 3. Repeat step and for the averages part, i.e. for the left upper corner 4. Do that recursively until only one global average is left over Remark: This technique works only for square images, i.e. n n pixels Page 79
80 D Haar Wavelet Basis There are two methods for the construction of a twodimensional basis:. The standard construction. The nonstandard construction Regarding V i which has i basis functions in D, namely i i φ ; ψ ; ψ, ψ ; ψ,..., ψ ;...; ψ,..., ψ 3 i we get i i basis functions for V i in D by combining the D basis functions with each other. Example for V where the Dbasis will consist of 4x4 = 6 basis functions (since there are 4 basis functions φ, ψ, ψ, ψ in V ) Page 8
81 The Standard Construction Let u,...,u be the D basis functions i i Then v j(x,y) = u j (x) u is a basis function for D images j (y) for j,j : There are i x i = i basis functions for V j in D. Example: y ψ ( ) y ψ ( ) y ψ ( x) ψ ( ) x y y +  x  + x x + means: + means:  Doing that for φ (x) ψ ψ ψ (x) ψ (x) ψ (x) ψ φ (y) (y) (y) (y) we get all the 6 basis functions for V in D Page 8
82 The Nonstandard Construction The nonstandard construction of a twodimensional basis defines a twodimensional scaling function φφ(x,y): = φ(x) φ(y) and three wavelet functions φψ(x,y): = φ(x) ψ(y) ψφ(x,y): = ψ(x) φ(y) ψψ(x,y): = ψ(x) ψ(y) Furthermore we define scaling by a superscript j and horizontal and vertical translations by a pair of subscripts k and l The nonstandard basis consists of φφ (x,y): = φφ (x,y) as well as of scaled and translated versions of the wavelet functions φψ, ψφ, and ψψ. The result is φψ (x,y): = φψ( x k, y l) j j j j kl φψ (x,y): = ψφ( x k, y l) j j j j kl φψ (x,y): = ψψ ( x k, y l) j j j j kl Page 8
83 Nonstandard Construction Of D Haar Wavelets φψ (x,y), φψ (x,y), ψψ (x,y), ψψ (x,y),  + φψ (x,y),  + φψ (x,y), ψψ (x,y), ψψ (x,y), φψ (x,y), function for global average + ψψ (x,y), +  ψφ (x,y), +  ψφ (x,y), +  φφ (x,y), ψφ (x,y), ψφ (x,y), ψφ (x,y), Page 83
84 The Standard Construction The standard construction of a D Haar wavelet basis for V We calculate the products of the onedimensional basis φ, ψ, ψ, ψ for both dimensions A detail: The portion of the square [:]x[:] where the basis function is different from zero is not a square for all 6 functions If we apply the standard construction to an orthonormal basis in one dimension, we get an orthonormal basis in two dimensions φ (x) ψ (y) ψ (x) ψ (y) ψ (x) ψ (y) function for global average φ (x) ψ (y) ψ (x) ψ (y) ψ (x) ψ(y) ψ(x) ψ +  ψ (x) ψ (y) (y) φ (x) ψ (y) ψ (x) ψ (y) ψ (x) ψ (y) ψ (x) ψ (y) φ (x) φ (y) ψ (x) φ (y) ψ (x) φ (y) ψ (x) φ (y Page 84
85 Comparison Lehrstuhl für Informatik 4 The standard decomposition of an m x m image requires onedimensional transforms on all rows and then on all columns. 4(m m) assignment operations are needed The nonstandard decomposition requires only (8/3) (m ) assignment operations In the nonstandard case, all the nonstandard basis functions have square support ( support = area where the function is nonzero). This is not the case for standard basis functions. Page 85
86 Decomposition Lehrstuhl für Informatik 4 Only to remember: decomposition of a sequence of h numbers together with a normalization. These procedures are needed in the dimensional decomposition. proc DecompositionStep(C:array[..h] of reals) for i:= to h/ do C [i]:=(c[i]+c[i])*sqrt() C [h/+i]:=(c[i]c[i])*sqrt() end for C:=C ; end proc proc Decomposition(C:array[..h] of reals) C := C/sqrt(h); // normalize input coefficients while h> do DecompositionStep(C[..h]); h:=h/; end while; end proc Page 86
87 Decomposition Lehrstuhl für Informatik 4 The standard decomposition of a D picture of h w pixels proc StandardDecompositionStep(C:array[..h,..w] of reals) for row:= to h do Decomposition(C[row,..w]); end for for col:= to h do Decomposition(C[..h,col]); end for end proc The nonstandard decomposition of a D picture of h w pixels proc NonstandardDecompositionStep(C:array[..h,..w] of reals) C:=C/h; // normalize input coefficients while h> do for row:= to h do DecompositionStep(C[row,..h]); end for for col:= to h do DecompositionStep(C[..h,col]); end for end while end proc Page 87
88 Decomposition Lehrstuhl für Informatik 4 Instead as of the unnormalized Haar Wavelet functions φ (x): = φ( x i) ψ φ ψ j j i where j j for x < i (x): = ψ( x i) ψ (x): = for x < j j j i (x): = φ( x i) j * j j i (x): = ψ( x i) φ(x): = for x otherwise otherwise we can use the normalized Haar wavelet functions As a compensation we have to multiply each unnormalized coefficient with superscript j by (j/) Example: Unnormalized version: coefficients = (6;;,;,,,) basis functions = ( φ ; ψ ; ψ, ψ ; ψ, ψ, ψ, ψ ) 3 Normalized version:   coefficients = (6;;, ;,,, ) normalized according to (* * * * * * * * * basis functions = ( φ ; ψ ; ψ, ψ ; ψ, ψ, ψ, ψ ) Page 88 3
89 Decomposition Lehrstuhl für Informatik 4 Comment on input normalization C [i]:=(c[i]+c[i])*sqrt(), i.e. the average is multiplied by At the beginning we normalize by Example: Let h = n, e.g. h = 4 ; basis functions Then the higher superscript is n , e.g. 3 We normalize first: C C C C: = = = n h n Then in the first run we multiply the first detail by, then the coefficient is normalized by C C = n n C: = C h ( Θ ; Ψ ; ; Ψ ) 3 7 The same holds for the averages. In successive runs the other detail levels and the global average are correctly evaluated by means of successive multiplications with Page 89
90 D Image Compression Dimage compression with DHaar basis functions (Generalization of the corresponding technique for Dimages) Step: Compute coefficients c,...,c m which represent an image in a normalized Haar basis. (Image = c u (x,y) + c u (x,y) c m u m (x,y)) Step: Sort c,...,c m in order of decreasing magnitude. Result: m Step3: Find the smallest k with c σ ε where ε is the allowed error regarding! i i= k+ the L norm,i.e. f f ε c σ,...,c σ m Page 9
91 Decomposition Lehrstuhl für Informatik 4 A fast procedure (by binary search) for finding the threshold which coefficients are negligible proc Compress(C:array[..m]of reals, ε: real) τ min :=min{ C[i] } τ max :=max{ C[i] } do τ:=(τ min + τ max )/ s:= for i:= to m do if C[i] τ then s:=s+(c[i])^ end for if s < ε^ then τ min := τ else τ max := τ until τ min τ max for i:= to m do if C[i] < τ then C[i]:=; end for end proc Page 9
92 Decomposition Lehrstuhl für Informatik 4 This procedure starts with coefficients C[],, C[m] and finds the smallest m for which m c ε where σ,, σ is a permutation of,,m i= m+ σ i m such that c c c σ σ σ m and where ε is a tolerable L error The algorithm works as follows:. It starts with a threshold τ which is the average between the smallest and the largest coefficient. It computes the L error if all coefficients smaller in magnitude than τ would be discarded 3. If squared error < ε then we continue with the right half of the interval 4. If squared error ε then we continue with the left half of the interval Page 9
93 Decomposition Lehrstuhl für Informatik 4 Let (without any loss of generality) c, c,, c m be already sorted such that c (τ max ) c c m (τ min ) Let s := squared error by discarding all coefficients smaller than τ, i.e. all coefficients on the left side of τ Case : s < ε We could even discard (possibly) more than those coefficients! Hence we discard the part on the left of τ. We set τ new,min := τ and restart with the new interval. Still very small error? τ min τ new,min = τ τ max Case : s > ε new interval By discarding the elements left of the actual τ we would discard too many elements. The elements left of τ must be kept anyway. Set τ new,max := τ and restart. Error too large? τ min new interval τ new,max = τ τ max Page 93
94 Multiresolution Analysis Multiresolution analysis means analyzing a signal at different frequencies giving different resolutions Consider a nested set of vector spaces V V V... Basis functions of V j are sometimes called scaling functions W j := orthogonal complement of V j in V j+ [orthogonal according to some definition of an inner product ] The functions which we choose as a basis for W j are called wavelets Example (Haar Wavelets): ψ ψ ψ ψ 3 Matrix notation: Φ (x): = [ φ (x),..., φ (x)]; Ψ (x): = [ ψ (x),..., ψ (x)] If the basis V j and W j consists of m j and n j elements respectively, then we may combine the basis functions into single row matrices j j j j j j j j m n Page 94
95 Matrix Formulation Some immediate consequences:. V j W j = V j+ ; W j orthogonal to V j m j + n j = m j+. V j V j The elements of V j must be expressible as linear combination of the finer functions of V j, i.e. the scaling functions must be refinable Example: Scaling functions of V : φ φ Scaling functions of V : φ φ We have: φ = φ + φ φ = φ + φ 3 φ φ 3 m j There must be a matrix P j with where P j is a m j m j matrix (a) Φ (x) = Φ (x) P j j j m j Page 95
96 Matrix Formulation 3. W j V j Elements of W j can be written as linear combination of scaling functions of V j There is a m j n j matrix Q j j j j with (b) Ψ (x) = Φ (x) Q Example: Φ (x) = Φ (x) P and Ψ (x) = Φ (x) Q with: P = and Q = 4. (a) and (b) can be combined to: j j j j j Φ Ψ = Φ P Q In our example: φφψ ψ φφ φφ = 3 Page 96
97 Matrix Formulation j 5. All functions of Φ (x) must be orthogonal to all functions of Ψ j (x). j j < Φ Ψ > = (Matrix whose (k,l) entry is < φ ψ > ) j j k l (from (b)): j j j < Φ Φ > = Q j j j 6. < Φ Φ > Q = is a homogeneous systems of equations. The set of j j solutions is called null space of < Φ Φ >. Page 97
98 Matrix Formulation The columns of Q j are a basis for the null space. There are many alternatives for Q j, thus there are many different wavelet bases for a given W j. The Haar wavelets are defined by the additional requirement that the number of consecutive nonzero values in Q j per column is minimal Q = Definition Orthogonal wavelet basis: all functions (wavelets + scaling functions) are orthogonal to each other Semiorthogonal wavelet basis: wavelets orthogonal to scaling functions but not to each other ˆ= Haar basis orthogonal; spline basis semiorthogonal ˆ= Page 98
99 Matrix Formulation A matrix notation for the approximation of functions We have shown earlier that (and how) it was possible to approximate a function f(x) in V j by f ( x ) j j = c u ( x ) c j u ( x ) j m m where u i (x) are basis functions of V j The coefficients c ij might be, for example, pixel colors or ycoordinates of the function f(x) j j j T We can write the coefficients c ij as C = c,...,c j m j j j f ( x ) = U C where U = [u (x),...,u (x)] j m Page 99
100 Matrix Formulation We may want to express f(x) within V j (a lower resolution version, i.e. less accurate, with lower number of coefficients, i.e. m j instead as of m j coefficients) The standard procedure of creating the m j coefficients of C j out of the m j coefficients of C j consists of: linear filtering the coefficients of C j down sampling This may be expressed as a matrix equation: C j = A j C j m j where A j is a m j m j matrix m j Since C j is smaller then C j this filtering process looses some detail. The lost detail may be expressed in a second matrix D j with D j = B j C j where B j is a n j m j matrix Page
101 Matrix Formulation C j = A j C j Low resolution of C j D j = B j C j Detail of C j which cannot be expressed by C j If A j and B j are appropriately chosen, C j may be recovered from C j and D j : C j = P j C j + Q j D j The process of calculating C j and D j from C j is called decomposition of C j or <analysis> The process of calculating C j from C j and D j is called reconstruction of C j or <synthesis> The global procedure looks as follows: A j A j A C j C j C j...c C B j D j B j D j B D This recursive procedure is called a filter bank The coefficients C j can be reconstructed from: C,D,D,D,...,D j This sequence is called wavelet transform. It has the same size as C j. Page
102 Example Lehrstuhl für Informatik 4 For the Haar basis of V we have A = B = C = A C D = B C A is the averaging operation B is the differencing operation (this explains the factor ½) A general relation which must be satisfied by the matrices A j and B j is : A Φ Ψ = Φ j j j j j B j j j j j Thus since Φ Ψ Φ P Q = (shown earlier) we must have A j j j P Q = j B j A i.e. must be invertible j B Page
103 Multiresolution Analysis How to choose a suitable technique for transformation? Suitable for a particular application. The following steps have to be made:. Select the scaling functions (this defines V j and P j ). Select an inner product for V, V,... (this defines the L norm as well as W j ) 3. Select a set of wavelets which are a basis for W j (j =,,...) (this defines Q j ) Ψ j (x) Remark: P j and Q j define A j and B j since Φ j ( x ) (j =,,...) A B j j j = P Q j Page 3
104 Multiresolution Analysis It is desirable for reasons of data compression that:. Wavelets are an orthogonal basis for W j. Wavelets have a small support (the support of a function f(x) is the number of points where f(x) ) But: orthogonality comes very often with large support. Thus it may be better to sacrifice the orthogonality. An example for that are the spline wavelets which: have minimum support are not orthogonal to each other (except for degree = ) Page 4
105 BSpline Wavelets Haar type wavelets have many advantages: simplicity orthogonality very small supports nonoverlapping scaling functions for a given level j nonoverlapping wavelets for a given level j However, Haar transforms are not well suited for animation and for curve editing since they do not have enough smoothness Therefore, we are interested in wavelets which have several continuous derivatives Such wavelets can be derived from piecewise polynomial splines (Bsplines) Bspline scaling functions and Bspline wavelets can be defined for different degrees d =, d =, d =,... The higher the degree the smoother the transform With degree d, the scaling functions have d  continuous derivatives d = is the Haar transform Page 5
106 NonUniform BSpline Scaling Functions The nonuniform Bspline basis functions for degree d are constructed as follows: Choose positive integers k and d (k d) and values x,...,x k + d + These values are called knots Then we define recursively the nonuniform Bspline basis functions of degree d: N (x): i = x x < x i i+ otherwise x x = + x x r i r xi+ r x r N i (x): N i (x) xi r x N i i (x) + + i+ r i (i =,..., k; r =,..., d) Remark: If the denominator is then the whole term is defined to be zero Page 6
107 Uniform BSpline Scaling Functions Endpointinterpolating Bsplines of degree d on the interval [,] are defined by setting the first d + knots to and the last d + knots to d d N ( x ),...,N k ( x ) form a basis for the space of piecewisepolynomials of degree d with d continuous derivatives Finally, uniformly spaced Bsplines are constructed by selecting k = j + d and by making the interior knots equally spaced. x d+,...,x k This gives j d d + d Bspline functions N ( x ),...,N j ( x ) which are a basis of a particular degree d and a level j, i.e. for V j + d (d). If V j (d) denotes the space which is spanned by the Bspline scaling functions of degree j with j uniform intervals, then the spaces V (d), V (d),... are nested: V (d) V (d) V (d)... Page 7
108 BSpline Scaling Functions Example J =, i.e. = subintervals: [ : ),[ :) Degrees d=, d=, d=, d=3 Bspline scaling functions 3 N N (x),...,n (x) N (x) d d d = + d + j d N N 3 N N N The functions N jd (x) have d  continuous derivatives. N d (x),..., N id (x) are a basis for V (d) N Degree N N Degree N N 3 Degree 3 N 3 N 3 3 N 4 Degree 3 Page 8
109 Scaling and Wavelet Functions for V 3 (d) In the following we show Bspline scaling functions for j = 3 and for degrees d = (Haar wavelet functions) d = (linear splines) d = (quadratic splines) d = 3 (cubic splines) We get j + d = 8 + d scaling functions and j =8 wavelets for each case j The wavelets are determined by matrices Q which satisfy The solution of this equation system is not unique j j j < Φ Φ > = Q Page 9
110 BSpline Scaling Functions for V 3 () Degree d = (Haar wavelets)  not a continuous function: lim f ( x + ε ) lim f ( x ε ) ε ε if x =,,..., scaling functions 8 wavelets Page
111 BSpline Scaling Functions for V 3 () Degree d = (linear Bspline wavelets) 9 scaling functions 8 wavelets Page
112 BSpline Scaling Functions for V 3 () Degree d = (quadratic Bspline)  can be continuously differentiated once scaling functions 8 wavelets Page
113 BSpline Scaling Functions for V 3 (d) Degree d = 3 (cubic Bspline)  can be differentiated continuously two times: scaling functions 8 wavelets There are j + d scaling functions if the unit interval [:) is subdivided into j subintervals. The scaling functions have d continuous derivatives Page 3
114 Wavelets in JPEG Using wavelets, images can better be compressed than with DCT. Used in compression here: nonstandard decomposition Page 4
115 Decomposition Lehrstuhl für Informatik 4 Coarser details Vertical information (fine details) Diagonal information (finest details) Horizontal information (fine details) Page 5
2.2: Images and Graphics Digital image representation Image formats and color models JPEG, JPEG2000 Image synthesis and graphics systems
Chapter 2: Representation of Multimedia Data Audio Technology Images and Graphics Video Technology Chapter 3: Multimedia Systems Communication Aspects and Services Chapter 4: Multimedia Systems Storage
More informationDigital Image Representation. Image Representation. Color Models
Digital Representation Chapter : Representation of Multimedia Data Audio Technology s and Graphics Video Technology Chapter 3: Multimedia Systems Communication Aspects and Services Chapter 4: Multimedia
More informationIMAGE COMPRESSION. Image Compression. Why? Reducing transportation times Reducing file size. A two way event  compression and decompression
IMAGE COMPRESSION Image Compression Why? Reducing transportation times Reducing file size A two way event  compression and decompression 1 Compression categories Compression = Image coding Stillimage
More informationIMAGE COMPRESSION. Chapter  5 : (Basic)
Chapter  5 : IMAGE COMPRESSION (Basic) Q() Explain the different types of redundncies that exists in image.? (8M May6 Comp) [8M, MAY 7, ETRX] A common characteristic of most images is that the neighboring
More informationCS 335 Graphics and Multimedia. Image Compression
CS 335 Graphics and Multimedia Image Compression CCITT Image Storage and Compression Group 3: Huffmantype encoding for binary (bilevel) data: FAX Group 4: Entropy encoding without error checks of group
More informationMultimedia Communications. Transform Coding
Multimedia Communications Transform Coding Transform coding Transform coding: source output is transformed into components that are coded according to their characteristics If a sequence of inputs is transformed
More informationAnatomy of a Video Codec
Anatomy of a Video Codec The inner workings of Ogg Theora Dr. Timothy B. Terriberry Outline Introduction Video Structure Motion Compensation The DCT Transform Quantization and Coding The Loop Filter Conclusion
More informationImage coding and compression
Image coding and compression Robin Strand Centre for Image Analysis Swedish University of Agricultural Sciences Uppsala University Today Information and Data Redundancy Image Quality Compression Coding
More informationCOMP 249 Advanced Distributed Systems Multimedia Networking. The Video Data Type Coding & Compression Basics
COMP 9 Advanced Distributed Systems Multimedia Networking The Video Data Type Coding & Compression Basics Kevin Jeffay Department of Computer Science University of North Carolina at Chapel Hill jeffay@cs.unc.edu
More informationcompression and coding ii
compression and coding ii OleJohan Skrede 03.05.2017 INF2310  Digital Image Processing Department of Informatics The Faculty of Mathematics and Natural Sciences University of Oslo After original slides
More informationThe PackBits program on the Macintosh used a generalized RLE scheme for data compression.
Tidbits on Image Compression (Above, Lena, unwitting data compression spokeswoman) In CS203 you probably saw how to create Huffman codes with greedy algorithms. Let s examine some other methods of compressing
More informationCompression Part 2 Lossy Image Compression (JPEG) Norm Zeck
Compression Part 2 Lossy Image Compression (JPEG) General Compression Design Elements 2 Application Application Model Encoder Model Decoder Compression Decompression Models observe that the sensors (image
More informationLossless Image Compression having Compression Ratio Higher than JPEG
Cloud Computing & Big Data 35 Lossless Image Compression having Compression Ratio Higher than JPEG Madan Singh madan.phdce@gmail.com, Vishal Chaudhary Computer Science and Engineering, Jaipur National
More informationComputer Faults in JPEG Compression and Decompression Systems
Computer Faults in JPEG Compression and Decompression Systems A proposal submitted in partial fulfillment of the requirements for the qualifying exam. Cung Nguyen Electrical and Computer Engineering University
More informationCOLOR IMAGE COMPRESSION USING DISCRETE COSINUS TRANSFORM (DCT)
COLOR IMAGE COMPRESSION USING DISCRETE COSINUS TRANSFORM (DCT) Adietiya R. Saputra Fakultas Ilmu Komputer dan Teknologi Informasi, Universitas Gunadarma Jl. Margonda Raya no. 100, Depok 16424, Jawa Barat
More informationA Image Comparative Study using DCT, Fast Fourier, Wavelet Transforms and Huffman Algorithm
International Journal of Engineering Research and General Science Volume 3, Issue 4, JulyAugust, 15 ISSN 912730 A Image Comparative Study using DCT, Fast Fourier, Wavelet Transforms and Huffman Algorithm
More informationLossy Coding 2 JPEG. Perceptual Image Coding. Discrete Cosine Transform JPEG. CS559 Lecture 9 JPEG, Raster Algorithms
CS559 Lecture 9 JPEG, Raster Algorithms These are course notes (not used as slides) Written by Mike Gleicher, Sept. 2005 With some slides adapted from the notes of Stephen Chenney Lossy Coding 2 Suppose
More informationTopic 5 Image Compression
Topic 5 Image Compression Introduction Data Compression: The process of reducing the amount of data required to represent a given quantity of information. Purpose of Image Compression: the reduction of
More information06/12/2017. Image compression. Image compression. Image compression. Image compression. Coding redundancy: image 1 has four gray levels
Theoretical size of a file representing a 5k x 4k colour photograph: 5000 x 4000 x 3 = 60 MB 1 min of UHD tv movie: 3840 x 2160 x 3 x 24 x 60 = 36 GB 1. Exploit coding redundancy 2. Exploit spatial and
More information15 Data Compression 2014/9/21. Objectives After studying this chapter, the student should be able to: 151 LOSSLESS COMPRESSION
15 Data Compression Data compression implies sending or storing a smaller number of bits. Although many methods are used for this purpose, in general these methods can be divided into two broad categories:
More informationWhat is multimedia? Multimedia. Continuous media. Most common media types. Continuous media processing. Interactivity. What is multimedia?
Multimedia What is multimedia? Media types +Text + Graphics + Audio +Image +Video Interchange formats What is multimedia? Multimedia = many media User interaction = interactivity Script = time 1 2 Most
More informationDigital Video Processing
Video signal is basically any sequence of time varying images. In a digital video, the picture information is digitized both spatially and temporally and the resultant pixel intensities are quantized.
More informationMultimedia. What is multimedia? Media types. Interchange formats. + Text +Graphics +Audio +Image +Video. Petri Vuorimaa 1
Multimedia What is multimedia? Media types + Text +Graphics +Audio +Image +Video Interchange formats Petri Vuorimaa 1 What is multimedia? Multimedia = many media User interaction = interactivity Script
More informationHYBRID TRANSFORMATION TECHNIQUE FOR IMAGE COMPRESSION
31 st July 01. Vol. 41 No. 00501 JATIT & LLS. All rights reserved. ISSN: 1998645 www.jatit.org EISSN: 18173195 HYBRID TRANSFORMATION TECHNIQUE FOR IMAGE COMPRESSION 1 SRIRAM.B, THIYAGARAJAN.S 1, Student,
More informationDIGITAL IMAGE PROCESSING WRITTEN REPORT ADAPTIVE IMAGE COMPRESSION TECHNIQUES FOR WIRELESS MULTIMEDIA APPLICATIONS
DIGITAL IMAGE PROCESSING WRITTEN REPORT ADAPTIVE IMAGE COMPRESSION TECHNIQUES FOR WIRELESS MULTIMEDIA APPLICATIONS SUBMITTED BY: NAVEEN MATHEW FRANCIS #105249595 INTRODUCTION The advent of new technologies
More informationIMAGE COMPRESSION. October 7, ICSY Lab, University of Kaiserslautern, Germany
Lossless Compression Multimedia File Formats Lossy Compression IMAGE COMPRESSION 69 Basic Encoding Steps 70 JPEG (Overview) Image preparation and coding (baseline system) 71 JPEG (Enoding) 1) select color
More informationRobert Matthew Buckley. Nova Southeastern University. Dr. Laszlo. MCIS625 On Line. Module 2 Graphics File Format Essay
1 Robert Matthew Buckley Nova Southeastern University Dr. Laszlo MCIS625 On Line Module 2 Graphics File Format Essay 2 JPEG COMPRESSION METHOD Joint Photographic Experts Group (JPEG) is the most commonly
More informationModule 7 VIDEO CODING AND MOTION ESTIMATION
Module 7 VIDEO CODING AND MOTION ESTIMATION Lesson 20 Basic Building Blocks & Temporal Redundancy Instructional Objectives At the end of this lesson, the students should be able to: 1. Name at least five
More informationMesh Based Interpolative Coding (MBIC)
Mesh Based Interpolative Coding (MBIC) Eckhart Baum, Joachim Speidel Institut für Nachrichtenübertragung, University of Stuttgart An alternative method to H.6 encoding of moving images at bit rates below
More informationJPEG. Wikipedia: Felis_silvestris_silvestris.jpg, Michael Gäbler CC BY 3.0
JPEG Wikipedia: Felis_silvestris_silvestris.jpg, Michael Gäbler CC BY 3.0 DFT vs. DCT Image Compression Image compression system Input Image MAPPER QUANTIZER SYMBOL ENCODER Compressed output Image Compression
More informationThe DCT domain and JPEG
The DCT domain and JPEG CSM25 Secure Information Hiding Dr Hans Georg Schaathun University of Surrey Spring 2009 Week 3 Dr Hans Georg Schaathun The DCT domain and JPEG Spring 2009 Week 3 1 / 47 Learning
More informationCHAPTER 4 REVERSIBLE IMAGE WATERMARKING USING BIT PLANE CODING AND LIFTING WAVELET TRANSFORM
74 CHAPTER 4 REVERSIBLE IMAGE WATERMARKING USING BIT PLANE CODING AND LIFTING WAVELET TRANSFORM Many data embedding methods use procedures that in which the original image is distorted by quite a small
More informationData Compression Fundamentals
1 Data Compression Fundamentals Touradj Ebrahimi Touradj.Ebrahimi@epfl.ch 2 Several classifications of compression methods are possible Based on data type :» Generic data compression» Audio compression»
More informationImage Compression. CS 6640 School of Computing University of Utah
Image Compression CS 6640 School of Computing University of Utah Compression What Reduce the amount of information (bits) needed to represent image Why Transmission Storage Preprocessing Redundant & Irrelevant
More informationJPEG IMAGE CODING WITH ADAPTIVE QUANTIZATION
JPEG IMAGE CODING WITH ADAPTIVE QUANTIZATION Julio Pons 1, Miguel Mateo 1, Josep Prades 2, Román Garcia 1 Universidad Politécnica de Valencia Spain 1 {jpons,mimateo,roman}@disca.upv.es 2 jprades@dcom.upv.es
More informationCHAPTER 6. 6 Huffman Coding Based Image Compression Using Complex Wavelet Transform. 6.3 Wavelet Transform based compression technique 106
CHAPTER 6 6 Huffman Coding Based Image Compression Using Complex Wavelet Transform Page No 6.1 Introduction 103 6.2 Compression Techniques 104 103 6.2.1 Lossless compression 105 6.2.2 Lossy compression
More informationCSCD 443/533 Advanced Networks Fall 2017
CSCD 443/533 Advanced Networks Fall 2017 Lecture 18 Compression of Video and Audio 1 Topics Compression technology Motivation Human attributes make it possible Audio Compression Video Compression Performance
More informationSPIHTBASED IMAGE ARCHIVING UNDER BIT BUDGET CONSTRAINTS
SPIHTBASED IMAGE ARCHIVING UNDER BIT BUDGET CONSTRAINTS by Yifeng He A thesis submitted in conformity with the requirements for the degree of Master of Applied Science, Graduate School of Electrical Engineering
More informationyintroduction to compression ytext compression yimage compression ysource encoders and destination decoders
In this lecture... Compression and Standards Gail Reynard yintroduction to compression ytext compression Huffman LZW yimage compression GIF TIFF JPEG The Need for Compression ymultimedia data volume >
More information7.5 Dictionarybased Coding
7.5 Dictionarybased Coding LZW uses fixedlength code words to represent variablelength strings of symbols/characters that commonly occur together, e.g., words in English text LZW encoder and decoder
More informationA NEW ENTROPY ENCODING ALGORITHM FOR IMAGE COMPRESSION USING DCT
A NEW ENTROPY ENCODING ALGORITHM FOR IMAGE COMPRESSION USING DCT D.Malarvizhi 1 Research Scholar Dept of Computer Science & Eng Alagappa University Karaikudi 630 003. Dr.K.Kuppusamy 2 Associate Professor
More informationRedundant Data Elimination for Image Compression and Internet Transmission using MATLAB
Redundant Data Elimination for Image Compression and Internet Transmission using MATLAB R. Challoo, I.P. Thota, and L. Challoo Texas A&M UniversityKingsville Kingsville, Texas 783638202, U.S.A. ABSTRACT
More informationISSN Vol.06,Issue.10, November2014, Pages:
ISSN 2348 2370 Vol.06,Issue.10, November2014, Pages:11691173 www.ijatir.org Designing a Image Compression for JPEG Format by Verilog HDL B.MALLESH KUMAR 1, D.V.RAJESHWAR RAJU 2 1 PG Scholar, Dept of
More informationNew Perspectives on Image Compression
New Perspectives on Image Compression Michael Thierschmann, Reinhard Köhn, UweErik Martin LuRaTech GmbH Berlin, Germany Abstract Effective Data compression techniques are necessary to deal with the increasing
More informationVideo Coding in H.26L
Royal Institute of Technology MASTER OF SCIENCE THESIS Video Coding in H.26L by Kristofer Dovstam April 2000 Work done at Ericsson Radio Systems AB, Kista, Sweden, Ericsson Research, Department of Audio
More informationImage Compression for Mobile Devices using Prediction and Direct Coding Approach
Image Compression for Mobile Devices using Prediction and Direct Coding Approach Joshua Rajah Devadason M.E. scholar, CIT Coimbatore, India Mr. T. Ramraj Assistant Professor, CIT Coimbatore, India Abstract
More informationPerformance analysis of Integer DCT of different block sizes.
Performance analysis of Integer DCT of different block sizes. Aim: To investigate performance analysis of integer DCT of different block sizes. Abstract: Discrete cosine transform (DCT) has been serving
More informationSteganography: Hiding Data In Plain Sight. Ryan Gibson
Steganography: Hiding Data In Plain Sight Ryan Gibson What Is Steganography? The practice of concealing messages or information within other nonsecret text or data. Comes from the Greek words steganos
More informationFPGA Implementation of 2D DCT Architecture for JPEG Image Compression
FPGA Implementation of 2D DCT Architecture for JPEG Image Compression Prashant Chaturvedi 1, Tarun Verma 2, Rita Jain 3 1 Department of Electronics & Communication Engineering Lakshmi Narayan College
More informationImage Processing. Application area chosen because it has very good parallelism and interesting output.
Chapter 11 Slide 517 Image Processing Application area chosen because it has very good parallelism and interesting output. Lowlevel Image Processing Operates directly on stored image to improve/enhance
More informationA Novel Approach for Image Compression using Matching Pursuit Signal Approximation and Simulated Annealing
A Novel Approach for Image Compression using Matching Pursuit Signal Approximation and Simulated Annealing Ahmed M. Eid Amin ahmedamin@ieee.org Supervised by: Prof. Dr. Samir Shaheen Prof. Dr. Amir Atiya
More informationIMAGE COMPRESSION I. Week VIII Feb /25/2003 Image CompressionI 1
IMAGE COMPRESSION I Week VIII Feb 25 02/25/2003 Image CompressionI 1 Reading.. Chapter 8 Sections 8.1, 8.2 8.3 (selected topics) 8.4 (Huffman, runlength, lossless predictive) 8.5 (lossy predictive,
More informationOverview: motioncompensated coding
Overview: motioncompensated coding Motioncompensated prediction Motioncompensated hybrid coding Motion estimation by blockmatching Motion estimation with subpixel accuracy Power spectral density of
More informationA realtime SNR scalable transcoder for MPEG2 video streams
EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science A realtime SNR scalable transcoder for MPEG2 video streams by Mohammad Alkhrayshah Supervisors: Prof. J.J. Lukkien Eindhoven
More informationREVIEW ON IMAGE COMPRESSION TECHNIQUES AND ADVANTAGES OF IMAGE COMPRESSION
REVIEW ON IMAGE COMPRESSION TECHNIQUES AND ABSTRACT ADVANTAGES OF IMAGE COMPRESSION Amanpreet Kaur 1, Dr. Jagroop Singh 2 1 Ph. D Scholar, Deptt. of Computer Applications, IK Gujral Punjab Technical University,
More informationStatistical Image Compression using Fast Fourier Coefficients
Statistical Image Compression using Fast Fourier Coefficients M. Kanaka Reddy Research Scholar Dept.of Statistics Osmania University Hyderabad500007 V. V. Haragopal Professor Dept.of Statistics Osmania
More informationΝΤUA. Τεχνολογία Πολυμέσων
ΝΤUA Τεχνολογία Πολυμέσων 3. Διάλεξη 3: Transform Coding Rate Distortion Theory D may be the Mean Square Error or some human perceived measure of distortion Types of Lossy Compression VBR Variable Bit
More informationCMPT 365 Multimedia Systems. Media Compression  Video
CMPT 365 Multimedia Systems Media Compression  Video Spring 2017 Edited from slides by Dr. Jiangchuan Liu CMPT365 Multimedia Systems 1 Introduction What s video? a timeordered sequence of frames, i.e.,
More informationIntroduction to Digital Image Processing
Fall 2005 Image Enhancement in the Spatial Domain: Histograms, Arithmetic/Logic Operators, Basics of Spatial Filtering, Smoothing Spatial Filters Tuesday, February 7 2006, Overview (1): Before We Begin
More informationFrom Wikipedia, the free encyclopedia
JPEG Page 1 of 9 From Wikipedia, the free encyclopedia (Redirected from JPEG file format) In computing, JPEG (pronounced JAYpeg) is a commonly used standard method of compression for photographic images.
More informationCoE4TN4 Image Processing. Chapter 8 Image Compression
CoE4TN4 Image Processing Chapter 8 Image Compression Image Compression Digital images: take huge amount of data Storage, processing and communications requirements might be impractical More efficient representation
More informationWavelet Transform (WT) & JPEG2000
Chapter 8 Wavelet Transform (WT) & JPEG2000 8.1 A Review of WT 8.1.1 Wave vs. Wavelet [castleman] 1 01 23 45 67 8 0 100 200 300 400 500 600 Figure 8.1 Sinusoidal waves (top two) and wavelets (bottom
More informationMultimedia Signals and Systems Motion Picture Compression  MPEG
Multimedia Signals and Systems Motion Picture Compression  MPEG Kunio Takaya Electrical and Computer Engineering University of Saskatchewan March 9, 2008 MPEG video coding A simple introduction Dr. S.R.
More informationTKT2431 SoC design. Introduction to exercises. SoC design / September 10
TKT2431 SoC design Introduction to exercises Assistants: Exercises and the project work Juha Arvio juha.arvio@tut.fi, Otto Esko otto.esko@tut.fi In the project work, a simplified H.263 video encoder is
More informationIn the first part of our project report, published
Editor: Harrick Vin University of Texas at Austin Multimedia Broadcasting over the Internet: Part II Video Compression Borko Furht Florida Atlantic University Raymond Westwater Future Ware Jeffrey Ice
More informationCh. 4: Video Compression Multimedia Systems
Ch. 4: Video Compression Multimedia Systems Prof. Ben Lee (modified by Prof. Nguyen) Oregon State University School of Electrical Engineering and Computer Science 1 Outline Introduction MPEG Overview MPEG
More informationData Compression. Media Signal Processing, Presentation 2. Presented By: Jahanzeb Farooq Michael Osadebey
Data Compression Media Signal Processing, Presentation 2 Presented By: Jahanzeb Farooq Michael Osadebey What is Data Compression? Definition Reducing the amount of data required to represent a source
More information1.Define image compression. Explain about the redundancies in a digital image.
1.Define image compression. Explain about the redundancies in a digital image. The term data compression refers to the process of reducing the amount of data required to represent a given quantity of information.
More informationIMAGING. Images are stored by capturing the binary data using some electronic devices (SENSORS)
IMAGING Film photography Digital photography Images are stored by capturing the binary data using some electronic devices (SENSORS) Sensors: Charge Coupled Device (CCD) Photo multiplier tube (PMT) The
More informationImage Processing. Blending. Blending in OpenGL. Image Compositing. Blending Errors. Antialiasing Revisited Computer Graphics I Lecture 15
15462 Computer Graphics I Lecture 15 Image Processing Blending Display Color Models Filters Dithering Image Compression March 18, 23 Frank Pfenning Carnegie Mellon University http://www.cs.cmu.edu/~fp/courses/graphics/
More informationData Representation and Networking
Data Representation and Networking Instructor: Dmitri A. Gusev Spring 2007 CSC 120.02: Introduction to Computer Science Lecture 3, January 30, 2007 Data Representation Topics Covered in Lecture 2 (recap+)
More informationMegapixel Video for. Part 2 of 4. Brought to You by. Presented by Video Security Consultants
rought to You by 2009 Video Security Consultants Presented by Part 2 of 4 A1 Part 2 of 4 How to Avert a Compression Depression Illustration by Jerry King While bandwidth is widening, larger video systems
More informationJPEG Syntax and Data Organization
JPEG Syntax and Data Organization Compressed image data SOI Frame EOI Frame [ Tables/ misc. [ Frame header Scan 1 [ DNL segment [ [ Scan 2 [ [Scan last [ Scan [ Tables/ misc. [ Scan header [ECS 0 RST 0
More informationCSE237A: Final Project MidReport Image Enhancement for portable platforms Rohit Sunkam Ramanujam Soha Dalal
CSE237A: Final Project MidReport Image Enhancement for portable platforms Rohit Sunkam Ramanujam (rsunkamr@ucsd.edu) Soha Dalal (sdalal@ucsd.edu) Project Goal The goal of this project is to incorporate
More informationFRACTAL IMAGE COMPRESSION OF GRAYSCALE AND RGB IMAGES USING DCT WITH QUADTREE DECOMPOSITION AND HUFFMAN CODING. Moheb R. Girgis and Mohammed M.
322 FRACTAL IMAGE COMPRESSION OF GRAYSCALE AND RGB IMAGES USING DCT WITH QUADTREE DECOMPOSITION AND HUFFMAN CODING Moheb R. Girgis and Mohammed M. Talaat Abstract: Fractal image compression (FIC) is a
More informationVideo Codec Design Developing Image and Video Compression Systems
Video Codec Design Developing Image and Video Compression Systems Iain E. G. Richardson The Robert Gordon University, Aberdeen, UK JOHN WILEY & SONS, LTD Contents 1 Introduction l 1.1 Image and Video Compression
More informationSpace Filling Curves and Hierarchical Basis. Klaus Speer
Space Filling Curves and Hierarchical Basis Klaus Speer Abstract Real world phenomena can be best described using differential equations. After linearisation we have to deal with huge linear systems of
More informationNOVEL ALGORITHMS FOR FINDING AN OPTIMAL SCANNING PATH FOR JPEG IMAGE COMPRESSION
NOVEL ALGORITHMS FOR FINDING AN OPTIMAL SCANNING PATH FOR JPEG IMAGE COMPRESSION Smila Mohandhas and Sankar. S Student, Computer Science and Engineering, KCG College of Engineering, Chennai. Associate
More informationDoes everyone have an override code?
Does everyone have an override code? Project 1 due Friday 9pm Review of Filtering Filtering in frequency domain Can be faster than filtering in spatial domain (for large filters) Can help understand effect
More informationA QUADTREE DECOMPOSITION APPROACH TO CARTOON IMAGE COMPRESSION. YiChen Tsai, MingSui Lee, Meiyin Shen and C.C. Jay Kuo
A QUADTREE DECOMPOSITION APPROACH TO CARTOON IMAGE COMPRESSION YiChen Tsai, MingSui Lee, Meiyin Shen and C.C. Jay Kuo Integrated Media Systems Center and Department of Electrical Engineering University
More informationOvercompressing JPEG images with Evolution Algorithms
Author manuscript, published in "EvoIASP2007, Valencia : Spain (2007)" Overcompressing JPEG images with Evolution Algorithms Jacques Lévy Véhel 1, Franklin Mendivil 2 and Evelyne Lutton 1 1 Inria, Complex
More informationPipelined Fast 2D DCT Architecture for JPEG Image Compression
Pipelined Fast 2D DCT Architecture for JPEG Image Compression Luciano Volcan Agostini agostini@inf.ufrgs.br Ivan Saraiva Silva* ivan@dimap.ufrn.br *Federal University of Rio Grande do Norte DIMAp  Natal
More informationCHAPTER 3 DIFFERENT DOMAINS OF WATERMARKING. domain. In spatial domain the watermark bits directly added to the pixels of the cover
38 CHAPTER 3 DIFFERENT DOMAINS OF WATERMARKING Digital image watermarking can be done in both spatial domain and transform domain. In spatial domain the watermark bits directly added to the pixels of the
More informationECE 499/599 Data Compression & Information Theory. Thinh Nguyen Oregon State University
ECE 499/599 Data Compression & Information Theory Thinh Nguyen Oregon State University Adminstrivia Office Hours TTh: 23 PM Kelley Engineering Center 3115 Class homepage http://www.eecs.orst.edu/~thinhq/teaching/ece499/spring06/spring06.html
More informationJPEG Compression Using MATLAB
JPEG Compression Using MATLAB Anurag, Sonia Rani M.Tech Student, HOD CSE CSE Department, ITS Bhiwani India ABSTRACT Creating, editing, and generating s in a very regular system today is a major priority.
More informationImage Processing Tricks in OpenGL. Simon Green NVIDIA Corporation
Image Processing Tricks in OpenGL Simon Green NVIDIA Corporation Overview Image Processing in Games Histograms Recursive filters JPEG Discrete Cosine Transform Image Processing in Games Image processing
More informationApplication of Huffman Coding in Lossless Video Compression
Application of Huffman Coding in Lossless Video Compression Mahessa Ramadhana  13511077 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. Ganesha
More informationEdge and local feature detection  2. Importance of edge detection in computer vision
Edge and local feature detection Gradient based edge detection Edge detection by function fitting Second derivative edge detectors Edge linking and the construction of the chain graph Edge and local feature
More informationIntroduction to Video Coding
Introduction to Video Coding o Motivation & Fundamentals o Principles of Video Coding o Coding Standards Special Thanks to Hans L. Cycon from FHTW Berlin for providing firsthand knowledge and much of
More informationRoll No. :... Invigilator's Signature : GRAPHICS AND MULTIMEDIA. Time Allotted : 3 Hours Full Marks : 70
Name : Roll No. :.... Invigilator's Signature :.. CS/MCA/SEM4/MCA402/2011 2011 GRAPHICS AND MULTIMEDIA Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates
More informationCHAPTER 6 A SECURE FAST 2DDISCRETE FRACTIONAL FOURIER TRANSFORM BASED MEDICAL IMAGE COMPRESSION USING SPIHT ALGORITHM WITH HUFFMAN ENCODER
115 CHAPTER 6 A SECURE FAST 2DDISCRETE FRACTIONAL FOURIER TRANSFORM BASED MEDICAL IMAGE COMPRESSION USING SPIHT ALGORITHM WITH HUFFMAN ENCODER 6.1. INTRODUCTION Various transforms like DCT, DFT used to
More informationData Compression. Guest lecture, SGDS Fall 2011
Data Compression Guest lecture, SGDS Fall 2011 1 Basics Lossy/lossless Alphabet compaction Compression is impossible Compression is possible RLE Variablelength codes Undecidable Pigeonholes Patterns
More informationCompression; Error detection & correction
Compression; Error detection & correction compression: squeeze out redundancy to use less memory or use less network bandwidth encode the same information in fewer bits some bits carry no information some
More informationImage compression. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year
Image compression Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Methods for Image Processing academic year 2017 2018 Data and information The representation of images in a raw
More informationIMAGE COMPRESSION USING HYBRID TRANSFORM TECHNIQUE
Volume 4, No. 1, January 2013 Journal of Global Research in Computer Science RESEARCH PAPER Available Online at www.jgrcs.info IMAGE COMPRESSION USING HYBRID TRANSFORM TECHNIQUE Nikita Bansal *1, Sanjay
More informationQuo Vadis JPEG : Future of ISO /T.81
Quo Vadis JPEG : Future of ISO 109181/T.81 10918/T.81 is still the dominant standard for photographic images An entire toolchain exists to record, manipulate and display images encoded in this specification
More informationEnhancing the Image Compression Rate Using Steganography
The International Journal Of Engineering And Science (IJES) Volume 3 Issue 2 Pages 1621 2014 ISSN(e): 2319 1813 ISSN(p): 2319 1805 Enhancing the Image Compression Rate Using Steganography 1, Archana Parkhe,
More informationDIGITAL IMAGE WATERMARKING BASED ON A RELATION BETWEEN SPATIAL AND FREQUENCY DOMAINS
DIGITAL IMAGE WATERMARKING BASED ON A RELATION BETWEEN SPATIAL AND FREQUENCY DOMAINS Murat Furat Mustafa Oral email: mfurat@cu.edu.tr email: moral@mku.edu.tr Cukurova University, Faculty of Engineering,
More informationImage Manipulation in MATLAB Due Monday, July 17 at 5:00 PM
Image Manipulation in MATLAB Due Monday, July 17 at 5:00 PM 1 Instructions Labs may be done in groups of 2 or 3 (i.e., not alone). You may use any programming language you wish but MATLAB is highly suggested.
More informationIMAGE DATA COMPRESSION
Draft Recommendation for Space Data System Standards IMAGE DATA COMPRESSION Draft Recommended Standard CCSDS 122.0P1.1 Pink Sheets July 2016 Draft Recommendation for Space Data System Standards IMAGE
More information