induction to wavelet transform and image compression

cung cấp kiến thức về biến đổi wavelet trong xử lý ảnh

Introduction to Wavelet Transform and Image Compression Student: Kang-Hua Hsu 徐康華 Advisor: Jian-Jiun Ding 丁建均 E-mail: r96942097@ntu.edu.tw Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC DISP@MD5 Outline (1)  Introduction  Multiresolution Analysis (MRA) - Subband Coding - Haar Transform - Multiresolution Expansion  Wavelet Transform (WT) - Continuous WT - Discrete WT - Fast WT - 2-D WT  Wavelet Packets  Fundamentals of Image Compression - Coding Redundancy - Interpixel Redundancy - Psychovisual Redundancy - Image Compression Model DISP@MD5 Outline (2)  Lossless Compression - Variable-Length Coding - Bit-plane Coding - Lossless Predictive Coding  Lossy Compression - Lossy Predictive Coding - Transform Coding - Wavelet Coding  Conclusion  Reference DISP@MD5 Introduction(1)-WT v.s FT Bases of the • FT: time-unlimited weighted sinusoids with different frequencies No temporal information • WT: limited duration small waves with varying frequencies, which are called wavelets WTs contain the temporal time information Thus, the WT is more adaptive DISP@MD5 Introduction(2)-WT v.s TFA • Temporal information is related to the time-frequency analysis • The time-frequency analysis is constrained by the Heisenberg uncertainty principal • Compare tiles in a time-frequency plane (Heisenberg cell): DISP@MD5 Introduction(3)-MRA • It represents and analyzes signals at more than one resolution • related operations with ties to MRA:  Subband coding  Haar transform • MRA is just a concept, and the wavelet-based transformation is one method to implement it DISP@MD5 Introduction(4)-WT • The WT can be classified according to the of its input and output  Continuous WT (CWT)  Discrete WT (DWT) • 1-D • DWT DISP@MD5 2-D transform (for image processing) Fast WT (FWT) recursive relation of the coefficients MRA-Subband Coding(1) • Since the bandwidth of the resulting subbands is smaller than that of the original image, the subbands can be downsampled without loss of information • We wish to select h ( n ) , h ( n ) , g ( n ) , g ( n ) so that the input can be perfectly reconstructed Biorthogonal Orthonormal DISP@MD5 1 MRA-Subband Coding(2) • Biorthogonal filter bank:          g [ k ] , h0 [ 2n − k ] = δ [ n ] g [ k ] , h1 [ 2n − k ] = g1 [ k ] , h1 [ 2n − k ] = δ [ n ] g1 [ k ] , h0 [ 2n − k ] = • Orthonormal (it’s also biorthogonal) filet bank:  g1 (n) = (−1) n g (2 K − − n)   i = {0,1} : time-reversed relation hi (n) = gi (2 K − − n),  ,where 2K denotes the number of coefficients in each filter • The other filters can be obtained from one prototype filter DISP@MD5 MRA-Subband Coding(3) • 1-D to 2-D: 1-D two-band subband coding to the rows and then to the columns of the original image • Where a is the approximation (Its histogram is scattered, and thus lowly compressible.) and d means detail (highly compressible because their histogram is centralized, and thus easily to be modeled) FWT can be implemented by subband coding! DISP@MD5 10 Lossless Compression It can be reconstructed without distortion • No quantizer involves in the compression procedure • Generally, the compression ratios range from to 10 • Trade-off relation between the compression ratio and the computational complexity DISP@MD5 33 Variable-Length Coding It assigns fewer bits to the more probable gray levels than to the less probable ones • It merely reduces the coding redundancy • Ex Huffman coding DISP@MD5 34 Bit-plane Coding A monochrome or colorful image is decomposed into a series of binary images (that is, bit planes), and then they are compressed by a binary compression method •It reduces the interpixel redundancy DISP@MD5 35 Lossless Predictive Coding It encodes the difference between the actual and predicted value of that pixel • It reduces the interpixel redundancies of closely spaced pixels • The ability to attack the redundancy depends on the predictor DISP@MD5 36 Lossy Compression It can not be reconstructed without distortion due to the sacrificed accuracy • It exploits the quantizer • Its compression ratios range from 10 to 100 (much more than the lossless case’s) • Trade-off relation between the reconstruction accuracy and compression performance DISP@MD5 37 Lossy Predictive Coding It is just a lossless predictive coding containing a quantizer • It exploits the quantizer • Its compression ratios range from 10 to 100 (much more than the lossless case’s) • The quantizer is designed based on the purpose for minimizing the quantization error • Trade-off relation between the quantizer complexity and less quantization error • Delta modulation (DM) is an easy example exploiting the oversampling and 1-bit quantizer DISP@MD5 38 Transform Coding(1) Most of the information is included among a small number of the transformed coefficients Thus, we truncate or coarsely quantize the coefficients including little information •The goal of the transformation is to pack as much information as possible into the smallest number of transform coefficients •Compression is achieved during the quantization of the transformed coefficients, not during the transformation DISP@MD5 39 Transform Coding(2) •More truncated coefficients Higher compression ratio, but the rms error between the reconstructed image and the original one would also increase •Every stage can be adapted to local image content •Choosing the transform: Information packing ability Computational complexity needed Practical! KLT Information packing ability Best WHT Not good DCT Good Computational complexity High Lowest Low DISP@MD5 40 Transform Coding(3) •Disadvantage: Blocking artifact when highly compressed (this causes errors) due to subdivision •Size of the subimage: Size increase: higher compression ratio, computational complexity, and bigger block size ?? ? ? How to solve the blocking artifact problem? Using the WT! ? DISP@MD5 ? ? 41 Wavelet Coding(1) Wavelet coding is not only the transforming coding exploiting the wavelet transform No subdivision! • No subdivision due to:  Computationally efficient (FWT)  Limited-duration basis functions Avoiding the blocking artifact! DISP@MD5 42 Wavelet Coding(2) • We only truncate the detail coefficients • The decomposition level: the initial decompositions would draw out the majority of details Too many decompositions is just wasting time DISP@MD5 43 Wavelet Coding(3) • Quantization with dead zone threshold: set a threshold to truncate the detail coefficients that are smaller than the threshold DISP@MD5 44 Conclusion The WT is a powerful tool to analyze signals There are many applications of the WT, such as image compression However, most of them are still not adopted now due to some disadvantage Our future work is to improve them For example, we could improve the adaptive transform coding, including the shape of the subimages, the selection of transformation, and the quantizer design They are all hot topics to be studied DISP@MD5 45 Reference [1] R.C Gonzalez, R.E Woods, Digital Image Processing, 2nd edition, Prentice Hall, 2002 [2] J.C Goswami, A.K Chan, Fundamentals of Wavelets, John Wiley & Sons, New York, 1999 [3] Contributors of the Wikipedia, “Arithmetic coding”, available in http://en.wikipedia.org/wiki/Arithmetic_coding [4] Contributors of the Wikipedia, “Lempel-ZivWelch”, available in http://en.wikipedia.org/wiki/Lempel-Ziv-Welch [5] S Haykin, Communication System, 4th edition, John Wiley & Sons, New York, 2001 DISP@MD5 46 47 ... prototype filter DISP@MD5 MRA-Subband Coding(3) • 1-D to 2-D: 1-D two-band subband coding to the rows and then to the columns of the original image • Where a is the approximation (Its histogram... represents and analyzes signals at more than one resolution • related operations with ties to MRA:  Subband coding  Haar transform • MRA is just a concept, and the wavelet- based transformation... (MRA) - Subband Coding - Haar Transform - Multiresolution Expansion  Wavelet Transform (WT) - Continuous WT - Discrete WT - Fast WT - 2-D WT  Wavelet Packets  Fundamentals of Image Compression