Lossless image compression via by lifting scheme


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Lossless image compression via by lifting scheme

  2. 2. General Concept of Lifting Oddj-1 aj split + P Evenj-1 - (even j-1;odd j-1)=split(s j-1) d j-1 =odd j-1 -P(even j-1) even j-1 =even j-1 +U(odd j-1) smooth U detail The lifting scheme allows for an in-place transformation on a signal: prediction replaces the odd values and update replaces the even using the prediction results.
  3. 3. Three Steps in lifting scheme a)Split step x[n] is split to two subsets, one even (s0[n]) and other odd (d0[n]). This splitting is called lazy wavelet transform, and is used for the ease of reconstruction. b)Predict step Now for a sparser approximation of one of x[n] subsets the next step with linear combination of elements in one subsequence is used to predict the values of the other subsequence. Here it is assumed that two subsequences produced in the splitting step are correlated. If correlation present in x[n] is high, the predicted values will be closer to the actual ones. Thus d[n] is predicted using samples in s[n] and replaces the samples in d[n] using prediction error i.e. given by d[n] ← d[n] - P(s[n]). The predict step results in the loss of some basic properties of the signal like mean value, which needs to be preserved. c)Update step The update step lifts the even sequence values using the liner combination of the predicted odd sequence so that the basic properties of the original sequence is preserved. Therefore the data in s[n] is updated using data in d[n] i.e. given by s[n] ← s[n] + U (d[n]).
  4. 4.  Lossless compressive image used in • • • • •    Medical images Seismic data Satellite images Manuscript images Heavily edited images. A common characteristic of most images is that the neighboring pixels are correlated, and therefore contain redundant information. The foremost task then is to find less correlated representation of the image. Spatial Redundancy or correlation between neighboring pixel values. Spectral Redundancy or correlation between different color planes or spectral bands. Temporal Redundancy or correlation between adjacent frames in a sequence of images (in video applications). Image compression research aims at reducing the number of bits needed to represent an image by removing the spatial and spectral redundancies as much as possible.
  5. 5. Image Compression System. Image Decompression System.
  6. 6. INTEGER FAST FOURIER TRANSFORM Basic butterfly structure for N-point FFT using split-radix structure. Lifting scheme that allows perfect reconstruction
  7. 7.  Converting complex numbers into real lifting steps Consider a complex number that is quantized to have a magnitude 1 then its represented as a=c+js were c and s are real numbers and c2 +s 2 = 1 and let aq be quantised version of a such that aq = cq + jsq where cq and sq are finite wordlength approximations of c and s. hence the reciprocal of aq is 1 = cq + s q aq [aq ]2 [aq ]2 In general, [aq ] is not one, although [ a]=1 . Instead, [aq ] -1 may not even be a finite word length complex number, even though [aq ] is. This is the reason why the conventional fixed-point arithmetic does not preserve the PR property. y =a*x=(cx r-sx i) +j(cx i +sxr) y=[1 j] c s -s -c xr xi where R = c -s s -c
  8. 8.  The PR property can be preserved via the lifting scheme. R can be decomposed into three lifting steps R= c s -s = 1 (c-1)/s 1 0 -c 0 1 s 1 Butterfly structure for implementing a complex multiplication. 1 (c-1)/s 0 1 Lifting structure for implementing a complex multiplication and its inverse.
  9. 9. ADVANTAGES OF LIFTING SYSTEM in IFFT  number of real multiplications is reduced from four to three  it allows for quantization of the lifting coefficients and the quantization of the result of each multiplication without destroying the PR property. To be specific,instead of quantizing a directly, the lifting coefficients c and (c-1)/s are quantized, and therefore, its inverse also consists of three lifting steps with the same lifting coefficients but with opposite signs.  nonlinear operators can also be applied to the product at each lifting step such as rounding or flooring operation. As an example Eight-Point Split-Radix IntFFT is shown.W18 and W38 are implemented using proposed conversion. There are 24 butterflies with coefficients 1, -1, j and –j that do not require any multiplication or addition. These 24 butterflies can be implemented usingcomplex adders or 96 real adders. The rest of the computation is based on the binary representation of the lifting coefficients + 1/(2) ½ and + ((2) ½ -1).
  10. 10. Lattice structure of eight-point IntFFT using split-radix structure
  11. 11.  The lifting scheme is a simple method for designing • • • • customized biorthogonal wavelets and offers several advantages: Allows a faster implementation of the wavelet transform, Saves storage by providing an in- place calculation of the wavelet transform, Simplifies determining the inverse wavelet transform, Provides a natural way to introduce and think about wavelets.
  12. 12. References:  Swanirbhar Majumder, N. Loyalakpa Meitei, A. Dinamani Singh, Madhusudhan Mishra,” Image Compression Using Lifting Wavelet Transform”  Wade Spires,”Lossless Image Compression Via the Lifting Scheme”  Soontorn Oraintara,Ying-Jui Chen,Truong Q. Nguyen,”Integer Fast Fourier Transform”  A.Alice Blessie, J. Nalini, S.C.Ramesh,”Image Compression Using Wavelet Transform Based on the Lifting Scheme and its Implementation”
  13. 13. The method used here is very simple. There are 5 steps mainly. Firstly the image is under gone lifting wavelet based DWT for the desired number of levels as per the size of the image. This is followed by undergoing zig-zag scan, to convert it to one dimensional format. Then it is uniformly quantized and encoded using the run length encoding (RLE). Finally the RLE encoded data is again encoded using Huffman coding such that the Huffman dictionary has a length equal to the one more than the number of quantization levels used. The decompression method is just the reverse of the compression method .Here compression is mainly achieved by removing spectral redundancy in the DWT domain, and achieving some amount of loss of data due to quantization. But as both the encoding methods used are lossless so no data loss is undergone due to the encoding and decoding steps. Moreover it has been seen that more amount of compression is achieved if Huffman encoding is undergone after RLE. Thus they have been used in this order.