TERM PAPER ON The Compressive Sensing Based on Biorthogonal Wavelet Basis Submitted By: Amrita Mishra 11104163 Manoj C 11104059 Under the Guidance of Dr. Sumana Gupta Professor Department of Electrical Engineering Indian Institute of Technology Kanpur
ABSTRACT: Compressive Sensing is one of the latest tools for simultaneous sensing and compression of data. It enables a significant reduction in the sampling and computation costs for signals having sparse representation in some basis. In this term paper, we use wavelet transformations such as Haar, db4, db6 and db8 wavelets for implementation of CS. We also test several wavelet bases from Biorthogonal family. The Error Ratio between the original coefficient and the reconstructed coefficient, the PSNR of the original image and reconstructed image, and the Elapsed Time were used as the measurement indexes. Lena along with CVX and l1-regularised least squares optimizer (in MATLAB) is used in experimental for obtaining the results. The section Discussion and Results talks about the observations and inferences in detail. INTRODUCTION: COMPRESSIVE SENSING Conventional approaches to sampling signals or images follow Shannon s celebrated theorem: the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate). In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, and so on. (For some signals, such as images that are not naturally bandlimited, the sampling rate is dictated not by the Shannon theorem but by the desired temporal or spatial resolution. However, it is common in such systems to use an antialiasing low-pass filter to bandlimit the signal before sampling, and so the Shannon theorem plays an implicit role.) In the field of data conversion, for example, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation: the signal is uniformly sampled at or above the Nyquist rate. The theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Sparsity expresses the idea that the information rate of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed in the proper basis Ψ. Incoherence extends the duality between time and frequency and expresses the idea that objects having a sparse representation in Ψ must be spread out in the domain in which they are acquired, just as a Dirac or a spike in the time domain is spread out in the frequency domain. Put differently, incoherence says that unlike the signal of interest, the sampling/sensing waveforms have an extremely dense representation in Ψ.
The crucial observation is that one can design efficient sensing or sampling protocols that capture the useful information content embedded in a sparse signal and condense it into a small amount of data. These protocols are non-adaptive and simply require correlating the signal with a small number of fixed waveforms that are incoherent with the sparsifying basis. What is most remarkable about these sampling protocols is that they allow a sensor to very efficiently capture the information in a sparse signal without trying to comprehend that signal. Further, there is a way to use numerical optimization to reconstruct the full-length signal from the small amount of collected data. In other words, CS is a very simple and efficient signal acquisition protocol which samples in a signal independent fashion at a low rate and later uses computational power for reconstruction from what appears to be an incomplete set of measurements. Mathematical Framework of CS technique 1. Acquire n << N measurements, using a special sampling matrix Φ, by computing for a signal x 2. Since the dimension of vector of the acquired samples y is substantially smaller than the dimension of the signal, we obviously obtain some initial compression, which can be further augmented by applying lossy or lossless compression to the vector y. 3. Similarly to standard transform-based compression techniques, the paradigm of CS is based on the assumption that the signal x has a sparse representation in some basis such as wavelets. This means that we assume that there exists a known fixed transform T, such that from the N (or more) transform coefficients c = Tx, only k < n coefficients are significant. Working under this sparsity assumption an approximation to x can be reconstructed from y by sparsity minimization, such as 1 l minimization A key assumption in the theory of CS is that the sampling process determined by the matrix Φ and the sparsity transform T are incoherent. Roughly speaking, this means that if a signal has a sparse representation in one, then it must have a dense representation in the other and visa versa, but a signal cannot have a sparse representation in both.
Figure 1: (a) Sparse wavelet representation of an image. Black- significant coefficient, white insignificant coefficient (b) JPEG2000 compressed image based on the sparse representation of (a) BIORTHOGONAL WAVELET FAMILIES The discrete wavelet transform can be represented in matrix form as equation, where Ψ is a matrix with columns corresponding to othonormal scaling and wavelet basis vectors. The functions that qualify as orthonormal wavelets, such as Daubechies wavelets, lack desirable symmetry properties. The Biorthogonal wavelets use two different wavelet bases, ψ(x) and. One is used for decomposition (analysis) and the other one for reconstruction (synthesis) i.e We then have, and for decomposition for reconstruction.
The two scaling functions given in the frequency domain are and the wavelets are and where The biorthogonal wavelets for the forward two-dimentional transform are given as For the inverse transform WAVELET TREE STRUCTURE COMPRESSIVE SENSING The wavelet coefficients may be represented in terms of a tree structure. To a real image, most of the significant wavelet coefficients are located in the vicinity of edges. Wavelets can be regarded as multi-scale local edge detectors. The absolute value of a wavelet coefficient corresponds to the local strength of the edge. The 3 level wavelet tree structure is depicted in Figure 2 and the wavelet tree structure of Lena is shown in Figure 3. The coefficients at the highest of the left in Figure 1 correspond to root nodes. And the coefficients at the bottom or right in the figure correspond to leaf nodes. The top-left block corresponds to the scaling coefficients which capture the coarse-scale representation of the image. Each wavelet coefficient has four children coefficients at the next level, and it has the statistical relationship between the parent and children coefficients. This could be exploited in the CS inversion model.
Figure 2: The 3 level wavelet tree structure Figure 3: The 3 level wavelet tree structure of Lena. (Two wavelet trees are shown in the figure) A wavelet coefficient has a small value, then its children coefficients are likely to also negligible. The statistics of the wavelet coefficients may be represented by the hidden Markov tree. The structure of the wavelet tree is exploited explicitly. Each wavelet coefficient is assumed to be drawn from one of two zero-mean Gaussian distributions in hidden Markov tree. These distributions define the observation statistics for two hidden states. One of the states is a low state, defined by a small Gaussian variance. And the high state is defined by a large variance. If a wavelet coefficient is relatively small, it is more likely to reside in the low state. A large wavelet coefficient has a high probability of coming from the high state. We do not get the wavelet coefficients directly in compressive sensing. We only get the projections of these coefficients. The form of the hidden Markov tree will be used in the compressive sensing inversion. If a given coefficient is negligible, we can scale its children coefficients as zero. The subtrees of these wavelet coefficients may all be set to zero with a little effect on the reconstruction accuracy.
Experiment: We performed compressive sampling on the standard Lena image. Due to computational limitations, we resized the Lena image to 64 64 pixels and performed compressive sampling on the resized image. The original Lena image used is shown in Figure 4. The experiments were carried out in MATLAB 7.11.0 running on PC with Intel Core i7 2.93 GHz CPU with 4 GB RAM. Figure 4: Original Lena image CS was performed using various wavelets Haar, Daubechies-4 (db4), Daubechies-6 (db6), Daubechies-8 (db8), biorthogonal 2.8 and biorthogonal 3.5 wavelets. We also used CVX and l1-regularized lease squares solver (l1_ls) for reconstructing the image from the compressed samples. The performance indicators Peak Signal to Noise Ratio (PSNR), time taken and the visual appearance of the reconstructed image were used for comparing the performance of various wavelets and the solvers. We reconstructed the 64 64 image (4096 coefficients) using 750, 1000, 1250, 1500, 1750, 200, 2500, 3000, 3500, 4000 measurements. Results: The experimental results obtained by compressing the Lena image using CVX are shown in Table 1. The performance indicators time taken in seconds, PSNR in db and the reconstructed image are tabulated. As expected, the PSNR and the image quality increase as the number of measurements increases. Among the several wavelets considered, we can observe that the biorthogonal wavelets give a good reconstruction. The time taken for reconstruction of biorthogonal 3.5 wavelets is the least compared to other wavelets. Table 1: Experimental results obtained using CVX. No. of measurements haar db4 db6 db8 bior2.8 bior3.5 Time 97 99 193 189 118 99 PSNR 12.88 13.17 13.11 12.88 13.03 12.88 750 1000 Time 162 178 305 303 200 182 PSNR 13.00 13.18 12.79 12.91 12.65 12.74
1250 1500 1750 2000 2500 3000 3500 4000 Time 270 279 389 352 300 273 PSNR 13.26 13.18 13.06 13.47 13.12 12.60 Time 383 397 52 529 467 386 PSNR 12.98 13.38 13.42 13.29 13.47 12.81 Time 513 551 714 648 605 529 PSNR 13.46 13.70 13.5 13.52 13.20 13.05 Time 723 722 847 871 733 712 PSNR 14.16 14.22 14.11 14.17 13.34 12.65 Time 1096 1167 1268 1397 1157 1185 PSNR 15.44 16.66 16.10 16.29 13.96 13.28 Time 1583 1695 1972 1981 1806 1844 PSNR 22.85 21.49 20.68 21.39 19.90 16.31 Time 2342 2236 2387 2423 2327 2219 PSNR 30.26 28.09 28.08 28.25 28.92 28.25 Time 2641 2597 2769 2728 2879 2687 PSNR 43.74 43.12 41.99 41.68 44.20 43.50 We also performed the same experiment using l1-regularised least squares solver (l1_ls) to reconstruct the image from compressed samples. The results corresponding to l1_ls are tabulated in Table 2.
Table 2: Experimental results obtained using l1-regularised least squares solver. No. of measurements haar db4 db6 db8 bior2.8 bior3.5 Time 142 178 231 178 168 375 PSNR 14.30 14.27 14.58 14.24 13.69 14.61 750 1000 1250 1500 1750 2000 2500 3000 Time 277 241 351 233 476 635 PSNR 15.23 15.00 15.36 15.29 14.36 15.23 Time 300 326 425 395 483 721 PSNR 15.86 15.78 15.75 15.51 15.69 15.73 Time 388 368 357 462 582 1505 PSNR 16.73 16.98 15.43 17.03 16.96 16.20 Time 469 564 431 447 596.85 1237 PSNR 18.31 18.04 17.70 17.81 17.03 17.24 Time 590 480 641 529 836 981 PSNR 19.78 18.29 18.04 17.46 18.61 16.00 Time 813 631 870 592 1155 24110 PSNR 21.38 20.09 21.33 20.00 20.60 20.18 Time 1103 1097 713 1128 1065 2784 PSNR 26.72 23.13 24.78 24.63 24.29 22.92
3500 4000 Time 1070 1211 1214 1278 2293 3192 PSNR 32.67 29.84 29.24 27.73 29.68 28.04 Time 962 1024 1023 1035 1293 3523 PSNR 44.87 42.05 41.77 40.57 41.58 40.68 We can observe that l1_s gives better performance in terms of both PSNR and time taken. We also conclude that the compression performance depends on the solver used to reconstruct the image from the compressed samples. This is the reason why the performance observed is different from that given in the paper. Contribution: The reconstructed image is not of very good quality when number of measurements is less. So, we tried denoising the image with by applying Total Variation denoising. Total variation denoising was applied on images reconstructed using l1_ls only and the results are shown in Table 3. Table 3: Experimental results obtained using l1-regularised least squares solver. No. of measurements haar db4 db6 db8 bior2.8 bior3.5 Time 142 178 231 178 168 375 PSNR 15.49 15.33 15.56 15.21 14.55 15.06 750 1000 1250 1500 Time 277 241 351 233 476 635 PSNR 15.72 15.74 16.62 15.82 15.97 15.90 Time 300 326 425 395 483 721 PSNR 17.68 17.04 17.05 17.19 16.72 16.69 Time 388 368 357 462 582 1505 PSNR 17.57 17.62 17.42 17.64 18.00 17.29
1750 2000 2500 3000 3500 4000 Time 469 564 431 447 596.85 1237 PSNR 19.22 19.16 18.89 18.61 18.01 17.89 Time 590 480 641 529 836 981 PSNR 20.26 19.79 19.62 19.99 19.79 18.79 Time 813 631 870 592 1155 24110 PSNR 21.27 19.82 20.47 20.43 19.79 20.61 Time 1103 1097 713 1128 1065 2784 PSNR 22.44 22.80 22.67 22.72 22.90 22.41 Time 1070 1211 1214 1278 2293 3192 PSNR 23.95 23.58 23.36 23.19 23.69 23.57 Time 962 1024 1023 1035 1293 3523 PSNR 24.13 24.14 24.12 24.11 24.11 24.11 From the results, we can infer that the total variation filtering improves the image quality and PSNR when number of measurements taken is less. The improvement in PSNR is around 1-2 db. When the number of measurements increases, the image reconstructed is almost similar to the original image. So, the total variation filtering has resulted in blurring of the image, resulting in a decreased PSNR. Since compression and compressive sampling aim at reducing the number of measurements taken, number of measurements taken in practice is very less. So, the PSNR can be increased by using denoising after reconstructing the image.
Conclusion: In this paper, we test the quality of Haar, db4, db6, db8,bior2.8 and bior3.5 wavelet basis for the implementation of CS and it is observed that Biorthogonal wavelets given a better reconstruction. We used both CVX and l1-regularised least squares optimizer (in MATLAB) to perform the optimisation step in CS and observe that l1-regularised least squares optimizer gave better performance with respect to both PSNR and time taken. We observed that the reconstructed image is not of very good quality when number of measurements is less. So, we tried denoising the image with by applying Total Variation denoising (results provided only for l1-regularised least squares optimized images) and observed the PSNR can be increased by using denoising after reconstructing the image. We also concluded that the type of wavelet basis used in CS depends upon the image to be compressed. For an image with more details like Lena, the best performance was achieved for Biorthogonal Wavelets. It can be seen that for a image with fewer details other wavelets like Daubechies also do a fair job. Even the number of pixels required for proper reconstruction would be less for an image with fewer details.