Denoising and Compression Using Wavelets

Transcription

1 Denoising and Compression Using Wavelets December 15,2016 Juan Pablo Madrigal Cianci Trevor Giannini

2 Agenda 1 Introduction Mathematical Theory Theory MATLAB s Basic Commands De-Noising: Signals De-Noising: Images De-noising: Video Compression Applications of Compression Final Comments

3 introduction 2 It is well known to any scientist and engineer who work with a real world data that signals do not exist without noise, either negligible or not negligible. However, there are many cases in which the noise corrupts the signals in significant manner, and it must be removed from the data in order to proceed with further data analysis. The process of noise removal is generally referred to as signal denoising or simply denoising.

4 introduction: Denoising 3 Since the 1990s, wavelets have been found to be a powerful tool for removing noise from a variety of signals (denoising). They allow to analyze the noise level separately at each wavelet scale and to adapt the denoising algorithm accordingly. Wavelet thresholding methods for noise removal, in which the wavelet coefficients are thresholded in order to remove their noisy part, were first introduced by Donoho in We consider a Gaussian additive noise model, meaning that if our noisy signal can be modeled as y(t) = x(t) + ɛ µ,σ, that is, our noisy signal (or image) is composed by a clean image plus some random noise with mean µ and standard deviation σ.

5 introduction: Denoising 4 The main idea is then to decompose the wave using the DWT. Then we use a threshold for the coefficients and as such get rid of some of them. After obtaining the coefficients that we decide to keep, we the recostruct our image using the IDWT

6 introduction: Compression 5 Now, let s talk about compression. The goal of compression is to store image data in as little space as possible in a file. Used in JPEG files. The compression method is as follows. 1. First, we use a wavelet transform. This will produce as many coefficients as there are pixels in the image. 2. These coefficients can then be compressed more easily because the information is statistically concentrated in just a few coefficients. 3. After that, the coefficients are quantized and the quantized values are entropy encoded and/or run length encoded.

7 introduction: Compression 6

8 Mathematical Aspects 7 {ψ n,m } is a Complete System L 2 (R 2 ) Let ψ n,m,j,k (x)(y) := ψ m,k (x)ψ m,j (y), where n, m, j, k Z. We know that ψ n,k (x) forms an orthonormal basis in L 2 (R). Thus we need to show that ψ n,m,k,j (x)(y) forms an orthonormal basis in L 2 (R 2 ). Thus we want to show that given a f L 2 (R 2 ), f (x, y) = f, ψ n,m ψ n,m (x, y), n Z m Z where ψ n,m (x, y) = ψ n (x)ψ m (y).

9 Mathematical Aspects 8 {ψ n,m } is a Complete System L 2 (R 2 ) Define f (x, y) := f x (y). We start by fixing an x R. Note that since f L 2 (R 2 ) then f (x, y) = f x (y) L 2 (R). for almost every x R. Let f x (y) = n Z f x, ψ m ψ m (y). (1) = f (x, y) = n Z f x, ψ m ψ m (y), (2) (3) where the last equality is made in the L 2 (R) sense. Moreover, let F m (x) = f x, ψ m = f (x, z)ψ m (z)dz L 2 (R), since f L 2 (R 2 ). (4)

10 Juan Pablo where Madrigal Cianci we Trevor used Giannini Cauchy-Schwarz Denoising and Compression Using fromwavelets equation (7) to equation (8) Mathematical Aspects 9 {ψ n,m } is a Complete System L 2 (R 2 ) Moreover, since ψ n (x) is a basis in L 2 (R) and f m (x) L 2 (R), we have that F m (x) = n Z F m, ψ n ψ n (x). (5) We need to show that F m (x) L 2 (R). To do so we compute F m 2 = F m (x) 2 dx = ( = f (x, z) 2 dz f (x, z)ψ m (z)dz 2 dx (6) ) ψ m (z) 2 dz dx (7) f (x, z) 2 dzdx = f (x, z) 2 <, (8) = F m (x) L 2 (R), (9)

11 Mathematical Aspects 10 {ψ n,m } is a Complete System L 2 (R 2 ) Also, note that F m, ψ n L 2 (R) = F m (x)ψ n (x)dx = (10) f (x, z)ψ m (z)ψ n (x)dzdx, (11) where we used Foubini s theorem between (11) and (12)

12 Mathematical Aspects 11 {ψ n,m } is a Complete System L 2 (R 2 ) Thus f (x, y) = ( ) F m, ψ n L2 (R)ψ n (x) ψ m (y) (By eq. (6)) (12) m Z n Z = f, ψ m ψ n L2 (R )ψ 2 n (x)ψ m (y) (13) m Z n Z = f, ψ n,m L2 (R )ψ 2 m,n (x, y) (14) m Z n Z (15) = ψ m,n (x, y) is a complete system in L 2 (R 2 ).

13 Mathematical Aspects 12 {ψ n,m } is a Complete System L 2 (R 2 ) Thus having show that it is indeed a basis, we want to show that it s also orthonormal, i.e, To do so consider ψ n,m, ψ p,q = = = ψ n,m, ψ p,q = δ (p,q),(n,m). (16) ψ p (x)ψ q (y)ψ n (x)ψ m (y)dxdy (17) ( ) ψ q (y)ψ m (y) ψ p (x)ψ n (x)dx dy (18) ψ q (y)ψ m (y)δ p,n dy = δ p,n ψ q (y)ψ m (y)dy (19) = δ p,n δ q,m = δ (p,q),(n,m) (20) Therefore, we have that {ψ n,m } is an orthonormal basis in L 2 (R 2 ).

14 Mathematical Aspects 13 Proof that they form an Orthogonal MRA Let V j = v j v j. Naturally, this means that V j+1 = v j+1 v j+1. We want to show V j V j+1. We assume that we are in an FIR. Thus it is sufficient to show that the basis elements of V j are in V j+1.

15 Mathematical Aspects 14 Proof that they form an Orthogonal MRA Let s examine the basis elements of V j. V j = span{φ j,k (x)φ j,n (y)} = span{φ j,k (x)φ j,n (y)} (21) = span{φ j,k,n (x, y)} = { n Z a j,k,n φ j,k,n (x, y) n Z a j,k,n 2 < }. (22) Where the right hand side of (22) is closed since we have all possible products here, including the limit of sequences.

16 Mathematical Aspects 15 Proof that they form an Orthogonal MRA Moreover, we have that φ j,k,b (x, y) are the basis elements. Without loss of generality, we assume that j = 0, otherwise the arguement is identical but the notation might vary slightly. As we know, the scaling functions in 1-D were φ(x) and φ(y). We define our scaling function in 2-D as φ(x, y) := φ(x)φ(y). Clearly, since φ(x), φ(y) V 0, then φ(x, y) V 0 V 0.

17 Mathematical Aspects 16 Proof that they form an Orthogonal MRA Recall that φ j,n (x) = 2 J/2 φ(2 J x n). This in turn implies that φ j,n,k (x, y) = 2 J φ(2 J x n)φ(2 J y k) = 2 J φ(2 j x m, 2 J y k) (23) = φ 0,n,k (x, y) = φ(x n, y k), (24) for j = 0. We thus need to show that φ 0,n,k (x, y) V 1 V 1. Recall that {φ 0,n (x)} is an orthonormal basis for V 1 and that the set of scaled-translates of φ(2x), {2 1/2 φ(2x n)} form an orthonormal basis for V 1.

18 Mathematical Aspects 17 Proof that they form an Orthogonal MRA Thus a basis for V 0 V 0 is given by {φ 1,k (x)φ 1,n (y)} = {φ 1,n,k (x, y)} = {2φ(2x n, 2y k)} (25) Thus, we need to show that φ(x n, y k) = 2 p Z h p,q φ(2x p, 2y p), which is a finite sum since we assumed that we had an FIR. Thus, φ(x) = φ 0,0 (x) = p Z h pφ 1,p (x) is a finite sequence. q Z

19 Mathematical Aspects 18 Proof that they form an Orthogonal MRA This in turn implies that φ 0,n (x) = φ(x n) = p Z h p φ 1,p (x n) = p Z h p 2 1/2 φ(2(x n) p) (26) = h p 2 1/2 φ(2x (2n + p)). Let r = 2n + p = p = r 2n (27) p Z h r 2n φ(2x r) = h r 2n φ 1,r (x) (28) r Z r Z = φ 0,n (x) = h r 2n φ 1,r (x). (29) r Z

20 Mathematical Aspects 19 Proof that they form an Orthogonal MRA Similarly, φ 0,k (x) = s Z h s 2kφ 1,s (y), which implies that φ 0,n,k (x, y) = φ 0,n (x)φ 0,k (y) = r Z h r 2n h s 2k φ 1,r (x)φ 1,s (y) (30) r Z = r Z h r,s,n,k φ 1,r,s (x, y) (31) r Z = φ 0,n,k V 1 V 1 (32)

21 Mathematical Aspects 20 Fast Wavelet Transform. The Fast Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain int o a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain.

22 Mathematical Aspects 21 Fast Wavelet Transform (Cont.). It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA). In the terms given there, one selects a sampling scale J with sampling rate of 2 J per unit interval, and projects the given signal f onto the space V J ; in theory by computing the scalar products s (J) n = 2 J f (t), φ(2 J t n). (33) where φ is the scaling function of the chosen wavelet transform

23 Mathematical Aspects 22 Fast Wavelet Transform (Cont.). in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so P J [f ](x) = n Z s (J) n φ(2 J x n) is the orthogonal projection or at least some good approximation of the original signal in V J. The MRA is characterized by its scaling sequence a = (a N,..., a 0,..., a N ) and its wavelet sequence b = (b N,..., b 0,..., b N ) (some coefficients might be zero). Those allow to compute the wavelet coefficients d (k) n, at least some range k = M,..., J 1, without having to approximate the integrals in the corresponding scalar products. Instead, one can directly, with the help of convolution and decimation operators, compute those coefficients from the first approximation s (J).

24 Mathematical Aspects 23 Mallat s Algorithm. In 1988, Mallat produced a fast wavelet decomposition and reconstruction algorithm. The Mallat algorithm for discrete wavelet transform (DWT). It is, in fact, a classical scheme in the signal processing community, known as a two-channel subband coder using conjugate quadrature filters or quadrature mirror filters (QMFs). Denote A i and D i as the i th approximate and detailed coefficients respectively.

25 Mathematical Aspects 24 Mallat s Algorithm (Cont.). The algorithm goes as follows: 1. the decomposition algorithm starts with signal s, next calculates the coordinates of A 1 and D 1, and then those of A 2 and D 2, and so on. 2. The reconstruction algorithm called the inverse discrete wavelet transform (IDWT) starts from the coordinates of A J and D J, next calculates the coordinates of A J 1, and then using the coordinates of A J 1 and D J 1 calculates those of A J 2, and so on. More precisely, the algorithm is given by

26 Mathematical Aspects 25 Mallat s Algorithm (Cont.).

27 MATLAB s Basic Commands 26 MATLAB has its own in-built functions to apply wavelets to signals, images, etc. The wavelet toolbox includes different wavelets to work with. Among them are Haar, Doubechies, symlets (almost symmetric wavelet), and many more.

28 MATLAB s basic Commands 27

29 Signals 28 Basics To denoise a signal (1D Wave) in MATLAB we can use the inbuilt function wden(x,tptr,sorh,scal,n,wname), where (X) is our noisy signal (TPTR) Is the threshold selection rule (SOHR) Specifies if soft or hard thresholding (SCAL) Scale (N) Wavelet level (WNAME) Wavelet Name So let s use it in an example.

30 Signals 29 Result

31 Signals 30 Let s consider two other examples that are not as manufactured: Sound denoising and denoising of a raw a raw signal. The idea is the same as before; we pick a wavelet, a threshold, a level and use MATLAB s Inbuilt function wden. The results are as follows

32 Signals 31 ECG We obtained raw data from an ECG. The data looks like

33 Signals 32 ECG As we can see it s very noisy! in practice, we would like to distinguish 4 peaks Thus, we denoise with a soft threshold and a symlet (modifyed version of a Daubechies wavelet with increased symmetry) and get the following:

34 Signals 33

35 Signals 34 Using different wavelets we get the following figure. Note the big difference between Haar and the other two!

36 Signals 35 Finally, we consider one last type of wave: A noisy audio file. Performing a denoising using a Biorthogonal wavelet and a soft thresholding we get the following:

37 Images 36 Algorithm The code for two dimensional case is a little more complicated. The steps are: 1. Convert our image into a matrix using I=imread( name of file.ext ). 2. Convert our image into gray scale by using I=rgb2gray(I). 3. Obtain the decomposition coefficients by [C,S] = wavedec2(j,level,wname). 4. Use thr = wthrmngr( dw2ddenolvl,... penalhi,c,s,1) in order the obtain the threshold 5. finally, use [XDEN,cfsDEN,dimCFS] = wdencmp( lvd,c,s,wname,level,thr,sorh) in order to perform the denoising.

38 Images 37 The process is as follows: Load the image file into MATLAB and convert it to a matrix using the command X=imread( imagename.ext Obtain the wavelet decomposition of said matrix using the command [C,S]=wavedec2(X,N, wavelet name ) Compute the thresholding using the function thr = wthrmngr( dw2ddenolvl, penalhi,c,s,1) Perform the denoising using wdencmp( lvd,c,s,wname,level,thr,sorh)

39 Image Denoising 38 Let s show some examples. Consider the following picture:

40 Image Denoising 39 Denoising the image using a bi-orthogonal wavelet we get the following result:

41 In which case the noisy image seems to be better than the denoised one! Image Denoising 40 Note, however that the denoised images can get very sensitive with respect to the wavelet we use. If we use a less-smooth wavelet, like Haar, we get the following

42 Image Denoising 41 Let s repeat the experiment with a nicer picture and a larger choice of wavelets. Consider the following picture:

43 Image Denoising 42 Denoising the image with a soft threshold and a varity of wavelets gives the following results

44 Video Denosing 43 Videos are made out of pictures and usually contain between 30 to 60 pictures per second and an underlying sound track. Thus, we can combine the image denoising code and the sound denoising code in order to denoise a video. To do so, we need to extract each of these frames and denoise them one by one. Clearly this is a process that takes time; we need to perform the denoising computation between 30 to 60 times per second of video. An alternative to this would be to implement this process in parallel!

45 Compression 44 Compression algorithm in MATLAB The compression algorithm is fairly similar to the image denoising algorithm. 1. Do the 2-dimensional wavelet composition using [c l] = wavedec2(x,n,w) 2. Perform the compression using [xd,cxd,lxd,perf0,perfl2] = wdencmp(opt,c,l,w,n,thr,sorh,keepapp); The code is implemented in the following slides.

46 Image Compression 45 Consider the following image to which we will perform compression with different threshold Figure: Image from the MATLAB.

47 Image Compression 46 Figure: Different thresholding levels

48 Juan Pablo Compressing Madrigal Cianci Trevor Giannini this image Denoising and for Compression higher Using thresholding Wavelets values we get that Image Compression 47 Now, let s consider a different picture with a higher resolution. Figure: A higher resolution picture of Audrey Hepburn.

49 Image Compression 48 Figure: Compression for different threshold levels.

50 Final Remarks 49 Some final comments: As we can see, we can apply wavelets in order to denoise a signal, an image or even a video! As mentioned in the beginning, Many (if not most) real problems in Engineering and science have some noise associated to them. As you might have noticed, all the denoising we did was un black and white pictures. This is not coincidental; it is more complicated to denoise colored images.

51 Selected References 50 Pereyra, María Cristina, and Lesley A. Ward. Harmonic analysis: from Fourier to wavelets. Vol. 63. American Mathematical Soc., Mallat, Stéphane G., and Zhifeng Zhang. "Matching pursuits with time-frequency dictionaries." IEEE Transactions on signal processing (1993): Rami Cohen. Signal Denoising Using Wavelets Technion, Technical Institute of Israel. Misiti, Michel, et al. "Matlab Wavelet Toolbox Userś Guide. Version 3." (2004).

52 Questions? Thank you for your time!