Wavelets & Mul,resolu,on Analysis

Transcription

1 Wavelets & Mul,resolu,on Analysis Square Wave by Steve Hanov More comics at Problem set #4 will be posted tonight 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 1

2 Changing Basis One of our primary tools for processing signals has been to transform them to an alternate representa8on basis. Our mo8va8ons were: Alternate basis beger supported certain signal opera8ons (i.e. convolu8on via Fourier) Under the new basis fewer parameters captured more of the signal s intrinsic varia8on (PCA) Parameters under the new basis beger represented natural constraints (CIE color space) 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 2

3 Basis Sets A basis consists of a minimum set of func8ons/vectors from which any other vector can be generated via linear combina8ons Today, we ll discuss these basis func8ons, denoted, φ i (t) explicitly and discuss some of their proper8es. The simplest basis set is merely the Kronecker delta basis φ i (t) = δ(t i) i Next we ll consider various proper8es of basis func8ons 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 3

4 Orthogonal Simply stated two vectors are orthogonal if their projec8ons onto each other are 0, i.e. they are perpendicular. For a vector N 1 φ i, φ j = φ i [d] φ j [d] = φ i φ j = 0 where i j and N is the vector' s dimension d = 0 For a func8on φ i (t), φ j (t) = b φ i (t) φ j (t)dt = 0 where i j and a,b a [ ] spans the domain of interest Pairwise orthogonal vectors are linearly independent 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 4

5 Orthogonal Examples All of the Fourier basis vectors/func8ons are orthogonal No8ce how repeated values in one vectors are always paired up with values with opposite signs in any other vector PCA bases vectors (factors) are also orthogonal This is a property of any eigenvector with a dis8nct Eigen values rela8ve to other eignevectors. 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 5

6 Normalized Basis A basis func8on is normalized if φ i (t), φ i (t) =1 Normalized basis vectors/func8ons are used to assure that the parameters are comparable, i.e. no one parameter has a greater influence due to its larger basis In the Fourier case all basis func8ons are normalized, implying iden8cal amplitudes of sinusoids with the same amplitudes. Allows us to compare coefficients (frequency contribu8ons) in a meaningful way. Likewise, PCA factors are normalized (unit length) If all basis func8ons are normalized and orthogonal the the basis set is orthonormal 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 6

7 Support & Localiza,on We ve discussed the concept of support previously, but not in the context of basis func8ons. Support refers to the region of the domain which encloses all nonzero values. If this region is finite, then the func8on is said to have compact support Compact support implies that the influence of a parameter is localized, i.e. if you change the value of a coefficient the func8on is modified in a bounded region The only basis that we ve discussed so far with compact support is the Kronecker delta basis set, not true of Fourier, or PCA factors in general 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 7

8 Mul,resolu,on Basis func8ons can also be related to each other by scaling over the domain, i.e. they are smaller/higher frequency/more compact copies of one another φ i (t) = φ k (σ t) This was the case with the Fourier basis (all sinusoids are related to each other by scaling their frequencies), but not for PCA factors The coefficients of a mul8resolu8on basis tell you something about similari8es across scales. sin(t) sin(2t) 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 8

9 Wavelet Basis A wavelet is a func8on/vector used to correlate a given series or con8nuous 8me signal locally and at a specific scale A wavelet transform is the representa8on of a func8on by a basis set of wavelets. Wavelets in a basis set is composed of scaled and translated copies (known as "daughter wavelets") of a finite length primary basis func8on (known as the "mother wavelet ). Conceptually we can decompose each basis into its scaling part, ϕ i (t), and its analysis waveform, ψ i (t), (mother wavelet) 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis Mother wavelet One daughter Another daughter 9

10 Simplest Possible Wavelet Let s return to the weather analysis example that we considered several weeks ago High temperatures over the last 8 days HighTemp = [61, 57, 69, 65, 73, 78, 77, 72] Recall how a local average and difference captured most of the informa8on in this signal? Ave2Day = [59, 67, 75.5, 74.5] Dif1Day = [2, 2, -2.5, 2.5] Ave4Day = [63, 75] Dif2Day = [-4, 0.5] Ave8Day = [69] Dif4Day = [-6] Note how that with a single sequence we can recover all of the informa8on of this mul8resoul8on decomposi8on: W = Ave8Day + Diff4Day + Diff2Day + Diff1Day = [69, -6, -4, 0.5, 2, 2, -2.5, 2.5] 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 10

11 Bases of this Decomposi,on Haar Wavelet Basis X[i] = c 0 φ c 1 ψ c 2 ψ c 3 ψ c 4 ψ c 5 ψ c 6 ψ c 1 ψ 3 2 = /21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 11

12 Haar Code Code has two parts decomposi8on and reconstruc8on def decompose(signal, maxlevel=-1): n = len(signal) c = 0.5 level = 0 while (n > 1) and (level!= maxlevel): l = [c*(signal[2*i]+signal[2*i+1]) for i in xrange(n/2)] h = [c*(signal[2*i]-signal[2*i+1]) for i in xrange(n/2)] signal = l + h + signal[n:] n >>= 1 level += 1 return signal Haar is orthognal (all wavelets aren t) Can be normalized by seong, and scaling the 1 returned signal by and returned wavelet by len(signal) def construct(wavelet): n = 2 c = 1.0 while n <= len(wavelet): r = [] for i in xrange(n/2): r.append(c*(wavelet[i]+wavelet[n/2+i])) r.append(c*(wavelet[i]-wavelet[n/2+i])) wavelet = r + wavelet[n:] n <<= 1 return wavelet c = 1 2 len(signal) 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 12

13 2 D Haar Basis func,ons? Under the standard basis, examples are: Gray = 0 13

14 Non standard 2D Haar basis 14

15 Image Wavelet Coding Haar wavelet applied to rows and then columns Coefficients near zero, can be quan8zed with minimal loss 11/21/08 Comp 665 Wavelets & Mul8resolu8on Analysis 15

16 Wavelet based Compression No need to to divide input images into blocks and its basis func8ons have variable lengths thus avoiding blocking ar8facts. More robust under transmission and decoding errors. BeGer matched to the HVS characteris8cs Good frequency resolu8on at lower frequencies, good localiza8on at high frequencies good match for natural images.

17 Wavelet Decomposi5on Filter bank Model

18 Energy Compactness

19 EZW: ZeroTree Coding Exploits mother daughter dependencies between subband coefficients that contribute to the same spa8al loca8on Bit plane coding: enables an embedded bitstream wrt distor8on

20 QuadTree Organiza,on Hierarchical decomposi8on 2D analog of a binary tree Coefficients can be thought of as having 4 descendants in the next higher res level. These descendants each also have four descendants in the next higher level. 20

21 Dominant Pass Determines if each coefficient is significant rela8ve to a given threshold T: c ij T In the dominant pass, all as yet insignificant coefficients are examined and declared as either: Significant posi8ve Significant nega8ve Root of a zero tree (Quadtree root and all its children are insignificant) Isolated insignificant Awer each pass, T T/2

22 Subordinate Pass All coefficients previously declared significant are refined by one bit: y es=mate quan8zed to + or T/4 Coefficients are visited by decreasing magnitude, then in some order

23 EZW Example T = 2 log 2 (Max) = 32 Dominant Pass Z 23 T Giving + ZT+TTTTZTTTTTTT+TT 23

24 EZW Example (cont) D1: + ZT+TTTTZTTTTTTT+TT Subordinate Pass Considers only +/ entries > 16 > 1 (writes = 15) < 16 > 0 (writes = 2) > 16 > 1 (writes = 1) < 16 > 0 (writes = 15) S1: 1010 Note all values are now < 32 T = 32/2 = 16 24

25 JPEG old vs JPEG2000 JPEG (old) is based on the Discrete Cosine Transform (DCT). Image is divided into 8x8 blocks At low bit rates, there is a sharp degrada8on with image quality. Peaks near 50:1 compression ra8o

26 JPEG Basics General block diagram of the JPEG (a) encoder and (b) decoder

27 Wavelets in Image Coding Orthogonal vs. Biorthogonal: JPEG 2000 uses biorthogonal filters Cohen Daubechies Feavau filters 9/7 Filters are symmetric/an8 symmetric Nearly orthogonal Symmetric extensions of the input data Representa8on is mul8resolu8on

28 DWT for Image Compression Block Diagram 2D Discrete Wavelet Transform Quantization Entropy Coding 2D discrete wavelet transform (1D DWT applied alternatively to vertical and horizontal direction line by line ) converts images into sub-bands Upper left is the DC coefficient Lower right are higher frequency sub-bands.

29 DWT for Image Compression Image Decomposi,on Parent Children Descendants: corresponding coeff. at finer scales Ancestors: corresponding coeff. at coarser scales LL 3 HL 3 HL 2 LH 3 HH 3 LH 2 HH 2 HL 1 LH 1 HH 1 Parent-children dependencies of subbands: arrow points from the subband of parents to the subband of children.

30 DWT for Image Compression Image Decomposi,on Feature 1: Energy distribu8on concentrated in low frequencies Feature 2: Spa8al self similarity across subbands LL 3 HL 3 LH 3 HH 3 HL 2 HL 1 LH 2 HH 2 LH 1 HH 1 The scanning order of the subbands for encoding the significance map.

31 DWT for Image Compression Differences from DCT Technique In conven8onal transform coding: Anomaly (edge) produces many nonzero coeff. significant energy TC allocates too many bits to trend, few bits lew to anomalies Problem at Very Low Bit rate Coding Block ar8facts DWT Trends & anomalies informa8on available Major difficulty: fine detail coefficients associated with anomalies the largest no. of coeff. Problem: how to efficiently represent posi=on informa8on?

32 Progressive Transmission Coefficients are ordered in the bit stream so that largescale and low resolu8on versions of the image arrive first

33 Region of Interest Coding Different image regions can be encoded at different quali8es Scale coefficients such that the bits corresponding to ROI are in the higher bit plane.

34 Scalability SNR scalability and spa8al scalability The ability to achieve coding of more than one quality Resilience to transmission errors, no need to know target bit rate/resolu8on, no need for mul8ple compressions

35 SNR Scalability The bit stream can be decompressed at different quality levels (SNR) Decompressed image bike at (a) b/p, (b) 0.25 b/p, (c) 0.5 b/p

36 Spa,al Scalability The bit stream can be decompressed at different resolu8on level

37 Set the Mean pixel value to 0 Applica,on: Edge Detec,on

38 Applica,on: Image Denoising Noisy Image: Denoise Image: Ignore all Φ Coefficients below some threshold

39 Denoising Images Transform the image to the wavelet domain Apply a coefficient threshold of two standard devia8ons of the image s variance Inverse transform Correct back to the mean by adding in a global constant

40 VisuShrink Level/Scale dependent thresholding Apply Donoho s universal threshold, M is the number of pixels represented at the given level. The threshold is usually high, overly smooth.

41 Image Enhancement Image contrast enhancement with wavelets, especially important in medical imaging Idea is to make the small coefficients very small and the large coefficients very large (i.e. increase the contrast) Applies a nonlinear mapping func8on to the coefficients.

42 Experiments

43 Denoising and Enhancement Apply DWT Shrink transform coefficients in finer scales to reduce the effect of noise Emphasize features within a certain range using a nonlinear mapping func8on Perform IDWT to reconstruct the image.

44 Examples

45 Other Applica,ons Computer vision Image processing in the human visual system has a complicated hierarchical structure that involves several layers of processing. At each processing level, the re8nal system provides a visual representa8on that scales progressively in a geometrical manner. Intensity changes occur at different scales in an image, so that their op8mal detec8on requires the use of operators of different sizes. Therefore, a vision filter have two characteris8cs: it should be a differen8al operator, and it should be capable of being tuned to act at any desired scale. Wavelets are ideal for this

46 FBI Fingerprint Compression A single fingerprint is about 700,000 pixels, and requires about 0.6MBytes.