1. Two-Dimensional Wavelets
ECE 802
Spring 2010

2. Two-Dimensional Wavelets
For image processing applications we need wavelets that are two-dimensional.
This problem reduces down to designing 2D filters.
We will focus on a particular class of 2D filters: separable filters (can be directly designed from their 1D counterparts)

3. 2D Scaling Functions
The theories of multiresolution analysis and wavelets can be generalized to higher dimensions.
In practice the usual choice for a two-dimensional scaling function or wavelet is a product of two one-dimensional functions. For example,
and the dilation equation assumes the form:

4. 2D Wavelet Functions Since both satisfy the dilation equation,
We can analogously construct the wavelets. However, now instead of 1 wavelet function, we have 3 wavelet functions:

5. The corresponding dilation equations are:
where g(I) (k,l)=h(k)g(l), g(II) (k,l)=g(k)h(l),
g(III) (k,l)=g(k)g(l).

6. Example: 2D Haar
2D Haar scaling:
2D Haar wavelets:

7. Subspaces V2j is the two-dimensional subspace at scale j.
As j increases, we get L2(R2).

8. 2D wavelet decomposition
The approximation and detail coefficients are computed in a similar way:
The reconstruction is:

9. Filterbank Structure: Decomposition

10. Filterbank Structure: Reconstruction

11. The Three Frequency Channels
We can interpret the decomposition as a breakdown of the signal
into spatially oriented frequency channels.
Arrangement of wavelet
representations
Decomposition of
frequency support

12. Wavelet Decomposition:Example
LENA

13. Wavelet Example 2

16. Applications: Edge Detection in Images

17. Application: Image Denoising Using Wavelets
Noisy Image:
Denoised Image:

18. Denoising Images Denoising Daubechies’ face:
Transform the image to the wavelet domain using Coiflets with three vanishing moments
Apply a threshold at two standard deviations
Inverse-transform the image.

19. Image Denoising Using Wavelets
Calculate the DWT of the image.
Threshold the wavelet coefficients. The threshold may be universal or subband adaptive.
Compute the IDWT to get the denoised estimate.
Soft thresholding is used in the different thresholding methods. Visually more pleasing images.

20. VisuShrink Apply Donoho’s universal threshold,
M is the number of pixels.
The threshold is usually high, overly smoothing.

21. SUREShrink
Subband adaptive, a different threshold is calculated for each detail subband.
Choose the threshold that will minimize the unbiased estimate of the risk:
This optimization is straightforward, order the wavelet coefficients in terms of magnitude and choose the threshold as the wavelet coefficient that minimizes the risk.

22. BayesShrink Adaptive data-driven thresholding method
Assume that the wavelet coefficients in each subband is distributed as a Generalized Gaussian Distribution (GGD)
Find the threshold that minimized the Bayesian risk.

23. GGD
Shape parameter β, std. σ.

24. BayesShrink Choose the threshold that will minimize the Bayesian risk.
There is no closed form for the threshold.

25. BayesShrink Empirical Threshold
An empirical threshold is used in practice that is very close to the optimum threshold.
Adapts to the SNR in each subband.

26. Comparisons

27. Image Enhancement
Image contrast enhancement with wavelets, especially important in medical imaging
Make the small coefficients very small and the large coefficients very large.
Apply a nonlinear mapping function to the coefficients.

28. Experiments

29. 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 function
Perform IDWT to reconstruct the image.

30. Examples
Original Denoised Denoising with Enhancement

31. Edge Detection
Edges correspond to the singularities in the image and are related to the local maxima of wavelet coefficients.
For edge detection, a smoothing function (such as a spline) and two wavelet functions are defined.
Wavelet functions are usually the first and second order derivatives of the smoothing function.
Examples:
Keep the detail coefficients and discard the approximation coefficients
Edges correspond to large coefficients

32. Applications 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 retinal system provides a visual representation that scales progressively in a geometrical manner.
Intensity changes occur at different scales in an image, so that their optimal detection requires the use of operators of different sizes.
Therefore, a vision filter have two characteristics: it should be a differential operator, and it should be capable of being tuned to act at any desired scale.
Wavelets are ideal for this

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

34. Image Compression and Wavelets

35. Why Compression?
Uncompressed images take too much space, require larger bandwidth for transmission and longer time to transmit
Examples:
512x512 grayscale image: 262KB
512x512 color image: 786KB
The common principle beyond compression is to reduce redundancy: spatial and spectral redundancy

36. Types of Compression
Lossy vs. Lossless: Lossy compression discards redundant information, achieves higher compression ratios. Lossless compression can reconstruct the original image.
Predictive vs. Transform Coding

37. Components of a Coder Source Encoder: Transform the image
DFT,DCT,DWT (linear transforms)
Quantizer: Scalar vs. Vector (lossy coding)
Entropy Encoder: Compresses the quantized values (lossless)

38. Original JPEG
Use DCT to transform the image (real part of DFT)

39. Original JPEG Transform each 8x8 block using DCT
Since adjacent pixels are highly correlated, most of the coefficients are concentrated at lower frequencies.
Quantize the DCT coefficients (uniform quantization) and then entropy encode for further compression

40. Disadvantages of DCT: Why wavelets?
DCT based JPEG uses blocks of image, there is still correlation across blocks.
Block boundaries are noticeable in some cases
Blocking artifacts at low bit rates
Can overlap the blocks Computationally expensive

41. Was JPEG not good enough?
JPEG is based on DCT.
Equal subbands.
At low bit rates, there is a sharp degradation with image quality.
43:1 compression ratio

42. Why Wavelets? No need to block the image
More robust under transmission errors
Facilitates progressive transmission of the image (Scalability)

43. Features of JPEG2000
Multiple Resolution: Decomposes the image into a multiple resolution representation.
Progressive transmission: By pixel and resolution accuracy, referred to as progressive decoding and signal-to-noise ratio (SNR) scalability: This way, after a smaller part of the whole file has been received, the viewer can see a lower quality version of the final picture.
Lossless and lossy compression
Random code-stream access and processing: JPEG 2000 supports spatial random access or region of interest access at varying degrees of granularity. This way it is possible to store different parts of the same picture using different quality.
Error resilience: JPEG 2000 is robust to bit errors introduced by noisy communication channels, due to the coding of data in relatively small independent blocks.

44. JPEG2000 Basics
General block diagram of the JPEG 2000 (a) encoder and (b) decoder

45. Wavelets in Image Coding
Orthogonal vs. Biorthogonal:
JPEG 2000 uses biorthogonal filters
Lossless and lossy compression
Cohen-Daubechies-Feavau filters 9/7
CDF 5/3 for lossless compression (integer)
Filters are symmetric/anti-symmetric
Nearly orthogonal
Symmetric extensions of the input data

46. Steps in JPEG2000
Tiling: The image is split into tiles, rectangular regions of the image that are transformed and encoded separately. Tiles can be any size. Dividing the image into tiles is advantageous in that the decoder will need less memory to decode the image and it can opt to decode only selected tiles to achieve a partial decoding of the image. Using many tiles can create a blocking effect.
Wavelet Transform: Either CDF 9/7 or CDF 5/3 biorthogonal wavelet transform.
Quantization: Scalar quantization
Coding: The quantized subbands are split into precincts, rectangular regions in the wavelet domain. They are selected in a way that the coefficients within them across the sub-bands form approximately spatial blocks in the image domain. Precincts are split further into code blocks. Code blocks are located in a single sub-band and have equal sizes. The encoder has to encode the bits of all quantized coefficients of a code block, starting with the most significant bits and progressing to less significant bits by EBCOT scheme.

47. DWT for Image Compression
Image Decomposition
Parent
Children
Descendants: corresponding coeff. at finer scales
Ancestors: corresponding coeff. at coarser scales
HL1
LH1
HH1
HH2
LH2
HL2
HL3
LL3
LH3
HH3
Parent-children dependencies of subbands: arrow points from the subband of parents to the subband of children.

48. DWT for Image Compression
Image Decomposition
Feature 1:
Energy distribution concentrated in low frequencies
Feature 2:
Spatial self-similarity across subbands
HL1
LH1
HH1
HH2
LH2
HL2
HL3
LL3
LH3
HH3
The scanning order of the subbands for encoding the significance map.

49. DWT for Image Compression
Differences from DCT Technique
In conventional transform coding:
Anomaly (edge) produces many nonzero coeff.
insignificant energy
TC allocates too many bits to “trend”, few bits left to “anomalies”
Problem at Very Low Bit-rate Coding : block artifacts
DWT
Trends & anomalies information available
Major difficulty: fine detail coefficients associated with anomalies  the largest no. of coeff.
Problem: how to efficiently represent position information?

50. Embedded Zerotree Wavelet Compression (EZW)
The zerotree is based on the hypothesis that if a wavelet coefficient at a coarse scale is insignificant, then all wavelet coefficients of the same orientation in the same spatial location at finer scales are likely to be insignificant.
Natural images in general have a low pass spectrum. When an image is wavelet transformed, the energy in the sub-bands decreases with the scale goes higher so the wavelet coefficient will, on average, be smaller in the higher levels.

51. EZW
We can set a threshold T, if the wavelet coefficient is larger than T, then encode it as 1, otherwise we code it as 0.
‘1’ will be reconstructed as T (or a number larger than T) and ‘0’ will be reconstructed as 0.
We then decrease T to a lower value, repeat 1 and 2. So we get finer and finer reconstructed data.
There are coefficients in different subbands that represent the same spatial location in the image depicted by a quad tree.
If a wavelet coefficient at a coarse scale is insignificant with respect to a given threshold T, i.e. |c|<T then all wavelet coefficients of the same orientation at finer scales are also likely to be insignificant with respect to T.

52. EZW Image Coding Embedded Coding
Having all lower bit rate codes of the same image embedded at the beginning of the bit stream
Bits are generated in order of importance
Encoder can terminate encoding at any point, allowing a target rate to be met exactly
Suitable for applications with scalability

53. EZW Algorithm
First step: The DWT of the entire 2-D image will be computed by FWT
Second step: Progressively EZW encodes the coefficients by decreasing the threshold
All the coefficients are scanned in a special order
If the coefficient is a zero tree root, it will be encoded as ZTR. All its descendants don’t need to be encoded – they will be reconstructed as zero at this threshold level
If the coefficient itself is insignificant but one of its descendants is significant, it is encoded as IZ (isolated zero).
If the coefficient is significant then it is encoded as POS (positive) or NEG (negative) depends on its sign.
Third step: Arithmetic coding is used to entropy code the symbols

54. EZW Image Coding Zerotree of DWT Coefficients An element of zerotree:
Significance map: Binary decision as to a pixel = 0 or not
Total encoding cost = cost of encoding significance map + cost of encoding nonzero values
An element of zerotree:
A coeff.: itself or some of its descendants are significant w.r.t. threshold T
Zerotree root: An element of zerotree, & not a descendant of a zero element at a coarser scale
Isolated zero: Insignificant, but has some significant descendant
Significance map can be efficiently represented as a string of four symbols:
* Zerotree root * Isolated zero
* Positive significant coeff * Negative significant coeff.

55. Set Partitioning in Hierarchical Trees (SPIHT)
Modify EZW.
Order coefficients by magnitude and transmit the most significant bits.
Progressive transmission.

56. EBCOT (Embedded Block Coding with Optimized Truncation)
Implements tiling, DC-level shifting, DWT, arithmetic coding
Compression of the image at different resolutions
JPEG2000 encoding process

57. Progression
Progression by Quality

58. Region of Interest Coding
Code different regions of interest with different quality
Scale coefficients such that the bits corresponding to ROI are in the higher bit plane.

59. Scalability SNR scalability and spatial scalability
The ability to achieve coding of more than one quality
Resilience to transmission errors, no need to know target bit rate/resolution, no need for multiple compressions

60. 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

61. Spatial Scalability
The bit stream can be decompressed at different resolution level