Some applications of wavelets Anna Rapoport

FBI Fingerprint Compression  Between 1924 and today, the US Federal Bureau of Investigation has collected over 200 million cards of fingerprints  Some come from employment and security checks, but million cards belong to some 29 million criminals ( bad guys tend to get fingerprinted more than once)

Some more facts…  This includes some 29 million records they examine each time they're asked to “round up the usual suspects.”  And to make matters worse, fingerprint data continues to accumulate at a rate of 30,000-50,000 new cards PER DAY

Let’s make simple calculations: The FBI is digitizing the nation's fingerprint database at 500 dots per inch with 8 bits of grayscale resolution. At this rate, a single fingerprint card turns into about 10 MB of data! Multiplied by 200 million cards – about 2,000 terabytes!

How to make checks faster? The FBI decided to adopt a wavelet-based image coding algorithm as a national standard for digitized fingerprint records. Some form of data compression is necessary!

The FBI standard - WSQ  The WSQ (Wavelet/Scalar Quantization) developed and maintained by the FBI, Los Alamos National Lab, and the National Institute for Standards and Technology involves: 2-dimensional discrete wavelet transform DWT Uniform scalar quantization Huffman entropy coding

Wavelets come to the stage! Lately, wavelets have made quite a splash in the field of image processing: FBI fingerprint image compression Next-generation image compression standard JPEG- 2000

Wavelets in Image Processing Problem:Area of application: How small can we compress our data without losing vital information? Wavelets work well for image compression What are essential features of the data, and what features are “noise”? Wavelet analysis lends itself well to denoising images

What are the principles behind compression?  Two fundamental components of compression are redundancy and irrelevancy reduction. Redundancy reduction aims at removing duplication from the signal source (image/video). Irrelevancy reduction omits parts of the signal that will not be noticed by the signal receiver, namely the Human Visual System (HVS).

Lossless vs. Lossy Compression LossyLossless contains degradation relative to the original numerically identical to the original image Reconstructed image high compression (visually lossless) 2:1 (at most 3:1)Compression rate

Image compression steps: Original image Source encoder linear transform to decorrelate the image data (lossless) Quantization of basis functions coefficients (lossy) Entropy Coding of the resulting quantized values (lossless) Compressed image (reconstructed) (inverse T) (dequantization) (decoding)

Basic ideas of linear transformation  We change the coordinate system in which we represent a signal in order to make it much better suited for processing (compression).  We should be able to represent all the useful signal features and important phenomena in as compact manner as possible.  Important to compact the bulk of the signal energy into the fewest number of transform coefficients.

The Good Transform Should: Provide good energy compaction Desirable to be orthogonal Decorrelate the image pixels

Which options do we have for linear transformation? A possible choice for the linear transformation are:  DFT  or, avoiding complex coefficients, the DCT  JPEG (decomposition into smaller subimages of size 8x8 or 16x16, followed by DCT as the compression algorithm)

Why Wavelet-based Compression?  No need to block the input image and its basis functions have variable length avoid blocking artifacts.  More robust under transmission and decoding errors.  Better matched to the HVS characteristics.  Good frequency resolution at lower frequencies, good time resolution at higher frequencies – good for natural images.

Transformation - FWT Original Image Wavelet coefficients Reconstructed

Example of DWT (Haar Basis) Let’s consider a 1D 4-pixel Image [ ] [ ] (9 + 7)/2 (3 + 5)/2 [ 84 ] (9 - 7)/2 (3 - 5)/2 [ 1-1 ] Average(smoothing) Detail coefficients (edge detection) (8 + 4)/2 6 (8 – 4)/2 2 [ ] WT

Mathematical Look at FWT  Assume that our 1D image is a piecewise constant function on the half-open interval [0,1)  One-pixel image is a const on the entire [0,1) – 1D vector  Denote V 0 to be the vector space of all such functions (1D space)  Two-pixel image is a function having two constant pieces in intervals [0,1/2] and [1/2,1), so it’s a 2D vector - their space V 1  In this manner, the space V j will include all piecewise-constant functions with constant pieces over each of 2 j equal-sized subintervals  Example: Our 4-pixel 1D image [ ] is a vector in V 2

Nested Spaces  Every vector in V j can be represented in V j+1 so spaces V j are nested V 0  V 1  V 2  …  The idea of nested spaces is one of the basic ingredients of the theory of multiresolution.

Basis for Vector space V j  Basis functions for V j are called scaling functions and are denoted by φ.  A simple basis for V j is given by the set of scaled and translated box functions: φ ij (x) : = φ (2 j x - i)i = 0,…, 2 j – 1 where φ(x) := 1, for 0  x <1 0, otherwise  Example basis for V 2 :

Definition of Wavelets  Define W j as the orthogonal complement of V j in V j+1, i.e. W j ⊕ V j =V j+1  A collection of linearly independent functions  ij (x) spanning W j are called wavelets:  The basis fun-s  ij (x) and φ ij (x) form a basis in V j+1  For each j  ij (x) orthogonal to φ ij (x)  Wavelets are orthogonal to each other

Haar Wavelets  The wavelets corresponding to the box basis are known as Haar Wavelets:  ij (x) : =  (2 j x - i) i = 0,…, 2 j – 1 where 1, for 0  x <1/2  (x):= -1, for 1/2  x <1 0, otherwise  Example: Haar Wavelets for W 1

Wavelets as a form of MA Wavelets work for decomposing signals (such as images) into hierarchy of increasing resolutions: as we consider more layers, we get more and more detailed look at the image.

What have we get till now?  We have matrix of coefficients (average signal and detail signals of each scale)  No compression has been accomplished yet, even the obtained representation can be longer than the original (since the decomposition uses a floating point representation, while the original signal could use an integer representation).  Compression is achieved by quantizing and encoding coefficients

 A quantizer simply reduces the number of bits needed to store the transformed coefficients by reducing the precision of those values. Since this is a many-to-one mapping, it is a lossy process and is the main source of compression in an encoder. Quantization can be performed on each individual coefficient, which is known as Scalar Quantization (SQ). Quantization The lossy step

The first idea of quantization: Coefficients that corresponds to smooth parts of data become small. (Indeed, their difference, and therefore their associated wavelet coefficient, will be zero, or very close to it). So we can throw away these coefficients without significantly distorting the image. We can then encode the remaining coefficients and transmit them along with the overall average value.

Quantization and Dequantization

Uniform Quantizer

Uniform Scalar Quantizer with Deadzone

Example of Uniform Quantization

Dequantization Rule

Example of Dequantization

Entropy encoding and decoding  Once the quantization process is completed, the last encoding step is to use entropy coding to achieve the entropy rate of quantizer. The Shannon entropy provides a lower bound in terms of the amount of compression entropy coding can best achieve. Examples:  Huffman  Arithmetic coding

Examples: FBI WSQ Results Original image bites 768x768 Reconstructed image (compressed file size 32702, compressed ratio 18:1)

WSQ vs. JPEG WSQ image, file size bytes, compression ratio JPEG image, file size bytes, compression ratio 19.6.

EPIC (Efficient Pyramid Image Coder) (by Euro Simoncelli and Edward Adelson) Based on biorthogonal wavelet decomposition and run-length/Huffman entropy coding Compressed 20: bytes Original image (512x512) bytes

Compressed 40: bytes

Original image (512x512) bytes Compressed 80: bytes

Wavelet Denoising

Wavelet denoising  DWT of the image is calculated  Resultant coefficients are passed through threshold testing  The coefficients < threshold are removed, others shrinked  Resultant coefficients are used for image reconstruction with IWT.

The Idea  The intuition behind this approach is that the neighboring pixels exhibit high correlation, which translates to only a few large wavelet coefficients. On the other hand, the noise is evenly distributed among the coefficients and is generally small. y i = x i + n i

Threshold techniques Hard threshold Soft threshold = σ (2log(N)) ½ N – is a block size in the WT σ – is the scale of the noise in SD scale

Advantages of Wavelet Denoising  It’s possible to remove the noise with little loss of details.  The idea of wavelet denoising based on the assumption that the amplitude, rather than the location, of the spectra of the signal to be as different as possible for that of noise.

Example 1

Example 2

References  “Wavelet Image Compression” Zixiang Xiong, Kannan Ramchandran  EPIC (Efficient Pyramid Image Coder)  “Filtering (Denoising) in the Wavelet transform Domain” Yousef M. Hawwar, Ali M. Reza et al wavelet.pdf  Wavelet Denoising with MatLab lmc.imag.fr/SMS/software/GaussianWaveDen/