Some applications of wavelets

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Some applications of wavelets

• Anna Rapoport

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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 114.5 million cards belong to some 29 million
  criminals (bad guys tend to get fingerprinted
  more than once)

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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

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Lets 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!
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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!
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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

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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
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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
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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).
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Lossless vs. Lossy Compression
Lossy Lossless
contains degradation relative to the original numerically identical to the original image Reconstructed image
high compression (visually lossless) 21 (at most 31) Compression rate
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Image compression steps
Original image
Source encoder linear transform to decorrelate
the image data (lossless)
(reconstructed)
(inverse T)
Quantization of basis functions coefficients
(lossy)
(dequantization)
Compressed image
Entropy Coding of the resulting quantized
values(lossless)
(decoding)
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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.

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

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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.

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Transformation - FWT

• Original Image

Reconstructed
Wavelet coefficients
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Example of DWT (Haar Basis)
Lets consider a 1D 4-pixel Image 9 7 3 5
Average (smoothing)
9 7 3 5
Detail coefficients (edge detection)
(9 7)/2 (3 5)/2 8 4
(9 - 7)/2 (3 - 5)/2 1 -1
WT
6 2 1 -1
(8 4)/2 6
(8 4)/2 2
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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 V0 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 its
  a 2D vector - their space V1
• In this manner, the space Vj will include all
  piecewise-constant functions with constant pieces
  over each of 2j equal-sized subintervals
• Example Our 4-pixel 1D image 9 7 3 5 is a
  vector in V2

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Nested Spaces

• Every vector in Vj can be represented in Vj1 so
  spaces Vj are nested
• V0?V1 ? V2?
• The idea of nested spaces is one of the basic
  ingredients of the theory of multiresolution.

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Basis for Vector space Vj

• Basis functions for Vj are called scaling
  functions and are denoted by f.

• A simple basis for Vj is given by the set of
  scaled and translated box functions
• fij(x) f (2j x - i) i 0,, 2j 1
• where f(x) 1, for 0?? x lt1
• 0, otherwise

• Example basis for V2

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Definition of Wavelets

• Define Wj as the orthogonal complement of Vj in
  Vj1, i.e. Wj ? VjVj1
• A collection of linearly independent functions
  ?ij(x) spanning Wj are called wavelets
• The basis fun-s ?ij(x) and fij(x) form a basis in
  Vj1
• For each j ?ij(x) orthogonal to fij(x)
• Wavelets are orthogonal to each other

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Haar Wavelets

• The wavelets corresponding to the box basis are
  known as Haar Wavelets
• ?ij(x) ? (2j x - i) i 0,, 2j 1
• where 1, for 0?? x lt1/2
• ?(x) -1, for 1/2?? x lt1
• 0, otherwise

• Example
• Haar Wavelets for W1

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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.

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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

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The lossy step
Quantization

• 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).

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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.

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Quantization and Dequantization
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Uniform Quantizer
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Uniform Scalar Quantizer with Deadzone
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Example of Uniform Quantization
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Dequantization Rule
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Example of Dequantization
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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

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Examples FBI WSQ Results
Reconstructed image (compressed file size 32702,
compressed ratio 181)
Original image 541832 bites 768x768
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WSQ vs. JPEG

• WSQ image, file size 30987 bytes, compression
  ratio 19.0.

JPEG image, file size 30081 bytes, compression
ratio 19.6.
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EPIC (Efficient Pyramid Image Coder)(by Euro
Simoncelli and Edward Adelson)
Based on biorthogonal wavelet decomposition and
run-length/Huffman entropy coding
Original image (512x512) 262144 bytes
Compressed 201 13103 bytes
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Original image (512x512) 262144 bytes
Compressed 401 6550 bytes
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Original image (512x512) 262144 bytes
Compressed 801 3275 bytes
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Wavelet Denoising
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Wavelet denoising

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

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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.

yi xi ni
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Threshold techniques

• Hard threshold

• Soft threshold

s is the scale of the noise in SD scale
N is a block size in the WT
? s (2log(N))½
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Advantages of Wavelet Denoising

• Its 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.

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Example 1
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Example 2
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References

• Wavelet Image Compression Zixiang Xiong, Kannan
  Ramchandran http//lena.tamu.edu/zx/
• EPIC (Efficient Pyramid Image Coder)
  http//www.cis.upenn.edu/eero/epic.html
• Filtering (Denoising) in the Wavelet transform
  Domain Yousef M. Hawwar, Ali M. Reza et al
  http//www.xilinx.com/products/logicore/dsp/denois
  e_wavelet.pdf
• Wavelet Denoising with MatLab http//www-lmc.imag.
  fr/SMS/software/GaussianWaveDen/