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					Wavelets

Fast Multiresolution Image
Querying
Jacobs et.al. SIGGRAPH95
Outline
   Overview / Background
     Wavelets

   2D Image matching
     L1,L2 metrics
     Wavelet metric
     Evaluation

   Use in 3D
Image matching
 2D analogue of 3D shape matching
 Raster instead of XYZ
What are we trying to match?
   Looking for different images of the same things?
     Different projections
     Different colors

   Looking for images that look the same?
     Look   for similar shapes and colors


   Metric to discern like human eye
Wavelets
   Decompose a signal into component parts
     Fourier analysis: a signal can be represented
      as a (possibly infinite) sum of sine and cosine
      functions
 Signal becomes a set of wavelet
  coefficients
 Coefficients represent features of signal
Wavelets II
 Signal can be completely reconstructed
  from all the coefficients
 Signal can be partially reconstructed from
  some coefficients
Wavelets for 2D Images
 Each color plane in image is signal
 Coefficients will represent visual features
  in the image
 Store as many coefficients as needed
     Image   compression (see next slide)
   c.f. Statistical shape descriptors
Wavelet Reconstruction




                SIGGRAPH 96 Course Notes: Wavelets in Computer Graphics
Comparing Images
 Develop a metric that describes how
  closely two images match
 Smaller difference in metric = images
  more similar
Image metrics
   Comparing images Q and T, with dimensions i,j
   L1-Norm
              Q, T 1   Q[i, j ]  T [i, j ]
                        i, j


   For each pixel in Q, calculate the difference
    between Q[i,j] and T[i,j]
   Add absolute value of differences of all i,j to form
    metric
Image metrics II
   L2-Norm
                                                      1/ 2
                                                    2
              Q, T   2
                            Q[i, j ]  T [i, j ] 
                                                     
                            i, j                     
   For each pixel in Q, calculate the square of the
    difference between Q[i,j] and T[i,j]
   Add for all i,j, and take square root
   Better than L1?
Image metrics III
   Problems with L1 and L2
     Expensive  to compute / compare: O(i*j)
     Not discriminating in cases with
        Color Shift
        Misregistration

        Noise / Dithering

   In general, not good descriptors
     c.f.   D1, D2 in 3D
Wavelets as image metrics
 Capture features of images e.g. edges in
  coefficients
 Use Haar wavelets
     Square  basis functions
     Easy to implement and compute



   Calculate coefficients, truncate, quantize
Truncation
 128x128 image has 1282 coefficients
 Truncation = only storing largest ‘n’
  coefficients
 ‘n’ ~ 40-60 depending on exact use
 Discarding smaller coefficients discards
  high frequency information i.e. detail
     Loss   of that information is desirable
Quantization
   Reduce precision of wavelet magnitude
     Large +ve  +1
     Large –ve  -1
     Else  0

   Turns out this works well for matching
Wavelet metric
                                                ~         ~
    Q, T  w0, 0 Q[0,0]  T [0,0]   wi , j Q[i, j ]  T [i, j ]
                                       i, j



 Q,T are query and target image
  coefficients
 w is weighting function
Weighting function
   Weighting function applied to give particular
    pairs of coefficients different significance in
    comparison
   Gives ability to statistically tune the metric
   Determined experimentally from dataset
    (Appendix A)
   Weights expensive to calculate
     Compute  fewer
     Bins to map range of i,j onto a weight
Wavelet metric II
                                                                    ~             ~
    Q, T  w0 Q[0,0]  T [0,0] 
                                        ~
                                                        wbin (i , j ) Q[i, j ]  T [i, j ]
                                   i , j:Q[ i , j ] 0



 For i=0, j=0 value in Q and T is
  proportional to the average overall color
 From quantization, use ≠
Calculating coefficients
 Standard two-dimensional Haar wavelet
  decomposition
 Decompose each row, then decompose
  each column of the result
 Trivial to implement
Wavelet metric III
   Final metric is
     T[0,0]
     Sign,   i and j of n largest coefficients in T
Faster Matching
   To speed up matching, use 6 arrays
     One  for each combination of R,G,B,+,-
     DR+, DR-, …
     Each i,j in Dx is a list of all images with a
      metric coefficient in that color range, with that
      sign
Evaluation
   Better than L1 and L2
   Matches to ~1% of database
   Compact
   Fast to compare: similar complexity to an 8x8 pixel
    image L1/L2 for any resolution
   More robust
       Misregistration
       Color shifting
       Dithering
       Different resolutions
Evaluation II
Evaluation III
   Limits
     Scaling ~1.5 times
     Rotation ~20 degrees
     Translation ~ 15% of width
Matching in 3D
 Can this be used in 3D as well
 Compares image rather than geometry
 Render 3D into voxels?
 Projection of 3D object into 2D?


   In general, other 3D specific methods
    probably much better