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Properties of Histograms and their Use for Recognition. Stathis Hadjidemetriou, Michael Grossberg, Shree Nayar Department of Computer Science Columbia University New York, NY 10027. Motivation. Histogramming is a simple operation:. Motivation. Histograms have been used for:

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properties of histograms and their use for recognition

Properties of Histograms and their Use for Recognition

Stathis Hadjidemetriou, Michael Grossberg,

Shree Nayar

Department of Computer Science

Columbia University

New York, NY 10027

motivation
Motivation
  • Histogramming is a simple operation:
motivation3
Motivation
  • Histograms have been used for:
    • Object recognition [Swain & Ballard 91, Stricker & Orengo 95]
    • Indexing from visual databases [Bach et al, 96, Niblack et al 93, Zhang et al 95]
  • Histogram advantages:
    • Efficient
    • Robust [Chatterjee, 96]
  • Histogram limitation:
    • Do not represent spatial information
overview
Overview
  • Image transformations that preserve the histogram
  • Image structure through the multiresolution histogram
  • Multiresolution histogram compared with other features
model for histogram

U

U

Histogram count for bin U

Area bounded by level sets U

Model for Histogram
continuous image transformations
Continuous Image Transformations
  • Vector fields, X, morph images [Spivak, 65]:
slide11

Original

Gradient Transformations

slide13

Condition 1: Histogram Preservation and Local Area

……

……

Histogram preservedLocal area preserved

[Hadjidemetriou et al, 01]

slide14

Small region

Local area preserveddivergence is zero

[Arnold, 89]

Condition 2: Local Area Preservation and Divergence

  • Divergence is rate of area change per unit area
slide15

Isovalue contours

Hamiltonian flow

Hamiltonian Fields

  • Fields along isovalue contours of an energy functionF
  • Flow of incompressible fluids[Arnold, 89]
slide16

Gradient of

Hamiltonian of

Computing Hamiltonian Fields

  • Compute gradient of F
  • Rotate gradient pointwise 900
slide17

Theorem

Transformations preserve histogram of all images  corresponding field is Hamiltonian

[Hadjidemetriou et al, CVPR, 00, Hadjidemetriou et al, IJCV, 01]

Condition 3: Divergence and Hamiltonian Fields

Divergence of field is zero Hamiltonian field[Arnold, 89]

examples of hamiltonian transformations

Original

Examples of Hamiltonian Transformations

Linear:Translations, rotations, shears

weak perspective projection

Planar object tilt(f) causes shearing and scaling

  • Depth (z) causes scaling

[Hadjidemetriou et al, 01]

Weak Perspective Projection
slide24

The Hamiltonian transformations is the complete class of continuous image transformations that preserves the histogram

previous work on features combining the histogram with spatial information
Previous work on Features combining the Histogram with Spatial Information
  • Local statistics:
    • Local histograms [Hsu et al, 95, Smith & Chang, 96, Koenderink and Doorn, 99, Griffin, 97]
    • Intensity patterns [Haralick,79, Huang et al, 97]
  • One histogram:
    • Derivative filters[Schiele and Crowley, 00, Mel 97]
    • Gaussian filter[Lee and Dickinson, 94]
  • Many techniques are ad-hoc or not complete
limitations of histograms
Limitations of Histograms

Database of synthetic images with identical histograms [Hadjidemetriou et al, 01]

matching with multiresolution histograms
Matching with Multiresolution Histograms

Match under Gaussian noise of st.dev. 15 graylevels:

matching with multiresolution histograms30
Matching with Multiresolution Histograms

Match under Gaussian noise of st.dev. 15 graylevels:

how is image structure encoded in the multiresolution histogram

Image

L

h(L*G(l))

Multiresolution histogram

Differences of histograms

?

Image structure

How is Image Structure Encoded in the Multiresolution Histogram?
histogram change with resolution and spatial information

Binj:

ill-conditioned

  • Averages of bins:
  • where Pj are proportionality factors

well-conditioned

Histogram Change with Resolution and Spatial Information

Spatial

information

histogram change with resolution and fisher information measures

Averages

=

Generalized Fisher information measures of order q[Stam, 59, Plastino et al, 97]

L is the image

=

D is the image domain

Histogram Change with Resolution and Fisher Information Measures
image structure through fisher information measures

P

Fisher information

measures (Analysis)

Jq

Image Structure Through Fisher Information Measures

Image

L

h(L*G(l))

Multiresolution histogram

Differences of histograms

?

Image structure

shape boundary and multiresolution histogram

Superquadrics:

h=6.67

h=1.00

h=1.48

h=2.00

h=0.56

Histogram change with l is higher for complex boundary

Shape Boundary and Multiresolution Histogram
texel placement and multiresolution histogram

Std. dev. of perturbation

Histogram change with l decreases with randomness

Texel Placement and Multiresolution Histogram
matching algorithm for multiresolution histograms

Cumulative histograms

Differences of histograms between

consecutive image resolutions

Concatenate to form feature vector

L1 norm

Matching Algorithm for Multiresolution Histograms

Burt-Adelson image pyramid

histogram parameters

5x5

44x44

89x89

179x179

Histogram Parameters
  • Bin width
  • Smoothing to avoid aliasing
  • Normalization:
    • Image size
    • Histogram size

……

database of synthetic images
Database of Synthetic Images

108 images with identical histograms [Hadjidemetriou et al, 01]

match results for brodatz textures
Match Results for Brodatz Textures

Match under Gaussian noise of st.dev. 15 graylevels:

database of curet textures

8,046 images with identical equalized histograms: 61 materials under different illuminations [Dana et al, 99]

Database of CUReT Textures
match results for curet textures
Match Results for CUReT Textures

Match under Gaussian noise of st.dev. 15 graylevels:

match results for curet textures47
Match Results for CUReT Textures

Match under Gaussian noise of st.dev. 15 graylevels:

sensitivity of class matching for curet textures
Sensitivity of Class Matching for CUReT Textures

100 randomly selected images per noise level

comparison of multiresolution histogram with other features
Comparison of Multiresolution Histogram with Other Features
  • Multiresolution histogram:
    • Variable bin width
    • Histogram smoothing
  • Fourier power spectrum annuli [Bajsky, 73]
  • Gabor features [Farrokhnia & Jain, 91]
  • Daubechies wavelet packets energies [Laine & Fan, 93]
  • Auto-cooccurrence matrix [Haralick, 92]
  • Markov random field parameters [Lee & Lee, 96]
comparison of class matching sensitivity of features54
Comparison of Class Matching Sensitivity of Features
  • Database of
  • CUReT textures
  • 100 randomly selected images per noise level
comparison of computation costs of features

Decreasing cost

Comparison of Computation Costs of Features

n- number of pixels

l- window width

l- resolution levels

summary and discussion
Summary and Discussion
  • Hamiltonian transformations preserve features based on:
    • Histogram
    • Image topology
  • Multiresolution histograms:
    • Embed spatial information
  • Comparison of multiresolution histograms with other features:
    • Efficient and robust
recognition of 3d matte polyhedral objects
Recognition of 3D Matte Polyhedral Objects
  • Face histograms:
    • Magnitude scaled by tilt angle (hi )
    • Intensity scaled by illumination (ai )
  • Total histogram: Sum of h(i) of visible faces
  • In an object database find [Hadjidemetriou et al, 00]:
    • Object identity
    • Pose (hi )
    • Illumination (ai )
slide60

Object 1:

Object 2:

Object 3:

Object 4:

A Simple Experiment

shape elongation and multiresolution histogram

Gaussian:

  • Pyramid:

St. dev. along axes:

sx, sy.

Sides of base :

rx, ry.

Elongation:

Elongation:

Histogram

change with l

(analytically)

Shape Elongation and Multiresolution Histogram
resolution selection with entropy of multiresolution histograms

l

…………………………………

….

  • Entropy-resolution plot [Hadjidemetriou et al, ECCV, 02]:
    • Global
    • Non-monotonic
Resolution Selection with Entropy of Multiresolution Histograms
slide66

Future Work

  • Histogram preserving fields:
    • Transformations over limited regions
    • Sensitivity of features to image transformation
  • Multiresolution histograms:
    • Color images
    • Rotational variance with elliptic Gaussians
  • Resolution selection:
    • Preprocessing step
    • Non-monotonic features
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