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Properties of Histograms and their Use for Recognition

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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:
- 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

- Image transformations that preserve the histogram
- Image structure through the multiresolution histogram
- Multiresolution histogram compared with other features

What is the complete class of continuous transformations that

preserves the histogram?

Continuous Image Transformations

- Vector fields, X, morph images [Spivak, 65]:

Gradient Transformations

Condition 1: Histogram Preservation and Local Area

……

……

Histogram preservedLocal area preserved

[Hadjidemetriou et al, 01]

Local area preserveddivergence is zero

[Arnold, 89]

Condition 2: Local Area Preservation and Divergence

- Divergence is rate of area change per unit area

Hamiltonian flow

Hamiltonian Fields

- Fields along isovalue contours of an energy functionF

- Flow of incompressible fluids[Arnold, 89]

Hamiltonian of

Computing Hamiltonian Fields

- Compute gradient of F

- Rotate gradient pointwise 900

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]

Planar object tilt(f) causes shearing and scaling

- Depth (z) causes scaling

[Hadjidemetriou et al, 01]

Weak Perspective ProjectionThe Hamiltonian transformations is the complete class of continuous image transformations that preserves the histogram

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

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

Matching with Multiresolution Histograms

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

Matching with Multiresolution Histograms

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

L

h(L*G(l))

Multiresolution histogram

Differences of histograms

?

Image structure

How is Image Structure Encoded in the Multiresolution Histogram?ill-conditioned

- Averages of bins:
- where Pj are proportionality factors

well-conditioned

Histogram Change with Resolution and Spatial InformationSpatial

information

=

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 MeasuresFisher information

measures (Analysis)

Jq

Image Structure Through Fisher Information MeasuresImage

L

h(L*G(l))

Multiresolution histogram

Differences of histograms

?

Image structure

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 HistogramHistogram change with l decreases with randomness

Texel Placement and Multiresolution HistogramDifferences of histograms between

consecutive image resolutions

Concatenate to form feature vector

L1 norm

Matching Algorithm for Multiresolution HistogramsBurt-Adelson image pyramid

44x44

89x89

179x179

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

……

Database of Synthetic Images

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

Match Results for Brodatz Textures

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

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

Database of CUReT TexturesMatch Results for CUReT Textures

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

Match Results for CUReT Textures

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

Sensitivity of Class Matching for CUReT Textures

100 randomly selected images per noise level

Embed spatial information into the histogram with the multiresolution histogram

How well does the multiresolution histogram perform compared to other image 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 Features

Database of Brodatz textures

Comparison of Class Matching Sensitivity of Features

- Database of
- CUReT textures
- 100 randomly selected images per noise level

The multiresolution histogram compared to other image features is robust and efficient

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

- 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 )

- Pyramid:

St. dev. along axes:

sx, sy.

Sides of base :

rx, ry.

Elongation:

Elongation:

Histogram

change with l

(analytically)

Shape Elongation and Multiresolution Histogram…………………………………

….

- Entropy-resolution plot [Hadjidemetriou et al, ECCV, 02]:
- Global
- Non-monotonic

The entropy of the multiresolution histogram can be used to detect significant image resolutions

- 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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