Paper:ModelTrans:miller-kde-transgroup

From Dahuawiki

Jump to: navigation, search

Back to Modeling Transformation

[edit]

Practical Non-parametric Density Estimation on a Transformation Group for Vision

E.G. Miller and C. Chefd'hotel.
CVPR (2003)
<Download from IEEE Xplore>
[edit]

Summary

This paper generalizes the kernel density estimation method in Euclidean space to generic transformation group. It introduces a general kernel function as

K(t; a) = f(D_G(t; a)), \quad D_G(t, a) = t^{-1} \circ a

It can be shown that the group difference DG is invariant to left group operation, and thus the kenerl.

D_G(b \circ t, \ b \circ a) = D_G(t,\ a)
K(b \circ t, \ b \circ a) = K(t,\ a)

Specially, the paper introduces an invariant kernel function for GL^+(n, \mathbb{R}) group (the groups with matrices of positive determinant) as follows

K_E(\mathbf{T}, \mathbf{A}) = \frac{1}{C |\mathbf{T}|^2} \exp\left(- \frac{1}{h} ||\log (\mathbf{T}^{-1} \mathbf{A})||_F^2 \right)

It is proved that this kernel not only satisfies invariance to left group operation, but also has invariant measure, which is crucial for consistent probability estimation.

With this kernel, we can readily obtain the density estimator as

f(\mathbf{U}; \ \mathbf{T}_1, \ldots, \mathbf{T}_N) = \frac{1}{N} \sum_{i=1}^N K(\mathbf{U}; \ \mathbf{T}_i)

This estimator is invariant

f(\mathbf{B}\mathbf{U}; \ \mathbf{B}\mathbf{T}_1, \ldots, \mathbf{B}\mathbf{T}_N) = \frac{1}{N} \sum_{i=1}^N K(\mathbf{B}\mathbf{U}; \ \mathbf{B}\mathbf{T}_i)

The paper also discusses the application of the estimator in transform-invariant classification under a simple Bayesian setting.

[edit]

My Comments

  • It is a simple and natural generalization of standard KDE to transformation group. The key is how to define the difference in a group. The generalization in this paper offers effective measurement of distance between transforms and density estimator as well.
  • Though theoretically appealing, whether it works well in complex reality is yet to be verified.
  • It may be difficult to directly sample from the distribution as the primary logarithm of matrix product is complicated.
  • This paper only discusses the density estimation with a set of sampled transforms given. It remains an open issue how to estimate the distribution of transformation when we are given the transformed images instead of the transforms themselves.
  • There are some other works in studying the orbits (their distances, density, and other relations) formed by applying group actions on a specific object. It may be interesting to explore the relation between those works and this paper.



Back to Modeling Transformation

Retrieved from "http://groups.csail.mit.edu/vision/dahuawiki/index.php/Paper:ModelTrans:miller-kde-transgroup"