Whitened Expectation Propagation: Non-Lambertian Shape from Shading and Shadow
Summary (2 min read)
1. Introduction
- Probabilistic inference for large loopy graphical models has become an important subfield with a growing body of applications, including many in computer vision.
- These methods have resulted in significant progress for several applications.
- The principal difference between BP and Gaussian EP can thus be summarized by a trade-off in their respective approximating families: BP favors flexible non-Gaussian marginals, while Gaussian EP favors a flexible covariance structure.
- Another possible explanation is that for a grid-based graphical model with D pixels, Gaussian EP requires O(D2) space and a run time of O(D3).
- Finally, the authors use the method to efficiently perform inference over large cliques produced by cast shadows and by global spatial priors.
2. Expectation Propagation
- The family P̃ is chosen so that EP̃ [τj( x)] can be estimated easily.
- EP achieves this goal by approximating each potential function φi( x) with an exponential family distribution P̃i( xi| θ(i)).
- Regardless of the rank of each potential, the covariance matrix of the posterior S remains full-rank, and must be stored as a D×D matrix.
- For large problems with tens of thousands of variables or more, this becomes limiting.
- When the underlying graphical model is highly sparse, such as a nearest-neighbor pairwiseconnected MRFs, each iteration can be performed in time O(D1.5) [2].
3. Whitened EP
- For many problems of computer vision, both the number of variables D and the number of potentials N grow linearly with the number of pixels.
- Low-rank potentials of large clique size have a wide array of promising applications in computer vision [17, 10].
- Expectation propagation can be made more efficient by limiting the forms of covariance structure expressible by S. Let S denote the covariance matrix for natural scenes.
4. Shape from Shading
- Whitened EP permits inference over images in linear time with respect to both pixels and clique size.
- In particular, the authors are interested in whether Gaussian message approximation will be effective when the potentials φi are highly non-Gaussian.
- In recent years, several methods have been developed that solve the classical SfS problem well as long as surface reflectance R is assumed to be Lambertian [19, 17, 6, 3, 7].
- For each pixel, one potential φR(p, q|i) enforces the surface normal to be consistent with the known pixel intensity i(x, y).
- Whitened EP provides two benefits for spatial priors.
5. Conclusions
- The methods in this paper reduce the run time of EP from cubic to linear in the number of pixels for visual inference, while retaining a run time that is linear in clique size.
- The computational expense of inference for large cliques has prohibited the investigation of complex probabilistic models for vision.
- The authors hope is that whitened EP will facilitate further research in these directions.
- Results for whitened EP on SfS shows that the sacrifice in performance for this approach is small, even in problems with highly non-Gaussian potentials.
- Performance remained strong for surfaces with arbitrary reflectance and arbitrary lighting, which is a novel finding in SfS.
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References
83 citations
"Whitened Expectation Propagation: N..." refers background or methods or result in this paper
...While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions....
[...]
...Other Lambertian SfS algorithms have reported image errors for the penny image of 0.0071 [9] and 0.0517 [13]....
[...]
...Subfigure c) shows the results of linear constraint node BP [17]....
[...]
...The SfS solution of [17] used p and q variables with 300 bins, and would thus sacrifice a 300-fold speed decrease to infer depth z directly....
[...]
...Lambertian SfS We first test our approach on Lambertian SfS, where it can be compared to past Lambertian SfS algorithms....
[...]
73 citations
"Whitened Expectation Propagation: N..." refers methods in this paper
...While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions....
[...]
...Other Lambertian SfS algorithms have reported image errors for the penny image of 0.0071 [9] and 0.0517 [13]....
[...]
...Lambertian SfS We first test our approach on Lambertian SfS, where it can be compared to past Lambertian SfS algorithms....
[...]
...We then test this approach on a problem with highly non-Gaussian potentials: non-Lambertian shape from shading (SfS)....
[...]
...SfS is one example, and non-Lambertian SfS produces especially non-Gaussian potentials....
[...]
66 citations
"Whitened Expectation Propagation: N..." refers methods in this paper
...While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions....
[...]
52 citations
"Whitened Expectation Propagation: N..." refers background or methods in this paper
...While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions....
[...]
...Other Lambertian SfS algorithms have reported image errors for the penny image of 0.0071 [9] and 0.0517 [13]....
[...]
...Lambertian SfS We first test our approach on Lambertian SfS, where it can be compared to past Lambertian SfS algorithms....
[...]
...We then test this approach on a problem with highly non-Gaussian potentials: non-Lambertian shape from shading (SfS)....
[...]
...SfS is one example, and non-Lambertian SfS produces especially non-Gaussian potentials....
[...]
31 citations
"Whitened Expectation Propagation: N..." refers methods in this paper
...SfS is one example, and non-Lambertian SfS produces especially non-Gaussian potentials....
[...]
...While there has been some success in applying methods such as Lax-Friedrichs and fastmarching to non-Lambertian reflectance [1, 23], these generalizations must proceed on a case-by-case basis for each class of reflectance functions....
[...]
...Other Lambertian SfS algorithms have reported image errors for the penny image of 0.0071 [9] and 0.0517 [13]....
[...]
...Lambertian SfS We first test our approach on Lambertian SfS, where it can be compared to past Lambertian SfS algorithms....
[...]
...We then test this approach on a problem with highly non-Gaussian potentials: non-Lambertian shape from shading (SfS)....
[...]