Informative descriptor preservation via commutativity for shape matching

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Changing the dimension of the reduced space

In our second range of experiments, we show the dependence of the results on the number of functions used in the basis for functional maps. Here, rather than changing the number of descriptor functions, we fix the descriptor set and change the dimensionality of the basis and evaluate the quality of the approximation of the point-to-point correspondences using the functional map pipeline. In Figure 2.4 we show the average correspondence error between a subset of shapes in the FAUST dataset [13], using the same pipeline described in the previous section, for a fixed number (in this case two) of descriptor functions. In particular, we used the Wave Kernel Signature for a single energy value, along with a single descriptor function that is aimed at segment preservation using the Wave Kernel Map with a fixed energy value. This gives us two descriptor functions, which we incorporate into the functional map energy using either the standard descriptor preservation constraints, as done in [69] or using our commutativity-based approach. We then convert the estimated functional map to a point-to-point correspondence and evaluate its accuracy using the distance to the ground truth. We plot the average pointwise map error, computed the same way as in the previous experiment, across the shape pairs for a varying number of basis functions number of basis functions in the functional map representation, for a fixed number (two in this case) descriptors. Unlike the standard approach of [69], which deteriorates when the size of the basis significantly exceeds the number of descriptors, our method continues to produce high-quality results even for a small number of descriptors and a large number of basis functions. The average error is computed as the average geodesic distance to the ground truth correspondence, symmetries allowed. As can be seen in Figure 2.4, compared to the basic method for descriptor preservation, our approach allows not only to improve quality of the correspondences significantly, without using any additional information, but also provides more resilience with respect to the choice of the number of basis functions, for a fixed descriptor set. This implies that our approach can potentially enable more accurate correspondence computation based on the functional map pipeline, without requiring any additional information, and supports the idea that using our formulation allows to extract additional information from descriptor functions, which in turns results in better pointwise maps.

Function approximation and transfer

In our first application, we evaluate the utility of our function transfer procedure using both ground truth (arising from known pointwise maps) and computed functional maps. Namely, given a set of pairs of (source, target) shapes and a collection of different functions on each source, we evaluate: i) the approximation of each function on the source shape, and ii) the transfer of a function between the source and the target shape. Figure 3.7 shows a qualitative example of approximation and transfer of a real-valued function, with the original function shown on the left. This function is generated as a combination of an indicator function (on the left leg) a gaussian around a point (top of the right leg) a sine function of the y coordinate (on the tail) and a continuously increasing function (on the ears). We compare the standard approach (std) vs. our extended method (prod), for both function approximation and transfer onto a different pose of the same shape. Note the improvement with our approach (Approach B) as captured by the mean squared error reported under each plot. In our quantitative experiments we consider the following families of functions:
hk k, hk K: the heat kernel functions ht(x; ) between a random point x and the rest of the shape approximated using kM + 1 eigenfunctions (for hk k) and using 200 eigenfunctions (for hk K). Note that in the former case, the function is contained in the span of the original basis, while in the latter 200 > kM.
HKS, WKS: the Heat and Wave Kernel Signatures [8,92] for 10 randomly time and energy values respectively.
Random: the function obtained as a linear combination of the extended basis using a random set of coefficients.
XYZ: the X, Y , Z, coordinates of vertices.
Indicator: the binary indicator function of a random region.

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HKS and WKS approximation and transfer

One interesting observation related to our method is that both the Heat Kernel Signature and the Wave Kernel Signature functions are of the form ht(x; x) = PkM i=1 i2i (x); for some scalar coefficients i. In other words, they are constructed explicitly using squares of eigenfunctions and therefore it must be possible to represent and transfer them exactly in our extended basis. Therefore, they provide a good test for the correctness of both our extended function approximation and transfer methods. To demonstrate the difference between the transfer of these functions using the standard and the extended basis we show in Figure 3.11 the transfer error using the ground truth functional map of size kN kM for increasing kN. Note that for large values of kN we can approximate the transfer of all kM basis functions from the source well, which means that we would
expect our extended transfer method to produce progressively better results. This can Figure 3.10 – Surface reconstruction via approximation of the 3-coordinates functions using two different bases: the standard manifold harmonics (MH) and localized manifold harmonics (LMH). On the left we show the original shape and the region (in red) in which are localized the LMH. On the right of each shape we report the reconstruction error. The colormap encodes reconstruction error, decreasing from dark red to white. clearly be seen in Figure 3.11 where our method converges to a very small value while the standard technique does not improve. This is because the functions 2icannot be well-approximated in the basis of the source shape, which means that regardless of the basis size on the target, we cannot achieve small error.

Table of contents :

1 Introduction to non-rigid shape matching 
1.1 Surface
1.1.1 Continuous setting
1.1.2 Discrete setting
1.1.3 Discretization of a function and its gradient
1.1.4 Area weights
1.2 Laplace-Beltrami operator
1.2.1 Continuous setting
1.2.2 Discrete setting
1.3 Goal
1.4 Related work
1.5 Partial correspondences
1.6 Overview of the Functional Maps Framework
1.6.1 Setup
1.7 Definitions and Notations
2 Informative descriptor preservation via commutativity for shape matching
2.1 Introduction
2.2 Related work
2.3 Overview
2.4 Novel Approach for Functional Correspondences
2.4.1 Motivation
2.4.2 Our constraints
2.4.3 Properties
2.5 Experiments
2.5.1 Using few descriptors
2.5.2 Changing the dimension of the reduced space
2.6 Conclusion, Limitations & Future work
3 Improved functional mappings via product preservation 
3.1 Introduction
3.2 Related Work
3.3 Motivation and Overview
3.4 Method Description
3.4.1 Function Representation
3.4.2 Extended Functional Basis
3.4.3 Extended Function Transfer
3.4.4 Function Comparison and Pointwise Map Recovery
3.5 Results
3.5.1 Function approximation and transfer
3.5.2 HKS and WKS approximation and transfer
3.5.3 Point-to-point map recovery
3.5.4 Joint quadrangulation
3.6 Conclusion, Limitations and Future Work
3.7 Appendix
3.7.1 Additional Results
3.7.2 Proof of Lemma 2:
3.7.3 Proof of Theorem 3.4.1
3.7.4 Future work
4 Deep learning for non linear function approximation and mapping 
4.1 Introduction
4.2 Related Work
4.3 Background
4.4 Main Idea
4.5 Description
4.6 Parameters
4.7 Results
4.8 Conclusion
4.9 Future work
5 Miscellaneous 
5.1 A new analysis method for evolutionary optimization of dynamic and noisy objective functions
5.2 On semiring complexity of Schur polynomials
5.2.1 Schur polynomial
5.2.2 Main result
5.3 Scheduling with gaps: New models and algorithms
6 General Conclusion 
6.1 Summary
6.2 Future work
6.3 Position of the work in the community
Bibliography

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