Reduced Representation of Deformation Fields for Effective Non-rigid Shape Matching
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Abstract
In this work we present a novel approach for computing correspondences betweennon-rigid objects, by exploiting a reduced representation of deformation fields.
Differently from existing works that represent deformation fields by training ageneral-purpose neural network, we advocate for an approximation based on meshfree methods.
By letting the network learn deformation parameters at a sparseset of positions in space (nodes), we reconstruct the continuous deformation fieldin a closed-form with guaranteed smoothness.
With this reduction in degrees offreedom, we show significant improvement in terms of data-efficiency and enablelimited supervision.
Furthermore, our approximation provides direct access tofirst-order derivatives of deformation fields, which facilitates enforcing desirableregularization effectively.
Our resulting model has a high expressive power andis able to capture complex deformations.
We illustrate its effectiveness throughstate-of-the-art results across multiple deformable shape matching benchmarks
Introduction
Shape correspondence is a central problem in computer vision and computer graphics as it facilitatesmany downstream tasks, such as tracking [1], texture transfer [2] and statistical modeling [3] to namea few.
Due to its ubiquitous applicability, a wide range of techniques have been developed over thepast several years [4].
While early approaches relied on axiomatic modeling, recent methods followdata-driven techniques based on different input signals [5, 6, 7, 8] within a shape collection.
A key question in this context is the choice of representation used to model the non-rigid shapematching problem.
Approaches based on intrinsic or pose invariant representations have establisheda gold standard in the context where surfaces are well-defined [9, 10, 11].
Such methods, however,strongly rely on the presence of clean shapes and struggle when acquisition comes from noisy andnon-uniform discretization [12].
In contrast, extrinsic techniques which directly operate on Euclideanspace ( R 3 ) show strong resilience to artifacts.
Unfortunately, this robustness of extrinsic methods often comes at the cost of having access to significant amounts of annotated training data [7, 5].
The main limiting factor arises in the representation ofthe deformation fields.
The standard approach is to use general-purpose MLPs to learn deformationfields that can fit an arbitrary shape deformation [13, 14, 5].
However, given the fact that MLPsare often biased towards low-frequency components in the input signal [15], they require abundanttraining data to generalize across high-frequency components [5].
To overcome this limitation, we propose to learn a coarse representation of deformation parametersat fixed positions in space called “nodes”.
By learning a reduced representation of deformation fields,intuitively, we restrict the learning process to low-frequency components of the input signal.
Then,we reconstruct the continuous deformation field function in a closed-form using a class of mesh-free approximation techniques [16].
This allows us to scale our approach to arbitrary resolution acrosshigh-frequency components with guaranteed smoothness.
Apart from being theoretically grounded and simple in practice, our reduced representation hastwo key advantages.
First, it is significantly more data-efficient and can learn to capture complexdeformations given only a small number of examples.
Secondly, it is more amenable to regularization,since it provides explicit access to first order derivatives of the deformation field in closed form.
Thisis especially useful in imposing geometric priors such us local rigidity and volume preservation.
Our contributions can be summarized as follows: (a) We propose to learn a compact representationof deformation parameters, that is data-efficient, resolution agnostic, and facilitates regularizationthrough direct access to deformation gradients (b) We show an efficient way of incorporating desirableregularization to promote a well-structured deformation space.
(c) Through extensive experiments,we demonstrate the generalization ability of our method over different down-stream applications suchas non-rigid shape matching, registration, unsupervised part segmentation and interpolation.
Ourcode will be publicly released.
Related Works2.1 Non-rigid shape Correspondence and Registration
Shape correspondence is a very well-studied area of computer vision and computer graphics and werefer interested readers to the recent survey [4] for a comprehensive overview.
Notable axiomaticapproaches in this category are based on the functional maps paradigm [9, 17, 18, 19], that solves anisometric mapping by estimating a linear transformations between functions represented in a reducedbasis.
This framework has been successfully adapted by learning techniques [6, 20, 21, 7, 10] whichdemonstrate near-perfect accuracy [11] on several shape correspondence benchmarks.
However, theseapproaches demonstrate weak resilience when exposed to artefacts arising from surface discretization,which are common in acquired data.
While registration techniques [22, 23, 24] are relatively a morereliable option, they either use strong human-centeric priors or require significant training data.
2.2 Template-based and Template-free Methods
Deforming a template shape to match a target geometry is a long-standing and well studiedproblem [25, 26, 27].
Such a template can be a polygonal mesh [28, 29], possibly parameterised [30, 31, 32] or an unordered point-set [33, 34, 35, 36, 37] or implicitly defined throughzero-level set of a Neural Field [13, 14, 38].
In the recent years, learning based model-free deformation techniques [33, 34, 24] have emerged successful for registration and correspondence taskswith copious availability of training data [33, 39].
Amongst them, the closest to our approach is3D-CODED [33] which learns deformation fields through point-wise MLPs.
However, since eachpoint in the shape is treated equally likely, this approach fails to generalize across high-frequencyfeatures, thereby requiring abundant training data to achieve optimal performance.
2.3 Deformation Field Representation and Shape Interpolation
Deformation between a pair of shapes can be represented as a simple displacement field at everyvertex.
However, such a representation can be unnecessarily complex, and costly to optimize.
As aresult, several alternatives have been proposed.
The most prominent ways to parameterize the spaceof deformations include handle-based [40, 41, 42, 43] or cage-based [44, 45, 46, 47] representations(see also [48, 49, 50] for an overview).
More recently, a common approach is to construct a reducedrepresentation via a learned latent embedding [51, 52, 53, 54].
Deformation Field
regularization Several geometric constraints have been proposed with the aimof preserving desirable properties of the shape by the deformation field, including elasticity [55, 56,57] and volume preservation[58, 59, 41, 60, 61].
Recently, these constraints have been successfullyadapted by data-driven methods [62, 52, 63, 38, 64] and more relevantly through the differentialof the map [52, 64, 64].
Distinct from such approaches, our approximation via mesh-free methodenables evaluating this map differential at fixed points in a closed-form, which significantly simplifiesthe deformation field regularization without additional computational overhead.
Shape Interpolation Shape interpolation refers to time parameterized deformation, where a sourceshape is continuously deformed to a target shape.
Our work is related to efforts which aim to enforceintermediate shapes to preserve certain intrinsic properties [53, 54, 62, 65, 66, 52].
Among themclosest to our approach is LIMP [53] which disentangles the latent space based on style and poseto preserve geodesic distance.
In contrast, our approach does not require such a priori information,which can be costly in terms of annotation efforts.
2.4 Reduced representations and Approximations
In this work we use mesh-free function approximation method [67, 68, 16], to approximate deformation fields.
Mesh-free methods have been successfully adapted in Smoothed Particle Hydrodynamics(SPH) modeling [69, 70], image processing [40], animation [41, 71, 72] and more recently in adata-driven framework [42].
Differently from [42], instead of learning the weights of the leastsquares function, we instead learn deformation values at nodes and demonstrate our method to beapplicable in wide-range of downstream tasks.
Alternatively, Eisenberger et al.
[61] have proposed touse a compact representation of deformation fields using the first k eigenfunctions of the Laplace
Beltrami Operator (LBO).
While their approach provides volume preserving deformation, it doesnot facilitate other regularizations such as as-rigid-as-possible deformation fields without requiringcorrespondence at inference time [73].
Motivation, Background and Notation3.1 Motivation:
Parametric models such as SMPL [30] have been tremendously useful over the recent years indigitizing and processing human models.
This success can largely be attributed to their expressivepower, allowing to generate a wide range of styles and poses using a small fixed set of deformationparameters.
While this efficacy with such a compact representation is remarkable, it also raises aninspiring question: what is the optimal amount of learnable parameters necessary to represent generaldeformations?
Today, general-purpose MLPs form the conventional way of representing deformationfields due to their simplicity and potential of being universal functional approximators [74, 75].
Unfortunately, the generic power of MLPs also comes at a cost of copious training efforts [33, 24].
Furthermore, representing a deformation field using a neural network makes access to certainquantities such Jacobian matrices of deformation fields cumbersome.
For these reason, we proposeto learn a reduced set of deformation parameters from which we approximate the deformation fieldfunction using a mesh-free method.
3.2 Mesh-free Approximation
Mesh-free methods are a class of approximation techniques which constructs a continuous functionbased on independent, potentially sparse and irregular observations.
Assume that our domain ofinterest R 3 is equipped with K fixed points q i ∈ R 3 along with some observations
u i at q i and apolynomial basis p ( · ) .
We refer to fixed points q i as “Nodes”.
Our main interest is to constructa continuous approximation of some real-valued function u ( . ) across regions in some subdomainΩ ⊂ R 3 of interest.
We let x ∈ Ω to be an arbitrary point in our region of interest.
The key ideabehind this approximation is to use a local weighted least-squares fitting (also referred to as Moving
Least Squares) approach [16], in the neighbourhood of q i , using a compactly supported weightingfunction given by:
[Equation]
The compactness of this weighting function is useful in preserving the local characteristics ofapproximation.
From this, a Shape Function Φ i associated with each node i , is constructed as:
[Equation]
Here M ( x ) is the Moment Matrix associated with the approximation defined as,
[Equation]
The shape function Φ i is a continuous function that describes how each node q i influences theapproximation of u ( . ) across points x ∈ Ω .
Jointly the the set of Φ i ’s enable the reconstruction ofarbitrary functions up to n th order consistency [68], where, n is the order of the polynomial p ( · ) .
Specifically, a smooth local approximation of u ( x ) is given as:
[Equation]
As the construction of Φ involves computing M − 1 (c.f Equation 2), it is a sufficient condition foreach point x to be compactly supported by 4 non-planar nodes q i for M to be non-singular.
It isimportant to note that Equation 3 is approximating and not interpolating , i.e u
i ̸
= u ( q i ) .
For instance,owing to the compact nature of w ( x ) , it is possible that u ( q i ) is undefined if q i / ∈ Ω .
For this reason,we sample the nodes a priori to have a well-supported domain Ω ⊆ R 3 where u ( x ) is well-defined.
Furthermore, an important advantage of using mesh-free approximations comes from an exact analytical expression for the gradient function of u ( x ) .
To the scope of our current discussion, consideringu ( x ) to be the approximation of deformation field function, the Jacobian of this deformation fieldonly depends on evaluation point and is independent of observed deformation parameters u i ,
[Equation]
This nice property helps us characterize the deformation fields with desired first-order regularizationin an efficient manner.
We refer interested readers to [68, 67] for a detailed summary.3.3 Notation:
As our training set, we consider a collection of shapes {S 1 . . .
S N } with correspondence Π S l S jbetween them.
They could be a triangular mesh S j := {V , E} or simply an unordered set of points(Point Clouds) S j := {V} .
From them, we pick one as a template T , and let [ T ] be the volumeenclosed by the boundary ∂ T .
We let D ( · ) : R 3 → R 3 be the deformation mapping .
We refer to U jas the nodal deformation parameters corresponding to j th shape and analogously define D j ( · ) .
Werefer to Q ∈ R K × 3 as nodal positions , which are K fixed points in space sampled from the templatevolume [ T ] .
Each node q i ∈ Q has a support radius r i and associated deformation parameter u i .
We use lower-case notation u i,j to refer to the values of deformation field at node q i correspondingto shape j .
We denote x ∈ Ω ⊂ R 3 as points in space which are supported by 4 non-planarnodes.
We refer to U j ( x ) as the continuous approximation of the deformation field, constructed fromdeformation parameters U j using Equation 3.
We re-iterate that U j ̸
= U j ( Q i ) .
The relation betweendeformation field function and mapping is given by D j ( x ) :
= x + U j ( x ) .
For the sake of consistency,we index nodes using i , shape collection using j, l and points within shape using k .
Method: Learning Nodal Deformation-Field
Overview.
Our network consists of PointNet [76] based Auto-Encoder as shown in Figure 1.
Ournetwork F θ ( · ) predicts nodal deformation parameters for each training shape S j , i.e U j = F θ ( S j )where U j ∈ R K × 3 .
As previously mentioned, the nodes Q are fixed a priori.
From the predicted U j ,we can reconstruct the shape-specific deformation mapping D j ( x ) and its Jacobian J j as
D j ( x
[Equation]
[Figure]
Where, ∇ x,y,z U j ( x ) is given in Equation 4.
4.1 Training
[Figure]
Intuitively, we would like to train a network so that S j ≈ { D j ( x ) | x ∈ T } , while ensuring D j ( x ) islocally rigid and volume preserving.
Although D j ( x ) can be approximated at an arbitrary x , whichis supported by 4 non-planar nodes, we restrict ourselves to x ∈ T for the ease of learning.
As F θ ( · )represents an auto-encoder, it can be decomposed as F θ ( S j ) =
Dec ( Enc ( S j )) =
Dec ( Z j ) where
Z j denotes the latent embedding.
Leveraging this fact, we provide a novel way to promote plausiblelatent deformation spaces Z j by enforcing first-order constraints over the intermediate shapes as well.
Consequently, the overall optimization objective of our network is given as,
[Equation]
For the unsupervised case, we replace L cor with L CD which denotes the Chamfer’s distance.
Correspondence Loss Given a set of C of corresponding points { x l , x k } , where x l ∈ T , x k ∈ S jour correspondence loss is given as
[Equation]
Where x jk denotes the k th point in j th shape.
Volume Preserving Field
A deformation field is volume preserving iff its Jacobian has unitdeterminant over the entire shape.
Consequently, our volume preservation constraint is given asfollows,
[Equation]
We empirically observe poor convergence when this constraint is enforced over the entire shape dueto its stringent nature.
This is because not all deformation are strictly volume preserving.
Thus, werestrict this regularization only at nodes.
As Rigid As Possible (ARAP) Deformation
Since rigid motions preserve pair-wise distances, adeformation field associated with such a transformation is characterized by an orthonormal Jacobianmatrix.
Thus, in order to promote locally rigid deformation field at nodes, we define our ARAPregularization as,
[Equation]
Structuring Latent
Deformation Space A well-known advantage of an auto-encoder architectureis the construction of the latent space, where each shape has an embedding Z j ∈ R D .
Then, aparameterized path in this latent space between two shapes Z l,j ( α )
=
αZ l + (1 − α )
Z j continuouslydeforms S j to S l with rate of change controlled by α .
This enables construction of a shape sequencebetween a pair of shapes, often referred to as interpolated shapes.
Since each Dec ( Z j ) =
U j , we canfurther constraint our network F θ to produce a plausible deformation between each pair of trainingshapes .
To that end, we introduce our latent smoothness loss as follows,
[Equation]
Where L arap , L vol are defined in Equations 8 9 and α ∈ (0 , 1) are sampled randomly.
4.2 Test-Time refinement
At test-time, given a pair of unseen shapes ( X , Y ) , we first independently extract the mappings( D X , D Y ) following Equation 5.
Then, the correspondence between two shapes Π XY is a simplenearest neighbour search between the mappings ( D X , D Y ) .
Additionally, we can further refinethe deformation mapping by optimizing the latent vector Z Y . This optimization is performedby minimizing bi-directional Chamfer’s Distance as a geometric constraint [33] while enforcingaforementioned first-order constraints as follows,
[Equation]
[Equation]
4.3 Extending sparse to dense Correspondence
An added advantage of our representation is the ability to retrieve dense shape correspondencebetween a pair of shape, given a few sparse key-point correspondences ( x l , y k ) , ∀ x l ∈ X , ∀ y k ∈ Y .
Distinct from a Functional Mapping paradigm [9], our representation is agnostic to discretization ofthe geometry.
Thus, to compute a dense-mapping, we first estimate the deformation parameter u i atthe nodes by solving an optimization,
[Equation]
Subsequently, a dense mapping can be computed by approximating the deformation field at anywell-supported points following Equation {+
+}
4.4 Implementation details
Analytical Gradients and Timing advantages
We leverage the advantage of inexpensive accessto Jacobians as mentioned in Equation 4.
Because our evaluation points are known a priori, due to theuse of a fixed template T , the matrix J can be pre-computed and re-used at training and evaluation.
In practice, we observe a 10 × speed-up at training time when enforcing our first-order constraintsand a 350 × speed-up incorporating the latent constraints (c.f. Eqn 10).
We provide more timingdetails in the supplementary.
Candidates
Nodes
Node Sampling: Since the deformation field is determined by its nodal neighbourhood, it is important to prevent a node from asserting its influence on semanticallydifferent regions.
Bearing this in mind, our node sampling strategy is divided into three main steps.
First, weconstruct a dense sampling of points in the volume andaround the boundary of the template ∂ T .
Second, we userejection sampling to exclude a node that exerts its influence in semantically different regions [30].
Finally, we perform Farthest Point Sampling (FPS) until each surface point is covered by 4 non-planarnodes.
Second and third steps are shown in the inset Figure.
Source [12] [33] [24]
Ours
GT
[Table]
Experiments
We consider four main tasks : Non-rigid 3D shape correspondence, Shape registration, Unsupervisedpart segmentation and Shape interpolation.
5.1 Shape Correspondence
We consider three challenging benchmarks, namely, SHREC’19, FAUST (PC), SCAPE (PC+N).
SHREC’19 [77] is a standardised benchmark consisting of 430 evaluation pairs with significantvariations in mesh resolution and connectivity.
FAUST (PC) denotes a more recent Non-Isometric
Point Cloud variant [12] of the FAUST dataset consisting of 1000 points with large variance inpoint sampling density.
Third, we evaluate on a variant of the recent SCAPE-Remesh dataset [78]consisting of 20 shapes of the same human in 20 distinct poses.
We further augment the challengeby adding random Gaussian noise and refer to as SCAPE (PC+N).
We evaluate correspondenceerror following the Princeton benchmark protocol [79].
We train our method on a subset of
SURREAL shapes [80] for 1000 epochs with data-augmentation along Y-axis.
Baselines We compare our method against data-driven correspondence methods broadly classifiedinto Spectral, Pair-wise and Template based.
We use GeoFMap [7] with the more robust featureextractor Diffusion-Net [11] as our spectral baseline, Diff-FMaps [12] and CorrNet3D [8] as ourpair-wise baseline.
For our template based baselines, we use 3D-CODED [33] and TransMatch [33].
For evaluation of baselines on our proposed SCAPE (PC+N), we use the author provided pre-trainedmodels upon desired pre-processing.
Discussion Our approach consistently outperforms baselines as summarised in Table 1.
While ourquantitative correspondence results are persuasive, it is remarkable to note that our method requiresan order of magnitude less training data in comparison to competing methods.
This supports ourpremise that characterizing articulated motions requires far fewer parameters than what is leveragedby existing data-driven methods.
We show qualitative correspondence results through color transferfor a challenging pair with “twisted” motion in Figure 2.
Src [5] [82]
Ours
5.2 Shape Registration
Shape registration is a special case of correspondence, wherein, given a possibly noisy scan, ourgoal is to find an optimal deformation between the scan and a fixed template.
For this, we considerthe recent SHREC’20 benchmark [81], consisting of 11 partial scans of stuffed toy rabbits to beregistered to a single scan.
This benchmark is particularly challenging due to granulated surfacedeformation, scanning aretefacts, and limited data and supervision.
Experiment We split this dataset into 7 training shapes and 4 shapes for evaluation.
The 4 shapesin our test set is made of “chickpea” material, which exhibits the largest magnitude of protrusions andindentations on the surface.
We stress that this category of deformation is unseen during training time.
We compare our method against 4 baseline methods namely, FARM [22], BCICP [78]+ZoomOut [83],3D-CODED [33] and TransMatch [24].
We summarize our quantitative and qualitative results in
Figure 3.
It is remarkable that our approach outperforms axiomatic and competing data-drivenbaselines by at least a two-fold margin.
Importantly, despite the fact that our approach shares thesame encoder [76] as 3D-CODED [33], there is a striking difference in performance.
We attributethis to our well-regularized deformation space.
5.3 Unsupervised Segmentation Transfer
In this section, we demonstrate the generalization ability of our approach to model deformation fieldsbetween shapes with considerable topological differences.
To that end, we consider the task of partlevel segmentation over point clouds consisting of table and plane categories from ShapeNet [84]dataset.
Apart from topological differences and large structural variance, absence of ground truthannotations exacerbates the challenge.
In this setting, we compare our method with two Deep Implicitnetworks, namely DIF-Net [13] and DIT [14], which model a volumetric deformation fields betweena learnt template and training shapes.
Our choice of baseline endows us with a fair ground ofcomparison between the two representations of deformation field - MLP based and ours.
Experiment and Discussion
We train our approach using the unsupervised loss mentioned in
Section 4.1 over 1000 objects sampled at random from each category.
We consider 190 evaluationpairs per-category from the prescribed validation set and measure the segmentation accuracy by the
[Figure]
[Table]
[Figure]
[Table]
[Figure]
[Table]
[Figure]
Qualitative result demonstrates color coded segmentation transfer across significant shape variability.
[Figure]
IoU metric [84].
In addition, we also measure the bi-directional Chamfer’s distance of reconstructedgeometries.
We summarize our quantitative observation along with a qualitative example in Figure 4.
We remark that while deformation fields between aforementioned categories are not strictly volumepreserving, we still observe a noticeable improvement over the baseline.
This is because ourdeformation priors help in structuring the space of deformations, which explicitly avoids degeneracysuch as collapsing shape parts.
This corroborated is
5.4 Shape Interpolation
A notable characteristic of a well-structured latent space is the ability to produce plausible intermediate shapes given a source and a target.
This task is commonly referred to as shape interpolation.
Sincethere exists no canonical path, interpolation sequences are gauged by the extent to which intrinsic metrics are preserved, in particular isometric distortion [85].
For this setting, we consider the FAUST [3]dataset, where, we train our method on the first 80 shapes and evaluate over the last 20 shape pairs.
Wecompare our method against four baselines namely 3D-CODED [5], NeuroMorph [62], LIMP [53]and ShapeFlow [52].
Since LIMP employs a fixed-size decoder and NeuroMorph uses a separateinterpolation module involving an explicit computation of correspondence matrix, both of theseapproaches are limited by shape resolution.
On the other hand, our approach is resolution agnosticand outperforms the baselines by a discernible margin as summarized in Figure 5.
This improvementover the baseline is due to the incorporation of our latent deformation priors in a computationallyfeasible manner, which we will be justified through an extensive ablation study in the supplementary.
Conclusion, Limitations and Future Work
We presented an effective representation of deformation fields by learning a reduced set of shapespecific deformation parameters while constructing the continuous deformation field function usingmesh-free approximation technique.
This elegant representation endowed us with access to firstorder derivatives in closed-form, thereby facilitating the use of strong first-order regularization.
Akey observation behind our method is that in many settings, the space of realistic deformations iswell-constrained and expressed with a small set of parameters.
To that end, we demonstrated that ourapproach can achieve significant improvement upon existing baselines across challenging downstreamapplications and remarkably reduce the dependence on training data.
Our approach still has somelimitations and leads to possible exciting future work.
Firstly, while our approach produces smoothdeformation field in principle, there is no guarantee on bijectivity or invertibleness.
Second, insteadof fixed nodal positions, optimizing with respect to our approximation function is also an interestingidea to ponder.
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