Face Recognition using Spherical Wavelets

Christian Lessig∗

Abstract

“If there is technological advance without social advance, there is, almostautomatically, an increase in human misery, . . . ”

Michael Harrington

In this report we evaluate the applicability of the SOHO wavelet basisfor face recognition. We demonstrate that a representation of 3D facemodels using spherical wavelets allows to distinguish between subjects,and that the intra-subject variability due to facial expressions is smallerthan inter-subject variations.

1 Introduction

Traditionally, two-dimensional images have been employed for face recognition. Re-cently however, approaches based on three-dimensional facial representations becamepopular [14, 2] due to improved capturing techniques (e.g. [12, 1]) and the promise ofhigher recognition rates by using signals in the native space R3 instead of projectionsR3 → R2 [15, 27, 28, 40, 48]. The last Face Recognition Vendor Test (FRVT) demon-strated the viability of 3D face recognition systems and these techniques provided there thesame accuracy as 2D approaches [43]. These results are particularly promising given theyoung history of the field and the relatively small number of approaches which have beenproposed in the literature.

In this report, we propose a novel technique for 3D face recognition which employs therecently developed SOHO wavelet basis [36, 37]. The input to our technique is a 3Dfacial mesh which is projected onto the sphere, and the resulting spherical signal is thentransformed using the SOHO basis. The basis function coefficients provide the templatewhich is used for recognition, and standard techniques for 1D digital signal analysis can beemployed to compare faces.

Work related to our technique can be found both in the face recognition literature and inComputer Graphics. In the following we will focus on 3D techniques [14, 2] and referthe interested reader to the surveys by Samal and Iyengar [46], Chellapa et al. [21], andZhao et al. [57] for a discussion of 2D approaches. Commercial systems, which outper-formed techniques submitted by academic institutions in the FRVT 2006 [43], will alsobe excluded from the following discussion because in general little is known about theunderlying technology.

∗lessig@dgp.toronto.edu

1

Gordon [26] was one of the first to employ 3D information for face recognition. He as-sumed that certain facial regions such as the forehead and the cheeks are almost invariantunder different poses and facial expressions, and concluded that these regions are particu-larly well suited for a face recognition system. He employed curvature to extract invariantregions and high-level shape descriptors projected into a feature space for comparison.Cartoux et al. [18] similarly used curvature to segment faces but these authors utilized theplane of bilateral symmetry for comparison. Tanaka et al. [49] projected the minimal andmaximal principal curvature directions of a face onto the sphere and employed these Ex-tended Gaussian Images for face recognition, and Moreno et al. [41] employed Gaussiancurvature to segment faces and compute feature vectors. As pointed out by Abate et al. [2],approaches which directly employ curvature are not scale-invariant and cannot recognizethe same face presented in different sizes.

Similar to the work by Gordon [26], Chua et al. [22] employed the forehead, the cheeks,and other regions invariant to pose and facial expressions. These author used howeverpoint signatures to identify invariant regions. Wang et al. [54] used sphere spin images,which are histograms derived from local shape descriptors computed at points of minimalcurvature, and performed face recognition by computing correlation coefficients betweenthe histograms of different subjects. Nagamine et al. [42] identified five feature points inthe face to standardize pose and then employed various curves and profiles to characterize aface. In similar work, Beumier and Acheroy [9] used vertical face profiles with gray valuesfor face recognition; and Wu et al. [55] employed horizontal profiles. Gaussian-Hermitemoments have been used by Xu et al. [56] as local shape descriptors to compare faces, andIrfanoglu et al. [30] employed the Point Set Distance, which is a discrete approximation ofthe volume between two models, for face recognition.

Blanz and Vetter [11, 13, 10] proposed a full face recognition system including the 3Dmodel generation. Their technique requires as input only a single 2D image and they esti-mate 3D mesh models using a morphing technique proposed in earlier work [12]. Differentapproaches to compare face models are discussed by these authors and Blanz and Vetteremployed for example a distance computed from the parameters of the morphable modelsof different subjects. In similar work, Lu et al. [38] also created a 3D model of a subjectfrom a single image using morphing. However they then generated synthetic images fromthe subject in different poses and with different illuminations, and employed the renderedimages for face recognition using standard 2D techniques. Ansari and Abdel-Mottaleb [7]employed a frontal and a side view to obtain a morphable model, and they used 29 prede-fined feature points for recognition, exploiting that a morphable model guarantees properalignment of the meshes. Approaches based on morphable models have been criticized fornot being accurate. Abate et al. [2] argued that a single image is not sufficient to uniquelyreconstruct a 3D representation, and it has been recognized that synthesized facial expres-sions might look realistic but often differ substantially from observed ones.

Similar to eigenfaces [35] which is the canonical algorithm for 2D face recognition, PCAhas also been employed by different authors for 3D face recognition [39, 28, 3]. Anothermachine learning technique, Gaussian Mixture Models, has been used by Cook et al. [23]to compare faces.

A variety of authors employed a combination of 2D and 3D techniques. Chang etal. [20, 19] performed PCA separately on grayscale and depth images and combined theresults using a weighted average. The authors demonstrated that using both modalities out-performs using either of them alone. In similar work, Tsalakanidou et al. [52] employedhidden markov models to combine 2D and 3D face recognition results. Wang et al. [53]used Gabor wavelets to characterize the 2D image and point signatures to compactly de-scribe the 3D shape.

Bronstein et al. [16, 15, 17] developed a technique based on isometric embedding [25]

2

which is in spirit similar to the ISOMAP algorithm [50] and Locally Linear Embed-ding [45]. Given the original geodesic distances of the samples they find a mapping ontoa low-dimensional space such that the total error of the geodesic distances after projec-tion is minimized. Multidimensional scaling on the metric tensor of the original surfaceis employed for the embedding. The authors demonstrate that a representation using thegeodesic distances is less sensitive to facial expressions and they therefore claim that theirapproach is (almost) expression invariant. Different embedding spaces are employed by theauthors and Bronstein et al. report that the sphere S2 provides the least distortion [15]. Ourwork has similarity with those of Bronstein et al. We also employ a projection of the facialmesh onto S2 but we directly resample the mesh onto the sphere along the normal directionand therefore do not require expensive nonlinear optimization. In contrast to the Spheri-cal Harmonics employed by Bronstein et al., we use spherical wavelets because the basisfunctions are localized, allow data-dependent approximation, do not suffer from ringingartifacts, and the transforms can be computed more efficiently. The SOHO wavelet basisis thereby the representation of choice because it is the only known spherical Haar waveletbasis which is symmetric and orthogonal, making it particularly well suited for the efficientrepresentation and approximation of spherical signals; the readers is referred to the paperby Lessig and Fiume [37] and the thesis by Lessig [36] for a more detailed discussion ofthe SOHO basis.

In Computer Graphics, different techniques have been developed to compare and identifyshapes. In early work, Horn [29] used the normal distribution across the surface as shapedescriptor, while Ankerst et al. [6] employed the distribution of points on the surface of themodel. Spherical Extent Functions, which describe the maximal extent of an object fromthe origin, have been employed by Saupe and Vranic [47] to characterize shapes. Kazhdanand coworkers [31, 32] employed a reflective symmetry descriptor which represents thesymmetry of a model with respect to all possible planes through the origin. In later work,the same authors employed a discrete shell representation of a model and used Spheri-cal Harmonics (SH) to compactly encode the shells [33, 34]. The SH basis is reflectionand rotation invariant and the approach therefore overcomes the alignment problem whichKazhdan et al. considered as severe for previous techniques. Our work has also similaritywith the shell technique by Kazhdan et al. We however do not require multiple shells. Aface or a head is already approximately an embedding S2 ⊂ R3 and we can therefore di-rectly project the mesh onto S2. Rotation- and reflection-invariance, which motivated theuse of Spherical Harmonics by Kazhdan et al., is no severe problem in our application sincefor example the ears and the mouth, or the eyes and the mouth, naturally span a referencecoordinate frame.

2 Face Recognition using the SOHO Wavelets

The input to our algorithm for 3D face recognition is a facial mesh model. Given the model,first the projection Mj → Mj of every mesh triangle Mj = {pj

1,pj2,p

j3}, with pj

i ∈ R3,onto the sphere is computed with

Mj ={pj

1, pj3, p

j3

}=

{pj

1

‖pj1‖

,pj

2

‖pj2‖

,pj

3

‖pj3‖

},

where ‖ · ‖ denotes the `2 norm, and pji ∈ S2. Next, the partition tree T = {Tl,j}, over

which the SOHO basis is defined, is constructed up to some level lk, and then the meshis resampled onto the domains Tlk,j . For every Tlk,j we generate M random samplessm ∈ S2 and then determine

dmlk,j = max

j∈Jsm

‖pjsm‖,

3

+ =

Figure 1: Left: Partition over which the SOHO basis is defined on level lk. Middle: Orig-inal mesh model. Right: Projection of the mesh onto the partitions on level lk. Visible arethe eyes, the nose and the mouth.

where Jsmis the set of all projected mesh faces such that sm ∈ Mj , and pj

smis the closest

point to sm in Mj . The normalized average

dlk,j =1M

M∑m=1

dmlk,j

dmax

over all samples sm yields the final depth value for Tlk,j . The maximal distance dmax isdefined as

dmax ≡ maxi,j

‖pji‖,

and leads to the scale-invariance of our representation. Note that the resampling of thefacial mesh onto the partitions makes the technique robust against variations in the meshmodel, for example a variable number of faces. An example for the projection of a headmodel is shown in Fig. 1.

The resampling of the facial mesh onto the Tlk,j yields a spherical signal with scalingfunction coefficients λlk,j ≡ dlk,j . The forward wavelet transform with the SOHO basisis employed to obtain wavelet basis function coefficients, and these coefficients providethe template which is used for recognition. Note that, according to the general theory ofwavelets, we can expect that most of the basis function coefficients are very small or zero,and that a very small number of the coefficients is sufficient to characterize a signal.

3 Evaluation

The 3D face recognition algorithm proposed in Sec. 2 has been evaluated with 16 facialmesh models from four different individuals and four different facial expressions per sub-ject. The models have been generated with Poser [24] and we employed Maya [8] totriangulate and simplify the quad meshes. The models employed for the experiments hadapproximately 1, 250 vertices for the man in the top row of Fig. 2; 1, 300 vertices for thewoman in the second row; 2, 600 vertices for the girl in the third row; and 1, 050 verticesfor the boy in the last row of Fig. 2. For the first two and the last model we employed avery low number of vertices to test the applicability of our algorithm for systems where theinput data comes from non-intrusive capturing techniques; and the third model has beenused to assess the influence of the mesh resolution on the algorithm.

For the SOHO wavelet basis an octahedron was used as base polyhedron, and we choselk = 5 resulting in 8, 192 partitions T5,j . The first four octants correspond to the upper

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hemisphere and the last four octants to the lower hemisphere. All experiments were per-formed with the Matlab programming environment [51].

In Fig 3 the magnitude of the basis function coefficients for the eighth octant before andafter approximation are shown for the models in the first column in Fig. 2. For the approx-imation we retained 512 or 6.25% of the 8, 192 coefficients. The results show that onlyvery small coefficients are affected by the approximation and that the peaks, which charac-terize the signals, are retained almost unaffected. The possibility of such an approximationprovides a three-fold advantage. Firstly, only a small fraction of the basis function coeffi-cients has to be stored for each template, which is important for large databases; secondly,the approximation makes comparisons of templates, and therefore the recognition problem,more efficient; and thirdly, the small coefficients correspond mostly to noise and using onlythe large coefficients improves the robustness of the system. In the following we will onlyconsider templates with the k = 512 largest basis function coefficients.

Fig. 4 shows the templates for all 16 models, and the basis function coefficient spectra ofthe last four octants are shown in detail in Fig. 5 to Fig. 8. In Fig. 4 it can be seen that thefirst four octants contain almost no information. With the projection of a face mesh ontoS2 shown in Fig. 1 it becomes clear that all features such as ears, eyes, and nose lie almostentirely in the lower hemisphere, and that the upper hemisphere therefore only encodesthe head shape. The very small number of significant basis function coefficients therebyconfirms that a sphere is a suitable embedding for a head. The results also demonstratethat the head shape is not discriminating between subjects, at least not with the projectioncurrently employed by us. The detail plots in Fig. 5 to Fig. 8 show the basis functioncoefficients corresponding to the characteristic features of the faces such as the eyes, thenose, and the mouth. The graphs demonstrate that the intra-subject variability is smallerthan the inter-subject variations and that each face has a characteristic spectrum of peaksin the set of wavelet basis function coefficients. Facial expressions thereby correspondto a local scaling of basis function coefficients but in general do not alter the position ofthe peaks. We attribute the discriminating power of our algorithm and its approximateinvariance to facial expressions to the localization of the SOHO basis functions in spaceand frequency.

Give the results presented in this section we conclude that the algorithm proposed in Sec. 2is in principal applicable for 3D face recognition.

4 Future Work

In Sec. 3 we demonstrated the applicability of the SOHO wavelet basis for 3D face recog-nition. Significant work remains however until the technique described in Sec. 2 can beemployed in a practical system.

Of particular importance is the development of an automatic algorithm to compare tem-plates. One approach, which is commonly used to compare digital signals, would be tocompute correlation coefficients. One could however also imagine to employ machinelearning techniques such as hidden markov models or neuronal networks. Another alter-native would be to use string matching algorithm which have been employed successfullyin Computational Structural Biology [4, 5]. It is thereby currently unclear to which extentthe magnitude of the coefficients should be included in the comparison and further researchis required to answer this question. A more thorough investigation of the approximate in-variance to facial expression, which we reported in Sec. 3, is also necessary. We currentlyassume that it occurs for similar reasons as in the work by Bronstein et al. [16, 15, 17] butwe would like to understand the observations in more detail.

In our experiments we employed the octahedron as base polyhedron for the SOHO basis. It

5

is also possible to use the tetrahedron or the icosahedron and previous results demonstratethat the choice of the base polyhedron can significantly affect the performance [36]. Futureexperiments should therefore include the tetrahedron and the icosahedron and determinewhich one is best suited for face recognition. Similarly, the method currently used to projectthe face model onto the sphere has not been compared to alternative approaches. Althoughits simplicity has advantages, in particular in terms of computation time, one could alsoimagine to employ the approach proposed by Bronstein et al. [16, 15, 17], or one of themany techniques for mapping mesh models onto the sphere which can be found in theliterature (e.g. [44]).

The results discussed in Sec. 3 demonstrate that at most four octants contain significantinformation about the face or more generally the head. In practice it therefore seems tobe sufficient to employ the octant(s) with the most significant and most discriminatinginformation. This would reduce storage requirements for the template and speed-up com-parisons. The efficacy of our face recognition system could most likely also be improvedby exploiting texture information, as done by many other state-of-the-art face recognitionsystems [43, 20, 19, 53, 52].

Crucial for every system is a meaningful evaluation. The data set employed for the FRVT2006 is available upon request and we believe this would provide a good basis for a com-parison to other systems.

5 Conclusion

In this report we investigated the applicability of the SOHO wavelet basis for 3D facerecognition. The proposed algorithm first projects a facial mesh model onto the sphere,and then performs a wavelet transform using the SOHO wavelets to obtain basis functioncoefficients which serve as template. Our results demonstrate that the template is charac-teristic for a face, and that intra-subject variability due to facial expressions is smaller thaninter-subject variations. We also showed that it is sufficient to employ only a small subsetof the k largest basis function coefficients, which significantly reduces the storage require-ments for a template, increases the robustness of the algorithm, and reduces the comparisontime. With the obtained results, we believe it would be worthwhile to further develop theproposed algorithm to a full 3D face recognition system.

Acknowledgement

We thank Patricio Simari for the Matlab OBJ loader.

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A Test Models

Figure 2: Test models used for the experiments presented in Section 3.

11

B Evaluation Results

Figure 3: Spectrum of the wavelet basis function coefficients for the last octant before(left) and after approximation (right) where 6.25% of all 8, 192 coefficients were retainednon-zero. Clearly visible is that the approximation only affects very small coefficientswhich can be identified as noise but that the peaks which characterize the signals are almostunaffected.

12

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15

Figu

re7:

Bas

isfu

nctio

nco

effic

ient

spec

trum

foro

ctan

t7fo

rthe

16m

odel

sin

the

test

set.

16

Figu

re8:

Bas

isfu

nctio

nco

effic

ient

spec

trum

foro

ctan

t8fo

rthe

16m

odel

sin

the

test

set.

17