Description of 3D-Shape Using a Complex Function on the Sphere Dejan Vranić and Dietmar Saupe Slides prepared by Nat for CS For more information: originally in Proceedings of the IEEE International Conference on Multimedia and Expo

Spherical Shape Descriptors Parameterize a shape onto the unit sphere – f( ,  ) – f(u) u is a 3-vector on the unit sphere, i.e, |u| = 1 f(u) may evaluate to – a Scalar (e.g. extented gaussian images) – a Vector (e.g. a geometry image) – a Complex number (this paper)

Summary There exist two pre-existing spherical shape descriptors x(u) and y(u) which are sensitive shape descriptors This paper presents a technique to combine both into a single descriptor – That works without weighting of the two functions (i.e  x(u) +  x(u) – That is spatially efficient – That is (mostly) resolution independent – That can be retrieved/compared hierarchically

Initial Conditions The input models have been – Converted into points – Aligned to principal axes using PCA (etc) An object has a center-of-mass c about which a unit sphere is oriented.

Basic Descriptors x(u): “how far is it from c to the surface?” – Let s(u) be the surface parameterized in terms of u – Let c be the centroid of the object – x(u) = s(u) – c Open question: In the case of multiple surface intersections, how is x(u) defined? y(u): “how much light would reach s(u) from c ?” – y(u) = u  n(u)

Example: x(u) and y(u) x(u) n(u) y(u)  1

Example: x(u) and y(u) x(u) n(u) y(u)  0

Example of x(u) Image couresy of Vranic et al, 2001.

Discussion: x(u) and y(u) as shape descriptors Both are sensitive to the parameterization from the surface to the unit sphere. x(u): ray based – Will capture shape but suffer when parameterizations differ across models y(u): shadow based – Captures “texture” lost by x(u)

A Complex Feature Descriptor Based on the idea, “good + good = better” – r(u) = x(u) + i y(u) Intuitively: now we describe a shape as both What is the feature vector?

From r(u) to a feature vector Goals: – Compact – Incrementally comparable 2D Analog: – How do you compare images? Represent them as a sum of basis functions. Solution: – Spherical harmonics representation of r(u)

Spherical Harmonics “As is well-known (see e.g. [52]), the spherical harmonics provide an orthonormal basis for L 2 (S 2 ).” [Healy et al, 2002] WTF?

Spherical Harmonics “As is well-known (see e.g. [52]), the spherical harmonics provide an orthonormal basis for L 2 (S 2 ).” [Healy et al, 2002] A sampling of r(u) can be expressed as The values correspond to r(u)’s forier coefficients Some top number of coefficients are used as the feature vector for shape matching

The Details (1) L 2 (S 2 ): – L 2 is a hilbert space – S 2 is the surface of the unit sphere – Complex numbers are a Hilbert space Eric W. Weisstein. "Hilbert Space." From MathWorld--A Wolfram Web Resource. Fast Spherical Harmonic Transforms Library: Any function in L 2 (S 2 ) can be represented as a sum of spherical-harmonic basis functions

The Details (2) Courtesy of: Eric W. Weisstein. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource.

The Details (3) Algorithms exist that can transform from n points in L 2 (S 2 ) to harmonic coefficients in O(nlog 2 n) time – Note: the Fourier transformation places restrictions on the location of samples in S 2.

“Complex” Feature Vector r(u) is transformed into another 2d function For a complex function like r(u), the coefficients look like: Interestingly, for a real-valued function like x(u), half of this table (all m>0) can be truncated.

Truncation of the feature vector The paper reports that they only need ~13 rows of coefficients to match well – i.e. they store for 0 < l < 13

Truncation of the feature vector The paper reports that they only need ~13 rows of coefficients to match well – i.e. they store for 0 < l < 13

Shape Matching Results (L 1 )

Shape Matching Results (L 2 )

Shape Matching Results (L 1 )

Shape Matching Results (L 2 )

Shape Descriptor Quality  Concise to stored=169 (and adjustable)  Quick to computeO(nlog 2 n)  Efficient to matchIncremental match is possible  DiscriminatingDependant on parameterization  Invariant to transformationsYes  Invariant to deformationsNope  Insensitive to noise  Insensitive to topology  Robust to degeneracies... if they do not affect the FFT. Better than x(u) and y(u) alone

Papers Referenced This paper: – D. V. Vranic and D. Saupe. Description of 3D-Shape using a Complex Function on the Sphere. In: Proceedings of the IEEE International Conference on Multimedia and Expo (ICME 2002), Lausanne, Switzerland, pp , August 2002.D. Saupe.Description of 3D-Shape using a Complex Function on the SphereICME 2002 Prior and related papers: – D. V. Vranic and D. Saupe. 3D Model Retrieval. In: Proceedings of the Spring Conference on Computer Graphics and its Applications (SCCG2000) (editor B. Falcidieno), Budmerice, Slovakia, pp , May 2000.D. Saupe.3D Model RetrievalProceedingsSCCG2000 – D. Saupe and D. V. Vranic. 3D model retrieval with spherical harmonics and moments. In: Proceedings of the DAGM 2001 (editors B. Radig and S. Florczyk), Munich, Germany, pp , September D. Saupe3D model retrieval with spherical harmonics and momentsDAGM 2001 Spherical Harmonics and the Transform: – Fast Spherical Harmonic Transforms. – Courtesy of: Eric W. Weisstein. "Spherical Harmonic." From MathWorld--A Wolfram Web Resource.