Hemispherical Harmonic Surface Description and Applications to MedicalImage Analysis

Heng Huang Lei Zhang, Dimitris SamarasDepartment of Computer Science Department of Computer Science

Dartmouth College SUNY at Stony BrookHanover, NH, 03755 Stony Brook, NY, 11794

Li Shen Fillia Makedon Justin PearlmanComputer and Information Science Department of Computer Science Department of Cardiology

Univ. of Mass. Dartmouth Dartmouth College Dartmouth Medical SchoolNorth Dartmouth, MA 02747 Hanover, NH, 03755 Lebanon, NH 03756

Abstract

The use of surface harmonics for rigid and nonrigid shapedescription is well known. In this paper we define a setof complete hemispherical harmonic basis functions on ahemisphere domain and propose a novel parametric shapedescription method to efficiently and flexibly represent thesurfaces of anatomical structures in medical images. Asthe first application of hemispherical harmonic theory inshape description, our technique differs from the previoussurface harmonics shape descriptors, all of which don’twork efficiently on the hemisphere-like objects that often ex-ist in medical anatomical structures (e.g., ventricles, atri-ums, etc.). We demonstrate the effectiveness of our ap-proach through theoretic and experimental exploration of aset of medical image applications. Furthermore, an evalu-ation criterion for surface modeling efficiency is describedand the comparison results demonstrated that our methodoutperformed the previous approaches using spherical har-monic models.

1. IntroductionThree dimensional (3D) shape description is a fundamentalissue in 3D computer vision with many applications, suchas shape registration, 3D object recognition and classifica-tion [1]. For a 3D object, its shape descriptor often consistsof a set of parameters that can capture its shape informa-tion well. A global surface shape descriptor uses explicit orimplicit functions to describe an entire surface shape intoa small set of parameters, where every parameter affectsthe whole surface representation, not just local parts. Asone important application in computer vision, medical im-age analysis also requires efficient and accurate shape de-scriptions. In particular many diseases resulting in or from

morphologic variations of structures (e.g., heart, brain,etc.)shows the importance of analysis of shape variability fordiagnostic classification and understanding of biomedicalprocesses. There has been growing interest in shape de-scription for medical applications [2]. Although, global sur-face description techniques have been proposed to addressthe biomedical anatomical structures reconstruction prob-lems, different types of shape descriptors are needed due tothe variety of different anatomical structures,i.e., the previ-ous surface harmonics shape descriptors cannot model thehemisphere-like objects efficiently. In this paper, we pro-pose a novel global parametric surface description methodthat is in the form of linear combination of hemisphericalbasis functions. We examine the effectiveness of our ap-proach through theoretic and experimental exploration ofa set of medical image applications and demonstrate thatour model works for simply connected objects and is espe-cially suitable for hemisphere-like objects such as medicalanatomical structures (ventricles, atriums,etc.).

In this paper, we apply our hemispherical harmonicshape description method on medical image analysis, how-ever, our method is a general technique to be applied intomany other areas in computer vision such as 3D face recog-nition, shape-based registration,etc.

1.1. Previous workMany shape description methods and their applications ingeodesy, biomedicine, molecular biology, and other fieldshave been explored [1]. Here we provide a summary ofalternative approaches as context for our contribution.

Mathematically there are different methods of describ-ing shapes including parametric models (such as harmonicfunctions [17], hyperquadrics [3], or superquadrics [4, 5]),medial axis (skeleton) [7, 8], distance distributions [6] and

1

landmark theory based descriptors [9]. Because sphericalharmonic descriptions are smooth, accurate fine-scale shaperepresentations with a sufficiently small approximation er-ror [18], they are widely studied and used in different ap-plications. In the shape matching field, Funkhouser andKazhdanet al. [10, 11] proposed a rotation invariant spher-ical harmonic representation for 3D shapes to avoid theregistration errors. Spherical harmonics also have a well-established standing in the molecular surfaces descriptionand docking [13, 14]. Meanwhile many spherical harmon-ics based shape descriptions have been developed for med-ical image analysis [12]. For instance, in [15], Chenet al.presented their spherical harmonic model to analyze the leftventricular shape and motion. In [22], similarly, Edvardsonand Smedby viewed a 3D object as a radial distance func-tion on the unit sphere and tested their method on a data setfrom magnetic resonance imaging (MRI) of the brain. Ma-theny and Goldgof [16] used 3D and 4D surface harmon-ics to reconstruct rigid and nonrigid shapes. Because theyused the radial surface function (r(θ, φ)) in all models, theirmethods are limited to represent only star-shape or convexobjects without holes.

Brechbuhleret al. [17] presented an extended spheri-cal harmonic (SPHARM) method to model any simply con-nected 3D object. A closed input object surface is assumedto be defined by a square surface parameter mesh convertedfrom an isotropic voxel representation. The key compo-nent of this method is the mapping of surfaces of volumet-ric objects to parametrized surfaces prior to expansion intoharmonics. SPHARM method have been applied in manymedical imaging applications,e.g., shape analysis of brain[19, 18] and cardiac [20, 21] structures. Since this methodstart the first order|Y 0

0 (θ, φ)| from an ellipsoid, it is suit-able to represent the surfaces with sphere topology, but notefficient in shape reconstruction of hemisphere-like objects.

In other research areas, spherical and hemisphericalharmonics are widely used for representation of BRDFs(bi-directional reflectance distribution function), environ-ment maps, illumination variation and image recognition[23, 24, 27],etc. Since they were applied on light/surfaceinteraction models, their models used the radial surfacefunction r(θ, φ). Because our paper focuses on shape de-scription, the review of these approaches is not provided.

1.2. Our contributionAs the first paper to introduce hemispherical harmonic the-ory into shape description, the contribution of this paperis to develop a set of complete hemispherical harmonicHm

l (θ, φ) with shifted associated polynomials and provetheir orthonormality property on a hemisphere domain. Wepropose the novel hemispherical harmonic surface descrip-tion method to meet the requirement of surface reconstruc-tion of hemisphere-like anatomical structures. Since our ba-

sis functionsHml (θ, φ) start from a hemisphere (order0,

|H00 (θ, φ)|), our new method can represent the hemisphere-

like objects efficiently by using less coefficients to describethe same surface. Furthermore, in order to evaluate the per-formance quantitatively, we provide a comparison criterionto the modeling efficiency and a shortcut to compute themodeling efficiency by using the hemispherical harmoniccoefficients directly. The result of comparison with the pre-vious SPHARM method shows our method outperformedthe previous approaches. The applications are vast suchas structures in cardiac MR image sequences [28]. Ourmethod performing on the left ventricular surface recon-struction and shape sequence description is shown in sec-tion 3.

As a smooth and accurate fine-scale shape representa-tion method, our hemispherical harmonic shape descriptorsare useful in shape classification for statistical analysisofanatomical shape differences. Because we map the surfacesof voxel objects to parametrized surfaces before expansioninto hemispherical harmonics, our method works well insimply connected objects (not only for star shapes). Sincethere is no shape descriptor for non-closed left ventricle inmedical image analysis, some constraint rules were addedinto parametrization step to generate the shape descriptorsfor non-closed left ventricle or the other hemisphere-likeanatomical structures. This non-closed shape representationis promising in the further shape analysis research.

This paper is organized as follows. Section 2 introducesthe hemispherical harmonic basis functions and new sur-face description technique. Section 3 shows medical imag-ing applications to demonstrate our new surface descriptor.Section 4 evaluates its performance by comparing surfacemodeling efficiency with SPAHRM method. Section 5 con-cludes the paper.

2. Hemispherical harmonic shape de-scriptor

In this section, we first propose a set of new hemispheri-cal harmonics (HemiSPHARM)Hm

l (θ, φ) and demonstratetheir orthonormality property. We then describe our surfaceparametrization method that generates the HemiSPHARMshape description result. At last, we will evaluate the mod-eling efficiency of our HemiSPHARM description by com-paring with spherical harmonics.

Previous SPHARM [17] represented the object surfaceas

v(θ, φ) =

x(θ, φ)y(θ, φ)z(θ, φ)

, (1)

whereθ ∈ [0, π] is the polar angle andφ ∈ [0, 2π) is theazimuthal angle. When the free variablesθ and φ rangeover the whole sphere,v(θ, φ) ranges over the whole object

2

−0.3−0.2

−0.10

0.10.2

0.3

−0.3−0.2

−0.10

0.10.2

0.3

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

(a) |H0

0(θ, φ)|

00.1

0.20.3

0.4

−0.2

−0.1

0

0.1

0.2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(b) |H−1

1(θ, φ)|

−0.6−0.4

−0.20

0.20.4

0.6

−0.6−0.4

−0.20

0.20.4

0.6

0

0.1

0.2

0.3

0.4

0.5

0.6

(c) |H0

1(θ, φ)|

−0.4−0.3

−0.2−0.1

0

−0.2−0.1

00.1

0.2

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(d) |H1

1(θ, φ)|

−0.4−0.2

00.2

0.4

−0.4

−0.2

0

0.2

0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(e) |H−2

2(θ, φ)|

−0.20

0.20.4

−0.20

0.2

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(f) |H−1

2(θ, φ)|

−0.5

0

0.5

−0.5

0

0.5

−0.8

−0.6

−0.4

−0.2

0

0.2

(g) |H0

2(θ, φ)|

−0.4−0.2

00.2

−0.20

0.2

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(h) |H1

2(θ, φ)|

−0.4−0.3

−0.2−0.1

00.1

0.20.3

0.4

−0.4

−0.2

0

0.2

0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(i) |H2

2(θ, φ)|

Figure 1: The visualization for hemispherical harmonics|Hml (θ, φ)|, for m = −l, ..., l, l = 0, 1, 2.

surface. The spherical harmonics expansion is used for allthree coordinates and the surfacev(θ, φ) is expressed asa linear combination of spherical harmonic basis functionsY m

l (θ, φ) of varying orderl and degreem,

v(θ, φ) =

∞∑

l=0

l∑

m=−l

cml Y m

l (θ, φ), (2)

wherec

ml =

(

cmlx, cm

ly , cmlz

)T. (3)

The spherical harmonic basis functions are single-valued,smooth (infinitely differentiable), complex functions of(θ, φ):

Y ml (θ, φ) =

√

2l + 1

4π

(l − m)!

(l + m)!Pm

l (cos θ)eimφ, (4)

where Pml (cos θ) are associated Legendre polynomials

(with argumentcos θ) that is defined by the differentialequation

Pml (x) =

(−1)m

2ll!(1 − x2)m/2 dm+l

dxm+l(x2 − 1)l, (5)

wherel andm are integers with−l ≤ m ≤ l. They areorthogonal over[−1, 1] with respect tol with the weightingfunctionw(x) = 1 [25]:

∫ 1

−1

Pml (x)Pm

l′ (x)dx =2(m + l)!

(2l + 1)(l − m)!δll′ . (6)

The coefficientscml are 3D vectors. Their components,

cmlx, cm

ly , andcmlz are usually complex numbers with a user-

desired degree. These coefficients can be calculated bysolving a set of linear equations in a least square fashion.As a result a set of coefficients is then used to express thesurface in compact.

As an important property, the orthonormality property ofspherical harmonic basis functions can be expressed as

∫ 2π

0

∫ π

0

Y ml (θ, φ)Y m′

l′ (θ, φ)sinθdθdφ = δmm′δll′ . (7)

Here,z denotes the complex conjugate andδij is the Kro-necker delta:

δij ≡

{

0 for i 6= j1 for i = j

(8)

2.1. HemiSPHARM basis functionThe standard definition of harmonic basis functions [24]starts from choosing the Legendre polynomialsPl(x) thatsatisfy the orthogonality relationship with the weight func-tion w(x) = 1. Based on the unassociated Legen-dre polynomialsPl(x), the associated Legendre polyno-mials Pm

l (x) are defined. At last the harmonics basisfunctions are constructed by combination ofPm

l (x) with{cos(mφ), sin(mφ)}.

As the first step in definition of HemiSPHARM we usethe shifted Legendre polynomials [25] that are a set of func-tions analogous to the Legendre polynomials, but definedon the interval[−1, 0]. For Legendre polynomialsPl(x),we use the linear transformationx = 2x′ + 1 to create thenew polynomials:

Pl(x′) = Pl(2x′ + 1). (9)

From [26], for orthogonal polynomialsPl(x), the lin-ear transformationx = kx′ + h, k 6= 0, carries over theinterval [a, b] into an interval[a′, b′] (or [b′, a′]), and theweight functionw(x) into w(kx′ + h). The polynomials(sgnk)l|k|

1

2 Pl(kx′ + h) are also orthogonal on the interval

3

[a′, b′] (or [b′, a′]) with the weight functionw(kx′ +h). Be-causex ∈ [−1, 1] in Legendre polynomialsPl(x), Pl(x

′)are also orthogonal on the interval[−1, 0] with the weightfunctionw(x) = 1. Thus, the orthogonal associated poly-nomialsPm

l (x′) are defined as:

Pml (x′) = (−1)m(1 − x′2)m/2 dm

dxmPl(x

′)

= Pml (2x′ + 1), (10)

and with respect tol,

∫ 0

−1

Pml (x′)Pm

l′ (x′)dx′ =

∫ 0

−1

Pml′ (2x′+1)Pm

l′ (2x′+1)dx′,

(11)using Eq. (6),

∫ 0

−1

Pml (x′)Pm

l′ (x′)dx′ =(m + l)!

(2l + 1)(l − m)!δll′ (12)

Therefore, we get a set of orthogonal associated polynomi-als Pm

l (cos θ) that are defined in the intervalθ ∈ [π2, π].

Their relationship to associated Legendre polynomials is:

Pl(cos θ) = Pl(2 cos θ + 1) onθ ∈ [π

2, π]. (13)

Based on our shifted associated polynomialsPml (cos θ),

we construct the HemiSPHARM basis functionsHml (θ, φ)

as:

Hml (θ, φ) =

√

2l + 1

2π

(l − m)!

(l + m)!Pm

l (cosθ)eimφ, (14)

with θ ∈ [π2, π] andφ ∈ [0, 2π). Here, in order to keep the

orthonormality ofHml (θ, φ), we change the normalization

value from√

2l + 1

4π

(l − m)!

(l + m)!to

√

2l + 1

2π

(l − m)!

(l + m)!.

As a result,Hml (θ, φ) are orthogonal over[π

2, π] × [0, 2π)

with respect to bothl andm:∫

2π

0

∫ π

π

2

Hml (θ, φ)Hm′

l′ (θ, φ)sinθdθdφ = δmm′δll′ . (15)

Figure 1 shows the visualization of HemiSPHARM basisfunctions forl = 0, 1, 2 andm = −l, ..., l.

2.2. Surface parametrizationIn order to describe a voxel surface (figure 2(b)) usingHemiSPHARM, we first need to create a continuous anduniform mapping from the object surface (see figure 2(c))to the surface of a half unit sphere (see figure 2(d)) so thateach vertex on the object surface can be assigned a pair

of spherical coordinates(θ, φ) in figure 2(a). This pro-cess is calledsurface parameterization, and the surface ofthe half unit sphere becomes our parameter space. We em-ploy a hemispherical parameterization approach that is sim-ilar to the spherical parameterization approach proposed byBrechbuhleret al. [17] to exploit a square surface mesh.

The parameterization is constructed by creating a har-monic map from the object surface to the parameter sur-face. For colatitudeθ two poles are selected in the surfacemesh by finding the two vertices with the maximum (forthe hemisphere-like object, it should be the center of topslice which includes many points with the same or closemaximumz values,e.g., our parametrization for left ven-tricular surface) and minimumz coordinate in object space.Then, a Laplace equation (Eq. (16)) with Dirichlet condi-tions (Eq. (17) and Eq. (18)) is solved for colatitudeθ:

∇2θ = 0 (except at the poles) (16)

θnorth =π

2(17)

θsouth = π (18)

Since our case is discrete, we can approximate Eq. (16)by assuming that each vertex’s colatitude (except at thepoles’) equals the average of its neighbours’ colatitudes.Thus, after assigningθnorth = π

2to the north pole and

θsouth = π to the south pole, we can form a system oflinear equations by considering all the vertices and obtainthe solution by solving this linear system. For longitudeφthe same approach can be employed except that longitude isa cyclic parameter. To overcome this problem, a “date line”is introduced. When crossing the date line, longitude is in-cremented or decremented by2π depending on the crossingdirection. After slightly modifying the linear system ac-cording to the date line, the solution for longitudeφ canalso be achieved.

2.3. Surface description

The parameterization result is a bijective mapping betweeneach vertexv = (x, y, z)T on a surface and a pair of spher-ical coordinates(θ, φ) (θ ∈ [π

2, π], φ ∈ [0, 2π)). We use

v(θ, φ) to denote such a mapping, meaning that, accordingto the mapping,v is parameterized with the spherical co-ordinates(θ, φ). Taking into consideration the x, y, and zcoordinates ofv in object space, the mapping can be repre-sented as:

v(θ, φ) =

x(θ, φ)y(θ, φ)z(θ, φ)

. (19)

We use the surface net representation to expand the sur-face of object into our HemiSPHARM basis functions with

4

x

θ

φ

z

y

(θ,φ)

(a) Spherical coordinates (b) Voxel object surface (c) Object surface inparametrization mapping

(d) Hemispherical surface inparametrization mapping

(e) Reconstructed surfacewith order 1

(f) Reconstructed surfacewith order 2

(g) Reconstructed surfacewith order 5

(h) Visualization of both ventricles

Figure 2: Surface parametrization and reconstruction. (a)shows the spherical coordinates(θ, φ); (b) shows the voxel surfaceof left ventricle before surface reconstruction; (c) and (d) show the parametrization mapping from the object surface (c) tothe surface of a half unit sphere (d); (e)-(g) show the reconstructed surfaces of left ventricle with order 1, 2, 5 respectively;(h) shows the reconstructed results of both ventricular (right and left) surfaces.

the coefficientscml = (cm

lx, cmly , cm

lz)T :

v(θ, φ) =

x(θ, φ)y(θ, φ)z(θ, φ)

=

∞∑

l=0

l∑

m=−l

cml Hm

l (θ, φ)

≈

L∑

l=0

l∑

m=−l

cml Hm

l (θ, φ), (20)

whereL is used to truncate the series into a desired finitenumber of terms. Figure 2(g) shows the reconstructed sur-face result of voxel object in figure 2(b). The coefficientsare calculated by forming the inner product ofv with theHemiSPHARM basis functions:

cml = 〈v(θ, φ), Hm

l (θ, φ)〉

=

∫ π

π

2

∫ 2π

0

v(θ, φ)Hml (θ, φ)dφ sin θdθ. (21)

3. Medical applications of HemiS-PHARM shape descriptors

In this section we will describe how to apply our HemiS-PHARM shape descriptors into shape analysis in medicalimage analysis applications. Based on segmented image

data of medical anatomical structures, we use the HemiS-PHARM method explained above for surface reconstruc-tion and our novel shape description allows researchers toperform further shape analysis or classification and accessmore functional details. In the following, we describe howthe HemiSPHARM shape descriptors can help the medicalapplications in details.

3.1. Surface reconstruction in cardiac MRIVisualization and modeling of cardiac shapes can providedirect and reliable indicators of cardiac function [28]. Sincethe shapes of hearts and ventricles are close to hemiellip-soid, it’s intuitive to apply our HemiSPHARM descriptorsto describe their shapes. Furthermore, in section 4, we willdemonstrate that our description is more efficient than pre-vious SPHARM method in hemisphere-like shape model-ing. By using the segmented cardiac MRI data, the HemiS-PHARM model can accurately visualize cardiac or ventriclegeometry and function and provide the shape descriptors.

Figure 2(e), 2(f), 2(g) show the surface reconstructionresult of left ventricle which is the most important func-tional structure in cardiac studies. By increasing the valueof order l from 1, 2 to 5, a more detailed surface is cre-ated. Meanwhile we reconstructed both ventricular (rightand left) surfaces and the result is shown in figure 2(h).

5

(a) (b) (c) (d) (e) (f)

Figure 3: The visualization result of shape sequence of a left ventricular inner surface during one heart cycle. The shapeschange from a diastolic phase (with largest volume) to next diastolic phase and the middle ones lie close to the systolic phase(with smallest volume).

This new surface description method also providesa promising visualization and representation method forstudying spatio-temporal cardiac structures. For a beatingheart, each surface model during a heart cycle can be gen-erated by HemiSPHARM method and a set of surfaces se-quence can describe the heart contraction and dilation inevery direction of 3D space. Based on this shape sequencevisualization result, more valuable diagnostic and prognos-tic information can be derived for helping make clinical de-terminations. Figure 3 describes a shape sequence recon-struction result of a left ventricular inner surface duringoneheart cycle. The shapes change from a diastolic phase (withlargest volume) to next diastolic phase and the middle oneslie close to the systolic phase (with smallest volume).

3.2. Surface reconstruction for non-closed ob-jects

During cardiac shape analysis study, since the ventricles andatriums are open objects, people only need the surface de-scriptions without the top parts. But the traditional sphericalharmonics methods work only for closed surface. Thus, theclosed shape descriptions introduce errors into shape anal-ysis and classification for cardiac shape studies. In order tosolve this problem, we added constraint rules into the sur-face parametrization step presented in section 2.2: 1) thepoints on the boundary of top slice (see figure 2(b)) areparametrized as[π/2, φ]; 2) the apex point of ventricle isparametrized as[π, φ]. The new parametrization result isshown in figure 4(a) and the surface reconstruction resultsare shown in figure 4(b).

After removing the top parts, these non-closed shapedescriptors are more accurate in the left ventricular shaperepresentation than the closed description method. Theycan provide more functional shape information for cardiacshape analysis.

4. Modeling efficiency analysisIn this section, we show the modeling efficiency (or accu-racy) of our HemiSPHARM description by comparing to

(a) (b)

Figure 4: (a) shows the new parametrized surface for non-closed shape representation; (b) shows the reconstructionresults for non-closed left ventricular surface.

the previous method. Since the methods [12, 15, 22, 16]that did harmonics expansion without mapping the surfaceof volumetric object to parametrized surface can only workon the star shape, we compare our method to the SPHARMmethod proposed by Brechbuhleret al. [17] on single con-nected objects.

In order to analyze the number of required coefficients,we use the energy fraction to determine the captured energyby a given number of coefficients:

Modeling Efficiency =EL

E

=

L∑

l=0

l∑

m=−l

(|cmlx|

2 + |cmly |

2 + |cmlz |

2)

∫ 2π

0

∫ π

π

2

v(θ, φ)2 sin θdθdφ

.

When the centroids of objects are moved to the origin, wecan compute the energyE in the definition of modeling ef-ficiency (or accuracy) as follows (it is different to the pre-vious definition in [27], because their definition only work

6

for radial surface function based spherical harmonics):

E =

∫ 2π

0

∫ π

π

2

v(θ, φ)2 sin θdθdφ

=

∫ 2π

0

∫ π

π

2

(∑

f∈{x,y,z}

f(θ, φ)2 sin θdθdφ

=∑

f∈{x,y,z}

(

∫ 2π

0

∫ π

π

2

f(θ, φ)2 × sinθdθdφ)

=∑

f∈{x,y,z}

(

∫ 2π

0

∫ π

π

2

(

L∑

l=0

l∑

m=−l

cmlfHm

l (θ, φ)

×L

∑

l=0

l∑

m=−l

cmlfHm

l (θ, φ))sinθdθdφ) (22)

By the orthonormality property of HemiSPHARM basisfunctions Eq. (15),

∫

2π

0

∫ π

π

2

v(θ, φ)2 sin θdθdφ =

∞∑

l=0

l∑

m=−l

(|cmlx|

2 + |cmly |

2 + |cmlz |

2)

>

L∑

l=0

l∑

m=−l

(|cmlx|

2 + |cmly |

2 + |cmlz |

2).

Figure 5: The comparison result of modeling efficiencybetween the HemiSPHARM description and SPHARMmethod in our experiment (left ventricle surface reconstruc-tion is used in comparison). The red line represents theHemiSPHARM modeling efficiency and the green line rep-resents SPHARM method. Thex-axis is the values of orderl.

Obviously if we increase the order numberL, the model-ing efficiency towards to 1. Figure 5 shows the comparisonresult of modeling efficiency between the HemiSPHARMdescription and SPHARM method in our experiment (leftventricle surface reconstruction is used with red line repre-senting the HemiSPHARM modeling efficiency and green

line representing SPHARM). Because our HemiSPHARMbasis functions start from a hemisphere (see Figure 1(a))and the SPHARM basis functions start from sphere, HemiS-PHARM method can describe the hemisphere-like objectsmore efficiently. In other words, HemiSPHARM requiresfewer coefficients (less orderl) than SPHARM to describethe same surfaces of hemisphere-like objects.

5. Conclusion and future workThe contribution of this paper is a novel shape descriptionmethod for the requirement of surface reconstruction forhemisphere-like anatomical structures. In order to proposethe new shape descriptors, we use a set of new orthogonalassociated polynomials to generate the orthonormal hemi-spherical harmonicsHm

l (θ, φ). The hemispherical harmon-ics are defined on a hemisphere domain and we map the sur-faces of volumetric objects to parametrized surfaces priortoexpansion into hemispherical harmonics. The success of thealgorithm is in its modeling efficiency by using fewer coef-ficients than previous spherical harmonic method to repre-sent the same surface. The surface reconstruction resultsof using segmented cardiac MRI clearly demonstrate theeffectiveness of our shape description method. Our newmethod is a general technique and can be applied into shapematching, 3D face recognition, molecular surfaces registra-tion and docking, or other research fields in computer vi-sion.

There are several opportunities for future work on thisnovel method. One of our future works is focused on addingthe optimized surface parametrization step to perform theequal area mapping for parametrizing the volumetric ob-jects. The shape representation for non-closed simple con-nected objects provides promising shape descriptors for thefurther shape analysis. Another research direction can bethe left ventricle shape classification for cardiac functionalanalysis on shape differences.

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