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Chapter 3Surface Smoothing Based onArea-Decreasing FlowThis chapter proposes a new surface smoothing method based on area-decreasing flow,which can be used for preprocessing raw range data or postprocessing reconstructedsurfaces. Although surface area minimization is mathematically equivalent to the meancurvature flow, area-decreasing flow is far more efficient for smoothing a discrete surfaceon which the mean curvature is difficult to estimate. A general framework of regularizationbased on area-decreasing flow is proposed and applied to smoothing range dataand arbitrary triangle meshes. Crease edges are preserved by adaptively changing theregularization parameter. The edge strength of each vertex on a triangle mesh is computedby fusing the tensor voting and the orientation check of the normal vector fieldinside a geodesic window. Experimental results show the proposed algorithm provides34
successful smoothing for both raw range data and surface meshes.The literature review of range image smoothing and surface smoothing is given inSection 2.1. In Section 3.1, the regularized energy function is formulated for surfacesmoothing based on area-decreasing flow. Section 3.2 derives the area-decreasing stabilizerfor calibrated range data and shows how to preserve the edges. The stabilizerfor the triangle mesh is introduced in Section 3.3 along with the discussion of edgepreservation based on the tensor voting approach. Experimental results are presentedin Section 7.1.3.1 Regularization Based on Area-Decreasing FlowThe 3D image smoothing problem corresponds to a constrained optimization problem.Regularization is the most widely used method to solve practical optimization problemswith one or more constraints.Major advantages of regularization include: (i)simple and intuitive formulation of the objective or energy function, and (ii) flexibilityin incorporating one or more constraints into the optimization process.When regularization is applied to surface smoothing, x is replaced by X, aparameterizedsurface which represents a differentiable map from an open set U ⊂ R 2 intoR 3 , that is X : U ⊂ R 2 → R 3 . Given a bounded domain D ⊂ U and a differentiablefunction l : D → R, where D represents the union of the domain D and its boundary,35
Page 1 and 2: To the Graduate Council:I am submit
Page 3 and 4: Copyright c○ Yiyong Sun, 2002All
Page 5 and 6: AcknowledgmentsFirst and foremost,
Page 7 and 8: search employs an implicit surface-
Page 9 and 10: 3.2.2 Edge Preservation for Range D
Page 11 and 12: 8.2 Future Research . . . .........
Page 13 and 14: List of Figures1.1 A framework for
Page 15 and 16: 7.14 Finding corresponding points o
Page 17 and 18: Chapter 1IntroductionThis introduct
Page 19 and 20: Range ImageSmoothingRegistrationCol
Page 21 and 22: obust to surface noise and registra
Page 23 and 24: more efficient than 2D image correl
Page 25 and 26: scending path, and region merging.L
Page 27 and 28: two-dimensional (2D) image, 2D lowp
Page 29 and 30: curvatures of points on a surface i
Page 31 and 32: smoothing on the surface mesh. Shri
Page 33 and 34: with different surface representati
Page 35 and 36: center point and every surface poin
Page 37 and 38: Table 2.1: Comparison of different
Page 39 and 40: Reconstruction from Unorganized Poi
Page 41 and 42: tional. An offset of the distance f
Page 43 and 44: surement confidence. However, in or
Page 45 and 46: triangle mesh. The triangle mesh is
Page 47 and 48: mesh segmentation based on the phys
Page 49: Rettmann et al. [70] proposed a mes
Page 53 and 54: where lim t→0 (R/t) =0,H represen
Page 55 and 56: the range r ij is estimated instead
Page 57 and 58: y minimizing the following energy f
Page 59 and 60: tions were used:r α = r i,j+1 −
Page 61 and 62: where κ represents a parameter tha
Page 63 and 64: inequality, such asI∑I∑S i ≤
Page 65 and 66: n ′ in iT ipqn ipw in ′ i−→
Page 67: The singular value decomposition is
Page 70 and 71: a limit on sharp edges. This limit
Page 72 and 73: of the smoothing.Surface smoothing
Page 74 and 75: fingerprint are presented. Section
Page 76 and 77: erroneously includes surface patch
Page 78 and 79: (ρ, θ)ρθT p (S)expSexp p (ρ,
Page 80 and 81: exponential map is introduced.4.2 F
Page 82 and 83: accurately than Dijkstra’s algori
Page 84 and 85: andθ = arccos ρ2 (A)+|AB| 2 −
Page 86 and 87: andy =((n × −→ pm) × n) · v
Page 88 and 89: images.4.3 Candidate Point Selectio
Page 90 and 91: (a)(b)(c)(d)Figure 4.6: Global fing
Page 92 and 93: and R ij is below a threshold. The
Page 94 and 95: If the singular-value decomposition
Page 96 and 97: Chapter 5Surface Reconstruction fro
Page 98 and 99: Figure 5.2: Two registered and over
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Figure 5.4: Intersecting triangles.
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Figure 5.7: Gap between surfaces af
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5.2 Volume-Based Surface Integratio
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2D texture coordinate of x is set t
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ABFigure 5.9: Hole filling illustra
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are presented in Section 7.4.6.1 A
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6.3 Applying the Fast Marching Wate
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Algorithm 6.2 (Fast Marching Waters
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15: end if16: if v i.h
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6.3.2 Geodesic ErosionThis step ass
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(a)(b)(c)(d)Figure 6.3: Segmenting
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Chapter 7Experimental ResultsThis c
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no discernible improvement. For fai
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(a)(b)(c)(d)Figure 7.4: Surface smo
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(a)(b)(c)(d)Figure 7.6: Smoothing s
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(a)(b)(c)(d)(e)(f)Figure 7.8: Adapt
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The real range images were scanned
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(a)(b)Figure 7.10: Registration of
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(a)(b)(c)(d)(e)(f)Figure 7.12: Matc
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(a) (b) (c) (d)(e) (f) (g) (h)(i)(j
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(a)(b)ICP Registration Error (mm)21
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(a)(b)(c)(d)(e)(f)Figure 7.17: Surf
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Table 7.1: Small parts 3D reconstru
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(a)(b)(c)(d)Figure 7.21: Surface mo
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(a)(b)Figure 7.23: Automatic hole f
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(a)(b)(c)(d)(e)(f)(g)(h)Figure 7.25
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(a)(b)(c)(d)Figure 7.27: Four-view
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(a)(b)(c)(d)Figure 7.28: Segmentati
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(a)(b)(c)(d)Figure 7.30: Surface se
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Table 7.2: Performance of the fast
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surface reconstructed from the raw
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to choose a proper flowing step siz
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of interest based on the shape irre
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8.2 Future ResearchThere are many o
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Bibliography[1] E. L. Allgower and
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[26] G. Dziuk and J. E. Hutchinson.
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[54] W. E. Lorensen and H. E. Cline
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[81] M. Soucy and D. Laurendeau. A
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[106] S. M. Yamany, A. A. Farag, an
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Appendix ALaser Range ScannersLaser
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(a)(b)Transmitter LensTargetDiodeLa
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A.2 Scanners Based on Laser Triangu
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LaserBaseline B-s0saOpticalCenter(a
Page 186 and 187:
Appendix BSimplification of theArea
Page 188:
VitaYiyong Sun was born in Beijing,
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