Bilateral Mesh Denoisingsheffa/dgp/student_ppts/BilateralMeshDenoising.pdf2d Gaussian Filter...
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1 Bilateral Mesh Bilateral Mesh Denoising Denoising S. Fleishman, I. S. Fleishman, I. Drori Drori , D. Cohen , D. Cohen- Or Or Tel Aviv University Tel Aviv University Presented by Derek Bradley Presented by Derek Bradley 2 Derek Bradley 2006 Outline Outline Motivation Motivation Previous Work Previous Work Bilateral Mesh Bilateral Mesh Denoising Denoising Image Processing Background Image Processing Background Bilateral Image Filtering Bilateral Image Filtering Transforming from Images to Meshes Transforming from Images to Meshes Mesh Mesh Denoising Denoising Results Results Discussion Discussion (Unless otherwise noted, all images are from Fleishman et al.) (Unless otherwise noted, all images are from Fleishman et al.) 3 Derek Bradley 2006 Motivation Motivation 3D scanning creates noisy meshes 3D scanning creates noisy meshes Smoothing can reduce high frequency noise Smoothing can reduce high frequency noise Challenge: how do you know Challenge: how do you know what is noise and what are what is noise and what are features? features? 4 Derek Bradley 2006 Previous Work Previous Work Laplacian Laplacian Smoothing Smoothing i i i v v v ∆ ⋅ + = λ ( ) ∑ − = ∆ ) , ( ) , ( 1 j i i j i v v j i v 5 Derek Bradley 2006 Previous Work Previous Work Weighted Weighted Laplacian Laplacian ( ) ∑ ∑ − = ∆ ) , ( ) , ( 1 j i i j ij j i ij i v v w w v i i i v v v ∆ ⋅ + = λ 6 Derek Bradley 2006 Previous Work Previous Work Laplacian Laplacian + Expansion + Expansion 2 ) ( i i i v v v v ∆ − ∆ − + = µλ λ µ ( ) ∑ ∆ − ∆ = ∆ ) , ( 2 ) , ( 1 j i i j i v v j i v
Transcript of Bilateral Mesh Denoisingsheffa/dgp/student_ppts/BilateralMeshDenoising.pdf2d Gaussian Filter...
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Bilateral Mesh Bilateral Mesh DenoisingDenoising
S. Fleishman, I. S. Fleishman, I. DroriDrori, D. Cohen, D. Cohen--OrOrTel Aviv UniversityTel Aviv University
Presented by Derek BradleyPresented by Derek Bradley
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OutlineOutline
MotivationMotivationPrevious WorkPrevious WorkBilateral Mesh Bilateral Mesh DenoisingDenoising
Image Processing BackgroundImage Processing BackgroundBilateral Image FilteringBilateral Image FilteringTransforming from Images to MeshesTransforming from Images to MeshesMesh Mesh DenoisingDenoising
ResultsResultsDiscussionDiscussion
(Unless otherwise noted, all images are from Fleishman et al.)(Unless otherwise noted, all images are from Fleishman et al.)
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MotivationMotivation
3D scanning creates noisy meshes3D scanning creates noisy meshesSmoothing can reduce high frequency noiseSmoothing can reduce high frequency noiseChallenge: how do you know Challenge: how do you know what is noise and what are what is noise and what are features?features?
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Previous WorkPrevious Work
LaplacianLaplacian SmoothingSmoothing
iii vvv ∆⋅+= λ
( )∑ −=∆),(),(
1ji
iji vvji
v
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Previous WorkPrevious Work
Weighted Weighted LaplacianLaplacian
( )∑∑−=∆
),(),(
1ji
ijij
jiij
i vvww
v
iii vvv ∆⋅+= λ
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Previous WorkPrevious Work
LaplacianLaplacian + Expansion+ Expansion
2)( iii vvvv ∆−∆−+= µλλµ
( )∑ ∆−∆=∆),(
2
),(1
jiiji vv
jiv
2
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Previous WorkPrevious Work
Implicit Fairing (IF) Implicit Fairing (IF) [[DesbrunDesbrun 1999]1999]
Implicit integration of the diffusion equationImplicit integration of the diffusion equation
Anisotropic FeatureAnisotropic Feature--Preserving Preserving DenoisingDenoising (AFP) (AFP) [[DesbrunDesbrun 2000]2000]
Features detected using local curvatureFeatures detected using local curvatureDenoiseDenoise using weighted mean curvature smoothingusing weighted mean curvature smoothing
Penalize vertices with large ratio between principle curvaturesPenalize vertices with large ratio between principle curvatures
nn XdtLIX )(1 λ+=+ nn XXdtLI =+ +1)( λExplicit Implicit
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Bilateral Mesh Bilateral Mesh DenoisingDenoising
Application of an image smoothing techniqueApplication of an image smoothing techniqueVertices are moved along their normal directionVertices are moved along their normal direction
Scalar value Scalar value dd to be computed for each vertexto be computed for each vertexFeature preservingFeature preservingCan be iterative or singleCan be iterative or single--passpass
But first … some image processing basicsBut first … some image processing basics
iii ndvv ⋅+=
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Image Processing BackgroundImage Processing Background
An image is an array of integers (0An image is an array of integers (0--255)255)
[Tomasi and Manduchi]
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Image Processing BackgroundImage Processing Background
v v is the current pixelis the current pixelN(v) is the set of neighbouring pixels of vN(v) is the set of neighbouring pixels of vI(v) is the intensity of vI(v) is the intensity of v
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Bilateral Image FilteringBilateral Image Filtering
Goal: Smooth the image intensities, but preserve Goal: Smooth the image intensities, but preserve strong edges (features)strong edges (features)New intensity = weighted average of New intensity = weighted average of neighboursneighboursTwo weights:Two weights:
GeometricGeometric: Closer pixels weighted higher : Closer pixels weighted higher (closeness smoothing filter)(closeness smoothing filter)PhotometricPhotometric: Strong changes in intensity penalized : Strong changes in intensity penalized (similarity weight function)(similarity weight function)
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Bilateral Image FilteringBilateral Image Filtering
1d Gaussian Function1d Gaussian Function
[wikipedia.org]
Closeness Smoothing Filter
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Bilateral Image FilteringBilateral Image Filtering
2d Gaussian Filter2d Gaussian FilterCloseness Smoothing Filter
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2)( c
x
c exW σ−
=
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Bilateral Image FilteringBilateral Image Filtering
Another Gaussian functionAnother Gaussian function
x x = absolute difference in intensity values= absolute difference in intensity valuesResult: pixels with large changes in intensity are Result: pixels with large changes in intensity are weighted lowerweighted lower
Similarity Weight Function
22
2)( s
x
s exW σ−
=
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Bilateral Image FilteringBilateral Image Filtering
Combining the weights and normalizing:Combining the weights and normalizing:
In practice, In practice, N(v)N(v) is defined by the set of points: is defined by the set of points:
( ) ( )( ) ( )∑
∑
∈
∈
−⋅−
⋅−⋅−=
)(
)(
)()(
)()()()(ˆ
vNpsc
vNpsc
pIvIWvpW
pIpIvIWvpWvI
{ } cii qvq σ2 where, <−
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Bilateral Image FilteringBilateral Image Filtering
Results:Results:
[Tomasi and Manduchi]
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Transforming from Images to Transforming from Images to MeshesMeshes
Vertices instead of pixelsVertices instead of pixelsNeighbourhoodNeighbourhood N(v)N(v), defined the same, defined the sameCloseness smoothing filter:Closeness smoothing filter:
3D Euclidean distance instead of 2D 3D Euclidean distance instead of 2D
Similarity weight function:Similarity weight function:Heights of neighbouring vertices = pixel intensitiesHeights of neighbouring vertices = pixel intensities
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Transforming from Images to Transforming from Images to MeshesMeshes
Dot product between normal and (Dot product between normal and (vv--qqii) used ) used instead of computing the height at instead of computing the height at qqii
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Mesh Mesh DenoisingDenoisingDenoisePointDenoisePoint(Vertex(Vertex v, Normal n)v, Normal n)
{{qqii} = } = neighbourhood(vneighbourhood(v))K = |{K = |{qqii}|}|sum = 0, sum = 0, normalizernormalizer = 0= 0forfor i := 1 i := 1 toto KK
t = ||v t = ||v -- qqii||||h = <n, v h = <n, v -- qqii>>WWcc = exp(= exp(--tt22/(2σ/(2σcc
22))))WWss = exp(= exp(--hh22/(2σ/(2σss
22))))sum += (sum += (WWcc * W* Wss) * h) * hnormalizernormalizer += += WWcc * W* Wss
endendreturnreturn Vertex v = v + n * (sum/Vertex v = v + n * (sum/normalizernormalizer))
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ResultsResults
Input ImplicitFairing
BilateralDenoising
MeanCurvature
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ResultsResults
Input Anisotropic Denoisingof Height Fields (AFP)
BilateralDenoising
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DiscussionDiscussion
Issues when using an imageIssues when using an image--based technique on based technique on a mesh:a mesh:
Only applies to manifold meshesOnly applies to manifold meshesIrregularity of meshesIrregularity of meshesShrinkageShrinkageVertex driftVertex drift
Handling boundariesHandling boundariesMirror Mirror neighboursneighbours at boundary vertices at boundary vertices Virtual vertices at infinity (used in this algorithm)Virtual vertices at infinity (used in this algorithm)
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DiscussionDiscussion
Setting the parameters (Setting the parameters (σσcc, , σσss, # iterations), # iterations)UserUser--assisted methodassisted methodσσcc and and σσss ::
User selects smooth point and radius on the meshUser selects smooth point and radius on the meshLarge Large σσcc = few iterations, small = few iterations, small σσcc = more iterations= more iterationsSmall Small σσcc makes sense makes sense
large values can cross features large values can cross features smaller smaller neighbourhoodneighbourhood leads to faster iterationsleads to faster iterations
< 6 iterations for all results in the paper< 6 iterations for all results in the paper
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DiscussionDiscussion
Independently, Jones et al. present the same Independently, Jones et al. present the same algorithm with minor differences:algorithm with minor differences:
Surface predictorSurface predictorSingle passSingle pass
Input Jones et al. BilateralDenoising
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DiscussionDiscussion
DisadvantagesDisadvantagesAssumes wellAssumes well--behaved meshesbehaved meshesCan result in selfCan result in self--intersectionintersection
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ConclusionConclusion
Simple, effective and fast algorithm for Simple, effective and fast algorithm for denoisingdenoising meshesmeshesEasy to implementEasy to implementTakes advantage of the success of an image Takes advantage of the success of an image processing techniqueprocessing technique
Would I implement this algorithm?Would I implement this algorithm?
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ReferencesReferences
S. Fleishman, I. S. Fleishman, I. DroriDrori, D. Cohen, D. Cohen--Or. Bilateral mesh Or. Bilateral mesh denoisingdenoising. . SIGGRAPH 2003.SIGGRAPH 2003.C. C. TomasiTomasi, R. , R. ManduchiManduchi. Bilateral filtering for gray and color . Bilateral filtering for gray and color images. ICCV 1998.images. ICCV 1998.T. Jones, F. Durand, M. T. Jones, F. Durand, M. DesbrunDesbrun. Non. Non--iterative featureiterative feature--preserving mesh smoothing. SIGGRAPH 2003.preserving mesh smoothing. SIGGRAPH 2003.M. M. DesbrunDesbrun, M. Meyer, P. , M. Meyer, P. SchroderSchroder, A.H. Barr. Implicit fairing , A.H. Barr. Implicit fairing of irregular meshes using diffusion and curvature flow. of irregular meshes using diffusion and curvature flow. SIGGRAPH 1999.SIGGRAPH 1999.M. M. DesbrunDesbrun, M. Meyer, P. , M. Meyer, P. SchroderSchroder, A.H. Barr. Anisotropic , A.H. Barr. Anisotropic featurefeature--preserving preserving denoisingdenoising of height fields and of height fields and bivariatebivariate data. data. Graphics Interface 2000.Graphics Interface 2000.www.wikipedia.orgwww.wikipedia.org
