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A Diffusion Wavelet Approach for 3-D Model Matching,[object Object],Authors: K.P. Zhu, Y.S. Wong, W.F. Lu, J.Y.H. Fuh,[object Object],Presented by: Raphael Steinberg,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],2,[object Object]
Introduction,[object Object],Currently - A larger than ever number of 3D Models in CAD, computer games, multimedia, molecular biology, computer vision and more,[object Object],There is a need for 3D Retrieval ,[object Object],3,[object Object]
Introduction (2),[object Object],Tagging are not always available or sufficient to describe the model we require,[object Object],Combine topological information with multi-scale properties,[object Object],4,[object Object]
Model Reusability (CAD/Animation),[object Object],Model Matching,[object Object],Video Retrieval (2.5D/Virtual environments),[object Object],Ecommerce,[object Object],Correcting defects,[object Object],Efficient Representation,[object Object],Many other uses…,[object Object],Motivation for 3D Retrieval,[object Object],5,[object Object]
Obstacles in Retrieval,[object Object],Partial retrieval - Non-transitive,[object Object],Functional description,[object Object],How to match text tags with vertices and texture?,[object Object],Orthonormal coordinate system,[object Object],6,[object Object]
7,[object Object]
3D Model Matching – Prior Art,[object Object],Feature vectors using wavelets to mesh vertices – localized in both space & frequency – ,[object Object],Paquet et. al. 2000,[object Object],Random sampling for comparison – Osada et. al. 2001,[object Object],8,[object Object]
Spherical harmonics (SH),[object Object],Global method in Euclidean space,[object Object],lacks multi-scale analysis ,[object Object],Legendre polynomials solve the Laplace equation in Spherical coordinates,[object Object],Vranic et. al. 2001,[object Object],9,[object Object]
Spherical Wavelets (SW),[object Object],Multi-scale in Euclidean space ,[object Object],Lacks connectivity on the manifold,[object Object],Tannenbaum et. al. 2007,[object Object],10,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],11,[object Object]
Diffusion Maps Introduction,[object Object],Originally suggested by Stephan Lafon and  R.R. Coifman from Yale Math, circa 2005,[object Object],Many other manifold learning techniques exist,[object Object],Data analysis based on geometric properties of the data set,[object Object],12,[object Object]
Manifold Learning algorithms,[object Object],13,[object Object],MANI - Manifold learning Matlab tool,[object Object]
Diffusion Maps,[object Object],Z,[object Object],Y,[object Object],X,[object Object],Coifman - 2005,[object Object],[object Object]
 Contains descriptive information about the 3D model14,[object Object]
Diffusion Maps,[object Object],Assumptions,[object Object],[object Object]
 Smooth manifold (no fractals in our case)
 Fixed boundary conditions
 Enough points = feature vectors (N→∞)15,[object Object]
Use RBF Gaussian Kernel to choose ε,[object Object],Normalize W to create a Stochastic Matrix,[object Object],Diffusion Maps Algorithm,[object Object],16,[object Object],Lu et. al. 2009,[object Object]
Diffusion Maps algorithm (2),[object Object],Diffuse by taking higher powers of t ,[object Object],	“The diffusion distance is equal to the Euclidean distance in the diffusion map space” , Nadler et. al. 2005,[object Object],Cut manifold according to dominant eigenvalues,[object Object],17,[object Object]
Diffusion Maps Code Example,[object Object],function checker();,[object Object],close all;,[object Object],tetha=2*pi*rand(1,500);,[object Object],z=[cos(tetha);sin(tetha)];,[object Object],figure(1);scatter(z(1,:),z(2,:),'b*');hold on;,[object Object],N=size(z,2);,[object Object],epsilon=linspace(0.01,.3,10);,[object Object],%epsilon=.3;,[object Object],W=nan(N);,[object Object],summer=nan(1,length(epsilon));,[object Object],for k=1:length(epsilon),[object Object],    for i=1:N,[object Object],        parfor j=1:N,[object Object],            W(i,j)=exp(-sum((z(:,j)-z(:,i)).^2)/2/epsilon(k));,[object Object],        end,[object Object],    end,[object Object],    summer(k)=sum(sum(W));,[object Object],end,[object Object],figure;scatter(log(epsilon),log(summer));title('Epsilon - linear region'),[object Object],p=polyfit(log(epsilon),log(summer),1);,[object Object],d=2*p(1);%manifold dimension,[object Object],M=W*diag(1./sum(W,2));,[object Object],[U V]=svds(M);,[object Object],sync=max(U(:,2));,[object Object],figure(1);scatter(U(:,2)./sync,U(:,3)./sync,'rd'),[object Object],title('Original manifold as stars and reconstructed manifold as diamonds'),[object Object],end,[object Object],18,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],19,[object Object]
Problems with Mesh Simplification,[object Object],20,[object Object]
Wavelets,[object Object],21,[object Object]
Novelty – Diffusion Wavelets,[object Object],Combination of Diffusion Maps and Wavelets,[object Object],Used for non-linear dimensionality reduction,[object Object],Extension of wavelets to the unit circle (just as diffusion maps extends the Fourier transform),[object Object],22,[object Object]
Diffusion Wavelets Intuition,[object Object],23,[object Object]
Example of Diffusion Wavelets,[object Object],24,[object Object],Wavelet basis ψ(2,2,3),[object Object],Scaling basis φ(1,1,1),[object Object],Wavelet basis ψ(4,2,5),[object Object],Wavelet,[object Object],basis ψ(3,2,3),[object Object]
Diffusion Wavelets,[object Object],Use an optimization scheme to construct the scaling functions,[object Object],Each scaling function should deal with a single dimension and be orthogonal to the other scaling functions,[object Object],Extension of wavelets to the sphere (or to any other manifold),[object Object],25,[object Object]
Diffusion Wavelets (2),[object Object],Better than LOD (Level of Detail - simplifies meshes),[object Object],Involved algorithm – very few implementations exist,[object Object],26,[object Object]
Wavelet decomposition example,[object Object],27,[object Object]
Wavelet coefficients,[object Object],28,[object Object],Scale 1,[object Object],Scale 2,[object Object],Scale 3,[object Object],Scale 4,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],29,[object Object]
Finding Shape Feature Vectors (X),[object Object],30,[object Object],[object Object]
Increase wavelet coefficient number from 0 to 450 and decreasescaling coefficient number from 1,450 to 1,000,[object Object]
Fisher’s Discriminant Ratio,[object Object],32,[object Object]
IRPR Curve,[object Object],Measure performance – use Princeton University 3D database,[object Object],IRPR – Information Retrieval Precision-Recall,[object Object],33,[object Object]
IRPR Curve,[object Object],m = relevant matches ,[object Object],r =  # of retrieved models,[object Object],1) Precision = ,[object Object],	2) Recall =,[object Object],34,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],35,[object Object]
3D Model Retrieval Procedure,[object Object],Compute the diffusionwavelet for each 3D model,[object Object],Obtain the model representing vector X,[object Object],Compute the 2nd order statistics of X for each scale,[object Object],36,[object Object]
1) Start with a coarsest scale comparison ,[object Object],2)Advance up to the finest scale ,[object Object],3) Stop on threshold or when finest scale reached,[object Object],* Use a threshold to determine if a model is from a certain class,[object Object],Model Matching Procedure,[object Object],37,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],38,[object Object]
Experimental Results,[object Object],39,[object Object],Differences in scaling levels,[object Object],DW gives better results than SH and SW,[object Object]
Visual Results,[object Object],40,[object Object]
Schedule,[object Object],Introduction,[object Object],Diffusion Maps,[object Object],Wavelets and Diffusion Wavelets,[object Object],Fisher’s Discriminant Ratio (FDR),[object Object],Retrieval Procedure,[object Object],Results,[object Object],Conclusions,[object Object],41,[object Object]
Authors’ Conclusions,[object Object],Surfaces with sharp peaks, grooves or holes contain high-frequency information which is not addressed by the wavelet multi-resolution (use diffusion wavelet packets instead?),[object Object],Possible to extend to partial matching,[object Object],DW presents better results than SH and SW,[object Object],42,[object Object]
My Conclusions,[object Object],Paper presents a novel solution,[object Object],[object Object]
Less novel solutions:
IRPR is a common measure in database retrieval
Fischer Discriminant Ratio is a common statistical measure43,[object Object]
My Conclusions (2),[object Object],Technically sound, feasible,[object Object],[object Object]
Seems like a reasonable solution to the problem of 3D object retrieval44,[object Object]
My Conclusions (3),[object Object],The diffusion wavelet part could be explained in more detail,[object Object],[object Object]
How are the wavelet functions affected when a new model is inserted?45,[object Object]
My Conclusions (4),[object Object],Not self-containing but reference papers are exceptionally good,[object Object],[object Object],46,[object Object]
“Would like to have” (Technical/1),[object Object], Non-rigid extensions,[object Object],[object Object]
Can we have an extension of Diffusion Wavelets for non-rigid manifolds?47,[object Object]
“Would like to have” (Technical/2) ,[object Object],	How to automatically choose the level of decomposition,[object Object],48,[object Object]
“Would like to have” (Technical/3),[object Object],	An intuitive explanation - why prefer Diffusion Wavelets over Diffusion Wavelet Packets?,[object Object],Wavelet Packets seem to give more information especially in high frequencies…,[object Object],49,[object Object]

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A Diffusion Wavelet Approach For 3 D Model Matching

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  • 49. IRPR is a common measure in database retrieval
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Hinweis der Redaktion

  1. Google has their own format –SketchUp (SU) and has been investing a lot of effort in 3D technologies. More examples from google include – google earth, google sketch-up and O3D platform for browser 3D display
  2. Other manifold learning techniques – PCA (+variants), Kernel PCA, LLE (locally linear embedding), ISOMAP, Hessian LLE, LaplacianEigenMaps and many more…
  3. Reference[9]
  4. Reference [2]
  5. See [1,6] for diffusion maps equations and [3] for graph
  6. Ignore λmax=1 since it is a trivial eigenvalue of the markov transition matrix and corresponds to the stationary distribution of the markov chain at t=∞Cut is promised to conform with min-cut max-flow algorithmSee reference [6] for a thorough description of diffusion maps
  7. We see that it is possible to reconstruct the manifold using just a single eigenvalue. Cuts can be made on the eigenvectors that represent the manifold (in this case, the second or the third eigenvectors corresponding to the second or third largest eigenvalue) – these cuts are meaningful since they are taking into account the geometric distribution of the original points. We see that diffusion maps approximate the Fourier series over the circle as the sine and cosine functions are the solution of the differential equation f’’=-f
  8. Animated objects can be more sensitive to mesh simplification algorithms than CAD models.
  9. Haar wavelet with 3 levels of decomposition to the Stanford Bunny image. By applying a threshold in the wavelet domain we can efficiently find similar images. The threshold is a very efficient way to remove noise. The high value coefficients correspond to edges at various scales. Collecting the high value coefficients to create a feature vector would ensure a good representation of the image. For example, check the lossy compression algorithm JPEG-2000.
  10. We take 4 levels of decomposition since there is no real advantage in taking more decomposition levels and the computational burden is heavy. This result is specific to Princeton’s database and can change when dealing with different databases.
  11. Take inverse of FDR to avoid numeric problems of overflow. Select model with minimum within cluster scattering and maximum within cluster scattering
  12. Can do training on the entire database…See [5] for database
  13. Chair – sparse structurePlane – smooth surface with local singularityKangaroo – smooth surfaceFlower – combines smooth surface with local singularity