Local Feature Extraction Using Scale-Space Decomposition Dmitriy Bespalov† Ali Shokoufandeh† William C. Regli†‡ Wei Sun‡ Department of Computer Science† Department of Mechanical Engineering & Mechanics‡ College of Engineering Drexel University 3141 Chestnut Street Philadelphia, PA 19104

Overview Introduction Overview of the approach Experimental results Current and future work

Project Goals Perform content- , feature- and shape-based retrieval of CAD data Develop systematic approaches of evaluating matching techniques Integrate new algorithms with the National Design Repository (http://www.designrepository.org)

Goals of this Work Feature extraction technique Models in polyhedral representation Using local information Tolerance to noise

Side Note: Compare Features Scale Space CAD/CAM

Selected Related Work Surface Feature Extraction on Shape Models (Computer Graphics) Scale-Space features [Bespalov et al., 03] Multi-resolutional Reeb Graphs [Hilaga et al., 01] Feature information on range data [Thompson et al., 96,99] Manufacturing Feature Extraction on Solid Models (Solid Modeling & Engineering) Feature relationship graphs [Elinson et al. 97; Cicirello and Regli 99,00,01,02] Automated generation of GT codes [Ames 91; Shah et al., 89 and Ham et al., 86] Feature Extraction on 2D images (Computer Vision) Scale-Space features using wavelets and Gaussian filters [Lindeberg, 90 and Shokoufandeh, 99]

Traditional CAD Representation Watertight boundary-representation solid Implicit surfaces Analytic surfaces NURBS, etc Topologically and geometrically consistent Produced by kernel modelers and CAD systems

Traditional Shape Representation Usually a mesh or point cloud Usually an approximate representation No explicit in/out Sometimes error prone STL files, acquired data Produced by CAD systems, animation tools, laser scanners, etc

What is a Scale Space Representation? Commonly used for Coarse-to-Fine representations of an object Very popular in computer Vision Constructed via spatial filters: Gaussian pyramids, Wavelets… Basic Idea: At each scale, topologically relevant components will decompose the object into so called salient parts Recursive application of this paradigm will create the object’s scale space hierarchy

Approach to Scale Space Feature Extraction Obtain polyhedral representation Convert CAD model or scan physical part Perform geometry-based decomposition Singular value decomposition Construct “feature” graph Obtain a segmentation into “features”

Algorithm Overview (I) Obtain mesh representation M Define measurement function: Our d is maximum angle on the shortest path between every two faces on M using angular measure. Distance matrix D is constructed. d(t1,t2)

Algorithm Overview (II) 3. Decompose M into relevant components using a singular value decomposition of distance matrix D Note: this creates a clustering based on the angle between a vector Oti and the basis vectors (ck, ck-1)

Algorithm Overview (III) The Eckart-Young Theorem: Given an n by m matrix X of rank r ≤ m ≤ n, and its singular value decomposition, ULV', where U is an n by m matrix, L is an m by m diagonal matrix of singular values, and V is an m by m matrix such that U'U = In and V'V = VV' = Im with the singular values arranged in decreasing sequence λ1 ≥ λ2 ≥ λ3 ≥ ... ≥ λm ≥ 0 then there exists an n by m matrix B of rank s, s ≤ r, which minimizes the sum of the squared error between the elements of X and the corresponding elements of B when B = UΛsV' where the diagonal elements of the m by m diagonal matrix Λs are λ1 ≥ λ2 ≥ λ3 ≥ ... ≥ λs > λs+1 = λs+2 = ... = λm = 0

Algorithm Overview (IV) Recursive feature decomposition using two principle components creates binary feature trees. Use leaf nodes as features.

Algorithm Overview (V) Stopping automatic feature decomposition process. Decomposition of M1 into M2 and M3 is significant if: angle across the border of the cluster is max on the path between most of the pairs of faces in M2 and M3.

Algorithmic Complexity Bisection process: SVD decomposition takes O(n3). Polyhedral representation creates a 3D lattice; if only neighboring vertices are used in construction of the distance matrix, SVD decomposition is faster and takes O(n2).

Experimental Results

Experimental Results – Partial Data

Experimental Results – Partial Data

Experimental Results – Partial Data

Experimental Results – Partial Data

Experimental Results – Partial Data

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of one-to-many correspondence

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of one-to-one correspondence

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of one-to-one correspondence

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of many-to-many correspondence

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of one-to-many correspondence

Experimental Results – Scanned Data From Exact Representation 360° Scan Single Scan An example of one-to-one correspondence

Contributions and Future Work Research Contributions Surface Feature extraction technique Performs on models in polyhedral representation Relatively stable Does not depend on global structure of the model Current and Future Work: Extract approximate B-Rep from polyhedral models Introduce matching technique for extracted features Use Scale-Space features for database indexing Use (partial) scanned data for database queries Explore techniques for feature extraction that resemble traditional CAD features

Q&A For more information http://gicl.cs.drexel.edu http://aal.cs.drexel.edu http://www.designrepository.org Sponsored (in part) by: ONR Grant N00014-01-1-0618 NSF CAREER Award CISE/IIS-9733545