Circular Augmented Rotational Trajectory (CART) Shape Recognition & Curvature Estimation Presentation for 3IA 2007 Russel Ahmed Apu & Dr. Marina Gavrilova Department of Computer Science University of Calgary

Brief Outline  Motivation  Shape Representation  Problems with current approach  Proposed Approach (CART)  R-Space Representation  Experimental Results

Motivation: Computer Graphics  Augmented Reality Can Vision algorithms in AR be improved so that objects can be inserted by recognizing more natures signs and shapes? Source:

Motivation: Computer Graphics  Markerless Motion Capture  Can we capture motion from body contours in natural images? Source:

Motivation: Artificial Intelligence  Aerial Robotics: Target Recognition Identify special shape/color for Automated Search and Rescue Operation

Ship Trajectory Analysis  MARIS Project: Risk Analysis How can we identify ship type and abnormal navigation patterns from the real-time GPS data? Source:

Key Problems in the area  Extraction of Shapes/contours: From noisy image with texture & clutters Overlapped, broken, faded & occluded Widely varying scale, rotation & transformation  Representation & Interpretation of Shapes, Regions & Contours Vector representation is much better than Raster (pixels) for interpretation Contour Models: Spline, points, lines or graphs Detection of invariant feature points  Analysis & matching of Shapes Shape matching and classification for distorted, transformed and often incomplete contour Detecting geometric properties in shapes despite local noise

Current Approaches  Active Contour (i.e. Snakes)  Edge Detectors  Segmentation  Normalized-Cuts (and it’s variants)  Corner Detector (I.e. Sift)  Kalman Filter (For noisy contours)  Gausian filters, Haugh Transform etc.

Problem Complexity…  Very difficult to extract shapes Object Contour ≠Edges  Effective methods are Computationally extensive  Some methods such as Active Contour have erratic convergence  Loss of detail in Kalman filter, Edge detector, Haugh transform etc.  Others: Does not work well to “Classify” shapes  Unable to cope with scale, rotation & distortion  Unable to detect geometric signatures

Difficulty in Contour Extraction  Intensity changes are not only observed in edges Texture Clutter Image artifacts  One solution is to smooth Smoothing destroys detail  Must Observe regions i.e. segmentation But region based methods are slow  When the Object shape is not just linear it is much harder I.e. noisy curved objects This edge gradient image shows that it is very difficult to ascertain actual contours from textures and clutters

Problem with current approaches  Active Contour (i.e. Snakes), Segmentation, Corner Detection are very slow to converge Not practical in most applications such as Augmented Reality  Edge detection is neither robust nor sufficient  Haugh transform is only good for Straight line Features

Extraction Anomaly Pixel Discretization artifacts is a notorious effect. It masks the actual shape of the object Often, shape extracted has erratic points which deviate from the curve Solution: Smoothing Then, how can we preserve linear features & sharp corners?

Curvature Interpretation Ambiguity  Which of the following interpretation is right?  Impossible to Ascertain by looking at a small local region  Shape can be: Part of a rotated rectangle Part of a curved surface There can be misleading noise

Circular Augmented Rotational Trajectory (CART)  A Curvature based Spline Model  Represents Rotation Invariant graphs  Main Idea: Estimate the curvature at a given point At what constant turn rate can we travel the furthest along a contour? Constraint: Cannot deviate from original curve more than Tolerance   Differs from Kalman Filter (or smoothing): No statistical assumption on noise distribution Does not smooth away sharp features  Differs from Haugh: CART works with both linear and curved objects  Differs from Active Contour & segmentaion: Convergence is guaranteed and bounded Much faster

CART: Main Concept  Estimation of d/dl  Linear Spline Model:  Problem: Not scale invariant Sensitive to Step resolution  Solution: Use Circular trajectory estimation Insensitive to rescaling (except that details are lost) At a constant turn rate, different stepsize generates the same exact curve  See Algorithm 1: Procedure Circular Projects a particle along a circular trajectory Estimate turn rate by linear/quadratic curve fitting Shape & Total Turn Varies depending on step resolution (Hard to perform Multiscale analysis)

Rotation Estimation  Define A Score Score= Distance = How far can a particle travel at constant turn rate without breaking the constraint  Initial Step: Estimate initial direction & turn-rate  Following Steps: Estimate Turn Rate only  Optimization Goal: Maximize distance and minimize deviation (distance gets priority)

Rotation invariant R-Space representation  Represent curve as a graph Length along curve VS rotation rate   Easy to detect geometric Signatures Convexity, Concavity Corners (sharp/smooth) Domes, Ovals Straight lines Circles/ellipses Polygons (sharp/cambered)  R-Space is Rotation invariant Same graph for any orientation Minimally affected by scaling Robust to noise and distortion R-Space conversion of shapes

R-Space Example (a) (b) (c) (d) Shapes and their representation in R-space. (a) Rectangles has four spikes (b) circles are horizontal lines (c) Distorted rectangular shape (d) Distorted circular shape

R-Space Example The object is a polygon with 12 sides (12 spikes in r- space). This is generated without CART by simple applying gaussian smoothing & differentiating

Discretization Anomaly and Noise  Gaussian smoothing no longer works when noise & anomalies are present The Object & tracked contour R-Space Graph without smoothing (too many false spikes) R-Space Graph with significant smoothing (false spikes still present and getting wider)

Using CART:  Anomalies are eliminated R-Space Graph with significant smoothing (false spikes still present and getting wider) R-Space Graph with CART: Shows linear segments and corners properly

Detection of Geometric Signatures (Invariant points) I. Natural Image II.Lots of Texture & clutter III.High Noise & anomaly present

Detection of Geometric Signatures (Invariant points) I. Presence of heavy noise II.Blurred image III.Misleading contour noise Easy to detect shape signatures in Region A,B,C & D

Conclusion  CART is simple and easy to implement  Very efficient and fast compared to other methods  Robust convergence & result  Robust to Noise & discretization error  Allow detection of Corners and other unique geometric signatures  Allow Geometric analysis (Convexity, linearity, global curvature etc.)  Invariant to rotation and scaling  Minimally affected by other distortions & transformations

Thank you :) Questions & inquiries?