Circular augmented rotational trajectory cart shape recognition curvature estimation
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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

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Circular Augmented Rotational Trajectory (CART) Shape Recognition & Curvature Estimation

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Circular augmented rotational trajectory cart shape recognition curvature estimation
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
Brief Outline

  • Motivation

  • Shape Representation

  • Problems with current approach

  • Proposed Approach (CART)

  • R-Space Representation

  • Experimental Results

Motivation computer graphics
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?


Motivation computer graphics1
Motivation: Computer Graphics

  • Markerless Motion Capture

  • Can we capture motion from body contours in natural images?


Motivation artificial intelligence
Motivation: Artificial Intelligence

  • Aerial Robotics: Target Recognition

Identify special shape/color for Automated Search and Rescue Operation

Ship trajectory analysis
Ship Trajectory Analysis

  • MARIS Project: Risk Analysis

How can we identify ship type and abnormal navigation patterns from the real-time GPS data?


Key problems in the area
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
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
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
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
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
Extraction Anomaly

  • Often, shape extracted has erratic points which deviate from the curve

  • Solution:

    • Smoothing

Then, how can we preserve linear features & sharp corners?

Pixel Discretization artifacts is a notorious effect. It masks the actual shape of the object

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

Curvature Interpretation Ambiguity

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

Circular Augmented Rotational Trajectory (CART)

Cart main concept

Estimation of d/dl

Linear Spline Model:

Problem: Not scale invariant

Sensitive to Step resolution


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

CART: Main Concept

Shape & Total Turn Varies depending on step resolution (Hard to perform Multiscale analysis)

Rotation estimation
Rotation Estimation

  • Define A Score

    • Score= <Distance , Sum(Deviation)>

    • 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


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

Rotation invariant R-Space representation

R space example
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 example1
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
Discretization Anomaly and Noise

  • Gaussian smoothing no longer works when noise & anomalies are present

R-Space Graph without smoothing (too many false spikes)

R-Space Graph with significant smoothing (false spikes still present and getting wider)

The Object & tracked contour

Using cart
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
Detection of Geometric Signatures (Invariant points)

  • Natural Image

  • Lots of Texture & clutter

  • High Noise & anomaly present

Circular augmented rotational trajectory cart shape recognition curvature estimation

Detection of Geometric Signatures (Invariant points)

  • Presence of heavy noise

  • Blurred image

  • Misleading contour noise

Easy to detect shape signatures in Region A,B,C & D


  • 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
Thank you :)

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