Circular Augmented Rotational Trajectory (CART)

Shape Recognition & Curvature Estimation

Presentation for 3IA 2007

Russel Ahmed Apu

& Dr. Marina Gavrilova

Department of Computer ScienceUniversity 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: http://www.artag.net/

Motivation: Computer Graphics Markerless

Motion Capture

Can we capture motion from body contours in natural images?

Source: http://www.toshiba.co.jp/rdc/mmlab/tech/w38e.htm

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: http://www.marin-research.ca/english/research/methods/spatial_statistics.html

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 , 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 Circles/ellipses Polygons (sharp/cambered)

R-Space is Rotation invariant Same graph for any

orientation Minimally affected by scaling Robust to noise and

distortionR-Space conversion of shapes

ShapeContour

15 Degrees Right

20 Degrees Right

+VE

0

-VE

ShapeContour

15 Degrees Right

20 Degrees Right

+VE

0

-VE

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?