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

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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

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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

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

- 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/

- Markerless Motion Capture
- Can we capture motion from body contours in natural images?

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

- Aerial Robotics: Target Recognition

Identify special shape/color for Automated Search and Rescue Operation

- 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

- 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

- 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.

- 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

- 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

- 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

- 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

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

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

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)

- 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)

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

(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

The object is a polygon with 12 sides (12 spikes in r-space).

This is generated without CART by simple applying gaussian smoothing & differentiating

- 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

- 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

- Natural Image
- Lots of Texture & clutter
- High Noise & anomaly present

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 :)

Questions & inquiries?