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
Dec 26, 2015
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?