Invariant GHTAdham Beyki Khoshk (abk2g11), MS Kapoor (msk1g11), Shashank Dhariwal (sd8g11)

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Introduction

• HOUGH TRANSFORM • IMPORTANT TERMS & DEFINITIONS• GENERAL HOUGH TRANSFORM (GHT)• INVARIANT GHT• IMPLEMENTATION• RESULTS (DEMO)• ADVANTAGES AND DISADVANTAGES• APPLICATIONS

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Hough Transform

• General Definition HT is a technique to extract required model features from Images and its purpose is primarily to find imperfect instances of objects within a certain class of shapes by a voting procedure. • Invented by R Duda and P Hart in year 1972 for

identifying position of arbitrary shapes [1]• Later in year 1981 , Dana H Ballard was responsible for

popularising GHT in Computer Vision community [1]• Example : HT of a line [ 1];

main idea – characterisation of line not as image points , but as certain parameters which define it.

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Various Definitions

• Hough Space/Accumulator space the space where the votes of an image point is defined in the terms of the selected parameters

• Image Space The 2D space that represents the image or (edge detected image)

• Parameterisation Selection of certain parameters that map a model point into the Hough space

• Mapping Process of defining a rule(using parameters) that binds the Hough space and model

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Various Definitions

• Accumulator array an array/matrix where result of HT is stored, its dimension is equal to the no of unknown parameters defining the transform. And the bin having the maxima is considered for extraction of the shape

• R table [2] Constructed by examining the boundary points of a shape. And later acts as look up table for obtaining parameter values of image points.

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• Finds application where shapes are arbitrary and the voting is carried out for the particular position, orientation and scaling, based on pre defined look-up tables

• general equation :

ω (θ,b,λ,ρ) = b(x,y) + λR(ρ)v(θ) [ 3]

• Mapping function : b = ωi - λR(ρ)v(θ) [3]

i.e mapping from the orientation of an edge point (ωi)to the reference point (b)

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Translation vector

Rotation MatrixScale factor

Model Shape

Image shape

General Hough Transform

• Evidence gathering – using 4D accumulator space

• Additional constraint – gradient direction information as we do not know v(θ)

φ΄i = φ΄ (θ) – ρ• For using the mapping function we need to know v(θ) and this

is determined by solving equation above, i.e we know φ΄i and find point v(θ) for which the gradient direction = 0.

• v(θ) is then used to compute λR(ρ)v(θ) using R- Table (where, rows = value of φ΄i )

• Therefore extraction = building R table + evidence gathering from the accumulator space.

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General Hough Transform

• Previously GHT improved evidence gathering by considering a point and its gradient direction (which was not invariant).

• This leads to good noise handling but high computational complexity.

• Therefore we need to replace gradient direction by a feature that is invariant. (e.g ‘angle’ : rotation & scale invariance)

• The feature can be colour of the point or any other point that does not change in model and image space.

• Now search for a point with same ‘gradient direction’ in GHT is replaced with search with same ‘invariant feature’ in IGHT

• Search is in 2D space rather than 4D space. (reduced computation) 8

Invariant GHT

• Q (ωi) = Q (v(θ)) – constraint equationwhere, Q – invariant function that computes a feature at

the point• Given ‘ωi ‘and a fixed angle ‘α’ we can determine ‘ωj’ (second

point)• third point ‘ωT’ – defined by intersection of ωi & ω j

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Invariant GHT

• The angle ‘ β’ obtained replaces the gradient angle in R table and is invariant to rotation and scaling• Now we have two parameters β and k (distance vector)• k changes with rotation and scale • therefore, k is redefined as invariant using the relation

k = φ΄i - φ΄’i (in terms of gradient angles)• This value of k is pre computed and stored in R table as

function of β.• Therefore now the solution is , given ωi we find v(θ) and φ΄’i

and β and using the look up table obtain k and carryout evidence- gathering procedure.

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Invariant GHT

(𝑥 𝑖 , 𝑦 𝑖 )𝑀𝑎𝑝⇒

(𝑐 ,𝑚 )

Hough Transform for a Line in Cartesian System

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(𝑥 𝑖 , 𝑦 𝑖 )𝑀𝑎𝑝⇒

(𝜌 ,𝜃 )

Hough Transform for a Line in Polar System

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Hough Transform for a Circle

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A model shape can be defined by a curve

Parameter Meaning

ω image shape

b translation vector (x0, y0)

λ scale factor

R(ρ) rotation matrix

v(θ) model shape

Generalised Hough Transform

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• Using extra constraint

• Representation in polar coordinates

Generalised Hough Transform

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Invariant Feature / New Constraint

Invariant Generalised Hough Transform

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Invariant Generalised Hough Transform

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• Efficient only if a high number of votes fall in the bin.

• Ambiguity when there are a large number of parameters.

• Becomes computationally expensive when number of parameters are more than three.

• Complexity increases at a rate of O(Am-2).

• Efficiency dependent on quality of input data.18

Limitations of Hough Transform

• Rotation and scale invariant.

• Computationally inexpensive.

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Advantages of Invariant GHT

• Produces noise.

• More sensitive to occlusion and noise.

• Reduced accuracy.

• Lack of longitudinal localization of lines in parameter space.

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Disadvantages of Invariant GHT

• Corner Detection.

• Detect 3D objects in laser range data.

• Artefact Matching.

• Inshore ships detection.

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Applications of Invariant GHT

Since it is better to see the output space in a 3D view, it is better to run .m files and use the Rotate 3D tool in MATLAB.

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Results

An image from a wrenchInput

HT Output – Accumulator Spaceusing Cartesian coordinates

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Results

An image from a wrenchInput

HT Output – Accumulator SpaceUsing polar coordinates

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ResultsThe Model Shape

The model shape in the image

GHT Output – Accumulator Space

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ResultsThe Model Shape

The model shape in the image

IGHT Output – Accumulator Space

1. Wikipedia, Hough Transform, http://en.wikipedia.org/wiki/Hough_transform2. D H Ballard , Generalising The Hough Transform to Detect Arbitrary Shapes, Pattern Recognition Vol 13, 2.pp .111-122,Pergamon Press Ltd, 1981.3. Mark Nixon & Alberto Aguado, Feature Extraction & Image Processing, Second Edition, Elsevier Ltd, 2008.4. Jian Xu, Kun Fu, Xian Sun, An Invariant Generalized Hough Transform Based Method of Inshore Ships Detection. International Symposium on Image and Data Fusion (ISIDF), 2011.5. E.R. Davies, Application of the generalised Hough transform to corner detection. IEE PROCEEDINGS, Vol. 135, Pt. E, No. 1, JANUARY 1988.

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References

6. Kourosh Khoshelham, EXTENDING GENERALIZED HOUGH TRANSFORM TO DETECT 3D OBJECTS IN LASER RANGE DATA. ISPRS Workshop on Laser Scanning 2007 and SilviLaser 2007, Espoo, September 12-14, 2007, Finland.7. Paul Lewis, Peter Durham, Stephen Shennan, Artefact Matching and Retrieval using Generalized Hough Transform. Proceedings of Computer Applications in Archaeology. 25—308. Jose A.R. Artolazabal and John Illingworth, LIGHT: Local Invariant Generalized Hough Transform. 18th International Conference on Pattern Recognition, 2006. ICPR 2006.

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References