2011. 1009 generalized hough 1

Generalized Generalized Hough Hough

TransformTransform

The Generalized Hough Transform

From Standard to Generalized From Standard to Generalized HTHT

1.1.Standard Hough Transform Standard Hough Transform requires parametric requires parametric representation for desired representation for desired curvecurve

2.2.This idea is generalized in This idea is generalized in the Generalized Hough the Generalized Hough TransformTransform

Example: Human Face Example: Human Face recognitionrecognition

• Is there some attribute of the structure of the head that we can exploit to help estimate pose estimation?

• Is this attribute invariant under change in pose?

– Or • “Can we model how this attribute varies with

pose?”

Hough Transform in General1. Technique to isolate curves of a

given shape in an image

2. Standard Hough Transform (HT) uses parametric formulation of curves

3. Generalized Hough Transform (GHT) extends for arbitrary curves

1. When we compute the correlation by voting, we spend most of the time casting bad votes.

2. Idea is to use extra shape information (e.g. gradientsgradients) to cast fewer votes:1. O(n) complexity: For each of O(n) points on the

boundary, cast O(1) votes.

Key Idea to improve Key Idea to improve correlation by votingcorrelation by voting

General Hough Algorithm IdeaGeneral Hough Algorithm Idea

• 1. explicitly list points on shape• 2. make table for all edge pixels for target• 3. for each pixel store its position relative to some

reference point on the shape– ‘if I’m pixel i on the boundary, the reference point is at ref[i]’

The Generalized Hough TransformThe Generalized Hough Transform

1.Technique to find arbitrary curves in a given image

2.Parametric equation no longer required

3.Look-up table used as transform mechanism

4.Two phases:

1.R-Table Generation phase

2.Object Detection phase

1. Standard Techniques allow for invariance to scale and rotation in the plane

2. In general, objects in the real world are 3-dimensional

3. Hence a single silhouette provides no invariance to pose (i.e. rotation out of the plane).

4. No pose estimation.

5. This is generalized to Surface Normal Hough Transform

The Generalized Hough TransformThe Generalized Hough Transform

Building the Building the R-Table R-Table in GHTin GHT

GHT: Building the R-TableGHT: Building the R-Table1. We are given the shape we want to localize

2. We build a lookup table for this shape, called R-Table

It will replace the need for a parametric equation in the transform stage

GHT: Building the R-TableGHT: Building the R-Table

GHT: Building the R-TableGHT: Building the R-TableGHT: Building the R-TableGHT: Building the R-Table

Object Object Localization in Localization in

the R-Table the R-Table in GHTin GHT

GHT: Object GHT: Object LocalizationLocalization

Conclusions on GHT1. Standard Techniques allow for

invariance to scale and rotation in the plane

2. In general, objects in the real world are 3-dimensional

3. Hence a single silhuette provides no invariance to pose (i.e. rotation out of the plane).

4. No pose estimation.

5. Now show more details

Conclusions on GHTConclusions on GHT

Generalized Generalized Hough Hough Transform Transform AlgorithmAlgorithm

Algorithm of the General Algorithm of the General Hough TransformHough Transform

Hough Transform for CurvesHough Transform for Curves

• The H.T. can be generalized to detect any curve that can be expressed in parametric form:– Y = f(x, a1,a2,…ap)– a1, a2, … ap are the parameters– The parameter space is p-dimensional– The accumulating array is LARGE!

Generalized Hough Generalized Hough TransformTransform

• Find all desired points in image• For each feature point

– for each pixel i on target boundary• get relative position of reference point from i

• add this offset to position of i

• increment that position in accumulator

• Find local maxima in accumulator• Map maxima back to image to view

algorithm

Generalizing the H.T.The H.T. can be used The H.T. can be used even if the curve has even if the curve has not a simple analytic form!not a simple analytic form!

1.1. Pick a reference point Pick a reference point (x(xcc,y,ycc))2.2. For i = 1,…,n :For i = 1,…,n :

1.1. Draw segment to PDraw segment to Pii on the on the boundary.boundary.

2.2. Measure its length rMeasure its length rii, and its , and its orientation orientation ii..

3.3. Write the coordinates of (xWrite the coordinates of (xcc,y,ycc) as a ) as a function of rfunction of rii and and ii

4.4. Record the gradient orientation Record the gradient orientation ii at at PPi.i.

3.3. Build a table with the data, Build a table with the data, indexed by indexed by ii . .

(x(xcc,y,ycc))

iirrii

PPii

ii

xxcc = x = xii + r + riicos(cos(ii))

yycc = y = yii + r + riisin(sin(ii))

Generalizing the H.T.

(x(xcc,y,ycc))

PPii

iirrii ii



Suppose, there were m Suppose, there were m differentdifferent gradient orientations: gradient orientations:(m <= n)(m <= n)

11

22

..

..

..

mm

(r(r1111,,11

11),(r),(r1122,,11

22),…,(r),…,(r11n1n1,,11

n1n1))

(r(r2211,,22

11),(r),(r2222,,11

22),…,(r),…,(r22n2n2,,11

n2n2))

..

..

..

(r(rmm11,,mm

11),(r),(rmm22,,mm

22),…,),…,(r(rmm

nmnm,,mmnmnm))

jj

rrjj

jj

H.T. tableH.T. table

Generalized H.T. Algorithm:



Finds a Finds a rotated, scaled, and translatedrotated, scaled, and translated version of the curve: version of the curve:

(x(x cc,y,y cc

))PP ii

ii

SrSr ii ii

PP jj

jj

SrSr jj jj

PP kk

ii

SrSr kk kk

1.1. Form an Form an A accumulator arrayA accumulator array of of

possible reference points (xpossible reference points (xcc,y,ycc), ),

scaling factor S and Rotation angle scaling factor S and Rotation angle ..

2.2. For each edge (x,y) in the image:For each edge (x,y) in the image:

1.1. Compute Compute (x,y)(x,y)

2.2. For each (r,For each (r,) corresponding to ) corresponding to

(x,y) do:(x,y) do:

1.1. For each S and For each S and ::

1.1. xxcc = x = xii + r( + r() S cos[) S cos[(() + ) +

]]

2.2. yycc = y = yii + r( + r() S sin[) S sin[(() + ) +

]]

3.3. A(xA(xcc,y,ycc,S,,S,) = ) = A(xc,yc,S,q) + 1A(xc,yc,S,q) + 1

3.3. Find maxima of A.Find maxima of A.

Another variant of the Generalized Another variant of the Generalized Hough TransformHough Transform

Find Object Center given edges

Create Accumulator Array

Initialize:

For each edge point

For each entry in table, compute:

Increment Accumulator:

Find Local Maxima in

),( cc yxA

),(0),( cccc yxyxA

),,( iii yx

1),(),( cccc yxAyxA

),( cc yxA

ik

ikic

ik

ikic

ryy

rxx

sin

cos

ikr

),( cc yx ),,( iii yx

Generalize HT applied for circuits

Properties of Generalized Hough Transform

• What can we do when the curve we want to detect is not easily described parametrically?

1. ~ By this, we mean, it cannot be captured in a relatively small number of parameters.

2. ~ Recall, the dimensionality of the Hough space equal the number of parameters!

• The GHT constructs a parametric description of an arbitrary shape based on a learning process.

• This parametric description is not, in general, compact.

• We will begin by assuming the size, shape, and rotation (orientation) of the region is known a priori. (Or that we want only to detect instances of a given size and orientation.

1. ~ The voting space is (equivalent to) image space, 2D, in the case of known size and rotation.

2. ~ We will see how to deal with unknown orientation and size shortly -- with a 4D Hough space.

The list of ( , ) pairs, for a given and constitutesa partial characterization of the shape.

X1

X2

XR

1 2

3 4 5

r 1r 4r 5

XR

r j

j

: An arbitrary reference point inside the shape.

: The length of the j-th line from the reference point to the shape perimeter, intersecting at a point of tangent angle ø.: The angle of the (current) tangent(s) to the perimeter.

: The orientation of the j-th line segment.

jr j XR

• By sweeping the tangent angle (ø) over the range (0,2π) in some reasonable quantization (!), we build what is called the R-table (reference table) description of the shape.

1 :

2 :

k :

(r11,1

1 ); (r12 ,1

2 ); .... (r1n1 ,1

n1 );

(r21, 2

1 ); (r22 , 2

2 ); .... (r2n2 , 2

n2 );

(rk1, k

1 ); (rk2 , k

2 ); .... (rknk , k

nk );

• Each pixel x (say, a detected edge point) with local orientation ø provides evidence (votes for) reference points at the set of locations indicated by the list in the R-table for that tangent direction...

{ x1 r ( ) cos[ ( )], x 2 r ( ) sin[ ( )]}

• A vote is cast for each (r , ) pair in the list for that ø value.

The voting space is isomorphic to image space.

• Again, this assumes known size and orientation for all appearances of the shape.

• After all the edge points have voted for all of their possible reference points, we interrogate the voting space for significant local maxima. These suggest possible detections of the shape of interest.

• If we have not prenormalized for size (S) and rotation ( )

then our voting space is four dimensional and the reference location

receiving the vote(s) for a given edge point and R-table entry is:

x1R x1 r()Scos[ ( ) ]

x2R x2 r()Ssin[() ]

• Now, we interrogate the 4D accumulator array to recover likely locations,

scale, and orientation for appearances of the shape.

• This is really a fancy form of a template match -- but one that is far more

robust than a straightforward template matching algorithm.

• Selecting among multiple possible shapes requires multiple R-tables,

multiple voting spaces.

• But, so does looking for lines and circles in the same image....

Generalized HT in biologically Generalized HT in biologically motivated roboticsmotivated robotics

Bimodal Active Stereo

Many simultaneous problems in robotics

Research Philosophy

The main concept of Radon Transform

Hough Transform: Comments

• Works on Disconnected Edges

• Relatively insensitive to occlusion

• Effective for simple shapes (lines, circles, etc)

• Trade-off between work in Image Space and Parameter Space

• Handling inaccurate edge locations:

• Increment Patch in Accumulator rather than a single point

H.T. Summary• H.T. is a “voting” scheme

– points vote for a set of parameters describing a line or curve.

• The more votes for a particular set– the more evidence that the corresponding curve is present

in the image.

• Can detect MULTIPLE curves in one shot.

• Computational cost increases with the number of parameters describing the curve.

end

2011. 1009 generalized hough 1

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