Design of Steerable Filters for Feature
Detection Using Canny-Like Criteria
Mathews Jacob,Member,IEEE,and Michael Unr,Fellow,IEEE Abstract—We propo a general approach for the design of2D feature detectors from a class of steerable functions bad on the optimization of a Canny-like criterion.In contrast with previous computational designs,our approach is truly2D and provides filters that have clod-form expressions.It also yields operators that have a better orientation lectivity than the classical gradient or Hessian-bad detectors.We illustrate the method with the design of operators for edge and ridge detection.We prent some experimental results that demonstrate the performance improvement of the new feature detectors.We propo computationally efficient local optimization algorithms for the estimation of feature orientation.We also introduce the notion of shape-adaptable feature detection and u it for the detection of image corners.
Index Terms—Steerable,feature,edge,detection,ridge,contours,boundary,lines.
æ
1I NTRODUCTION
I N his minal paper on computational edge detection, Canny identified the desirable qualities of a feature detector and propod an appropriate optimality criterion. Bad on this criterion,he developed a general approach to derive the optimal detector for specific image features such as edges[1].This work had a great impact on the field and stimulated further developments in this area,particularly on alternate optimality criteria and design strategies[2],[3].
All of the above authors considered the derivation of optimal1Doperators.For2Dimages,theyapplied theoptimal 1D operator orthogonal to the feature boundary while smoothing in the perpendicular direction(along the bound-ary).This extension is equivalent to computing inner-products between the image and a ries of rotated versions of a2D reference template(tensor product of the optimal 1D profile and the smoothing kernel).With this detector,the rotation angle of the template that yields the maximum inner product,gives the feature orientation.Since the optimal 1D template did not have explicit formulae,they were typically approximated by simple first or cond order differentials of a Gaussian.In practice,they were extended using Gaussian kernels of the same variance since the resulting2D template could be applied in a directional manner inexpensively via the computation of smoothed image gradients or Hessians.
An alternative to the differential approaches to rotation independent feature detection is provided b
y the elegant work of Freeman and Adelson on steerable filters[4].The underlying principle is to generate the rotated version of a filter from a suitable linear combination of basis filters;this ts some angular bandlimiting constraints on the class of admissible filters.Perona et.al.,Manduchi et al.,Simoncelli and Farid,and Teo and Hel-Or ud this framework to approximate and design orientation-lective feature detec-tors[5],[6],[7],[8].The concept of steerablity was also applied successfully in other areas of image processing such as texture analysis[9],[10]and image denoising[11].
In this paper,we propo to reconcile the two methodo-logies—computational approach and steerable filterbanks—-by prenting a general strategy for the design of2D steerable feature detectors.We derive the filter directly in2D as oppod to the1D schemes(1D optimization followed by an extension to2D)of Canny and others.Moreover,in contrast with the work of Perona[5],we do not approximate a given template within a steerable solution space,but arch for the filter that gives the best respon according to an optimality criterion.Our filter is specified so as to provide the best compromi in terms of signal-to-noi ratio,fal detections, and localization.We illustrate the method with the design of optimal edge and ridge templates.The detectors that we obtain analytically have better performance and improved orientation lectivity,yet they are still computationally quite attractive.
The paper is organized as follows:In Section2,we introduce the concept of steerable matched filtering and reinterpret some of the classical detectors within this framework.In Section3,we propo an optimality criterion and show how to determine the best filter from a class of steerable functions.In Section4,we concentrate on specific 2D feature detectors and demonstrate their u in different applications.Though our algorithm is general,in this paper, we focus only on the detection of edge and ridge features.In Section5,we introduce the concept of shape adaptive feature extraction and illustrate it with an example.
2O RIENTATION I NDEPENDENT M ATCHED F ILTERING 2.1Detection by Rotating Matched Filtering Suppo our task is to detect some feature in an image fðx;yÞat some unknown position and orientation.The detection procedure can be formulated as a rotated matched filtering.It involves the computation of inner-products with the shifted
.The authors are with the Biomedical Imaging Group,Swiss Federal
Institute of Technology Lausanne,CH-1015,Switzerland.
E-mail:michael.unr@epfl.ch,Mathews.Jacob@ieee.
Manuscript received7Feb.2003;revid20July2003;accepted13Aug.
麒麟座v838
2003.
Recommended for acceptance by W.Freeman.
For information on obtaining reprints of this article,plea nd e-mail to:
tpami@computer,and reference IEEECS Log Number118259.
0162-8828/04/$20.00ß2004IEEE Published by the IEEE Computer Society
and rotated versions of a 2D feature template f 0ðx;y Þ¼h ðÀx;Ày Þat every point in the image.A high magnitude of the inner-product indicates the prence of the feature and the angle of the corresponding template gives the orientation.Some simple examples of templates are shown in Fig.1.Mathematically,the estimation algorithm is
Ãðx Þ¼arg max
ðf ðx ÞÃh ðR x ÞÞ
ð1Þr Ãðx Þ¼f ðx ÞÃh ðR Ãx Þ;
ð2Þ
where r Ãis the magnitude of the feature and Ãits orientation at the position x ¼ðx;y Þ;R is the rotation matrix
R ¼cos ð Þsin ð Þ
Àsin ð Þcos ð Þ !ð3Þ
and u Ãv stands for the convolution between u and v .Equations (1)and (2)correspond to the matched filter detection.They give the maximum-likelihood estimation of the angle and weight r for the signal model
f ðx Þ¼r Áf 0ðR ðx Àx 0Þþx 0Þþn ðx 0Þ;
where n ðx Þdenotes Gaussian white noi.However,this scheme of detection is not very practical,for it requires the implementation of a large number of filters (as many as the quantization levels of the angle).
2.2Steerable Filters
To cut down on the computational load,we lect our detector within the class of steerable filters introduced by Freeman and Adelson [4].The filters can be rotated very efficiently by taking a suitable linear combination of a small number of filters.Specifically,we consider templates of the form
h ðx;y Þ¼
X M k ¼1X k i ¼0
k;i @k Ài @x @i
@y
茶叶推广方案g ðx;y Þ;
ð4Þ
where g ðx;y Þis an arbitrary isotropic window function.We call such a h ðx;y Þan M th order detector.
Proposition 1.The filter h ðx;y Þis steerable.In other words,the convolution of a signal f ðx;y Þwith any rotated version of h ðx;y Þcan be expresd as
f ðx ÞÃh ðR x Þ¼
X M k ¼1X k i ¼0
b k;i ð Þf k;i ðx Þ;ð5Þ
where the functions f k;i ðx;y Þare filtered versions of the signal f ðx;y Þ
f k;i ðx;y Þ¼f ðx;y ÞÃ@k Ài @x k Ài @
i @y i
g ðx;y Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
g k;i ðx;y Þ
:ð6Þ
The orientation-dependent weights b k;i ð Þare given by
b k;i ð Þ¼
X k j ¼0 k;j X l;m 2Sðk;j;i Þ
k Àj l j m ðÀ1Þm cos ð Þj þðl Àm Þsin ð Þðk Àj ÞÀðl Àm Þ!
;
ð7Þ
where Sðk;i;j Þis the t Sðk;i;j Þ¼f l;m j 0<¼l <¼k Ài ;0<¼m <¼i ;k Àðl þm Þ¼j g .
The proof is given in the Appendix A.A graphical reprentation of the implementation is given in Fig.2.Once the f k;i ðx;y Þare available,f ðx ÞÃh ðR x Þcan be evaluated very efficiently via a weighted sum with its coefficients that are trigonometric polynomials of .Since the number of partial differentials in (5)for a general M th order template is M ðM þ3Þ=2,h ðx Þis steerable in terms of as many individual parable functions.Using some simplification,we can show that such a general h ðx Þcan also be rotated using 2M þ1nonparable filters 1(an example of such a simpli-fication is given by (39)-(42)).
A ca of special interest corresponds to g ðx Þbeing the Gaussian;indeed,the Gaussian is optimally localized in the n of the uncertainty principle and the corresponding filters in (6)are all parable.
Interestingly,the Gaussian family is equivalent to the class of moment filters (poly-nomials multiplied by Gaussian window)discusd in [4],but the filters are not identical.We will now show that the family described by (4)includes some popular feature detectors as particular cas.
Fig.1.Examples of feature templates.Feature detection is performed by
convolution of the rotated versions of the template with the image.(a)Idealized edge template.(b)Idealized ridge template.(c)Popular edge template.(d)Popular ridge template.
1.This is the minimum number of filters required to steer a general M th order
tempate.
Fig.2.Implementation of steerable filtering (cf.(5)).
2.3Conventional Detectors Revisited
2.3.1Canny’s Edge Detector
As already obrved by Freeman and Adelson,the widely-ud Canny edge detection algorithm can be reinterpreted in terms of steerable filters[4].This algorithm involves the computation of the gradient-magnitude of the Gaussian-smoothed image.The direction of the gradient gives the orientation of the edge.Mathematically,
üarctan
ðfÃgÞy
ðfÃgÞx
ð8Þ
rü
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðfÃgÞx
ÀÁ2
þðfÃgÞy
2
;单身久了
r
ð9Þ
where g x¼@g=@x and g y¼@g=@y;g is a2D Gaussian of a specified variance.The above t of equations can be shown to be the solution of(1)and(2),with h¼g x.Substituting M¼1; 1;0¼1; 1;1¼0in(7),we get b1;0ð Þ¼cosð Þ; b1;1ð Þ¼sinð Þ.Thus,
ÃðxÞ¼arg max
ðfðxÞÃg xðR xÞÞð10Þ
¼arg max
ðfÃðg x cosð Þþg y sinð ÞÞÞ:ð11ÞHere,we ud the steerability of g x from(5).To compute the maximum of the above expression,we t the differential of (11)with respect to to zero:
ðfÃg xÞsinð ÞÀðfÃg yÞcosð Þ¼0;ð12Þwhich results in(8)and(9).The corresponding feature template is shown in Fig.1c.
2.3.2Ridge Detector
Less well-known is the fact that a popular ridge estimator bad on the eigen-decomposition of the Hessian matrix[12], [13],[14]can also be interpreted in terms of steerable filters. Assuming the template to be g xx(the cond derivative of a Gaussian),ridge detection can be formulated exactly as(1) and(2).The corresponding detector is shown in Fig.1d.In this ca,the steerability relation(5)can be expresd in a matrix form as
g xxðR xÞ¼u T
g xxðxÞg xyðxÞ
g xyðxÞg yyðxÞ
!
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
H g
u ;ð13Þ
where H g is the Hessian matrix and u ¼ðcosð Þ;sinð ÞÞ. Using the linearity of convolution,fðxÞÃg xxðR xÞ¼
u T H fÃg u .We would like to obtain the maximum of
u T H fÃg u ,subject to the constraint u T u ¼1.We solve this
constrained optimization problem using Lagrange’s multi-plier method by tting the gradient of u T H fÃg u þ u T u to zero:
H fÃg u ¼À u :ð14ÞThis implies thatÀ is an eigenvalue of H fÃg;the corresponding normalized eigenvectors are the possible solutions to the problem.Since we are looking for the maximum of u T H fÃg u ,the optimal respon and the angle are given by
rü maxð15Þ
共享人生
u üv max:ð16ÞHere, max and v max are the maximum eigenvalue and the corresponding eigenvector,respectively.
It can be en from Figs.1c and1d that the classical detectors do not have a good orientation lectivity.In the next ction,we propo a new approach for the design of detectors that attempts to correct for this deficiency.
3D ESIGN OF S TEERABLE F ILTERS FOR F EATURE
D ETECTION
The widely-ud contour extraction algorithm[1]has three steps:1)feature detection,2)nonmaximum suppression, and3)thresholding.In this ction,we prent a general strategy for the design of steerable filters for feature detection,while keeping in mind the subquent steps. We propo a crite
rion similar to that of Canny and we analytically derive the optimal filter—or,equivalently,the optimal weights—within our particular class of steerable functions specified by(4).
3.1Optimality Criterion
We now review Canny’s criterion and modify it slightly to enable analytical optimization.To derive the optimal 2D operator,we assume that the feature(edge/ridge)is oriented in some direction2(say,along the x axis)and derive an optimal operator for its detection.As the operator is rotation-steerable by construction,its optimality properties will be independent of the feature orientation.
The three different terms in Canny’s criterion are as follows.
3.1.1Signal-to-Noi Ratio
The key term in the criterion is the signal-to-noi ratio.The respon of a filter hðxÞto a particular signal f0ðxÞ(e.g.,an idealized edge)centered at the origin is given by
S¼
Z
R2
f0ðx;yÞhðÀx;ÀyÞdx dyð17Þ
S is given by the height of the respon at its maximum.If the input is corrupted by additive white noi of unit variance,then the variance of the noi at the output is given by the energy of the filter:
Noi¼
Z
R2
j hðx;yÞj2dx dy:ð18ÞWe desire to have a high value of S for a given value of
Noi;S2
Noi
is the amplification of the desired feature provided by the detector.
3.1.2Localization
The detection stage is preceded by nonmaximum suppres-sion.The estimated feature position corresponds to the location of the local maximum of the respon in the direction orthogonal to the feature boundary(y axis in our ca).The prence of noi can cau an undesirable shift in the estimated feature location.The direct extension of Canny’s
JACOB AND UNSER:DESIGN OF STEERABLE FILTERS FOR FEATURE DETECTION USING CANNY-LIKE CRITERIA1009
2.In2D,the features of interest have boundaries of dimension1.
expression for the shift-variance (due to white noi of unit variance)to 2D gives
E ðÁy Þ2h i
¼R R 2j h y ðx;y Þj 2dx dy j R R 2f 0ðx;y Þh yy ðÀx;Ày Þdx dy j 2:ð19Þ
Canny has propod to maximize the reciprocal of this term.The numerator of (19)is a normalization term which will be small automatically if the impul respon of the filter is smooth along the y axis (low norm for the derivative).Since we are imposing this type of smoothness constraint elwhere via an additional regularization term (e next ction),it is not necessary to optimize this term here,whic
h also keeps the effects well parated.Therefore,we propo to maximize the cond derivative of the respon,orthogonal to the boundary,at the origin
Loc ¼Àd 2
dy ðf 0Ãh Þ
¼À
Z
R 2
f 0ðx;y Þh yy ðÀx;Ày Þdx dy ð20Þ
which is the square-root of the denominator in (20).The above expression is ensured to be positive becau the cond derivative of the respon is negative at the maximum (assuming S >0).Note that the new localization term is a measure of the width of the peak.The drift in position of the maximum due to noi will decrea as the respon becomes sharper.In this work,we are neglecting the effect of neighboring features in deriving the localization term.
3.1.3Elimination of Fal Oscillations
Canny obrved that when the criterion is optimized only with the SNR and the localization constraint,the optimal operator has a high bandwidth;the respon will be oscillatory and,hence,have many fal maximas.In 2D,we desire that the respon be relatively free of oscillations orthogonal to the feature boundary.This can be achieved by penalizing the term:
R o ¼Z R 2
j h yy ðx;y Þj 2dx dy:ð21Þ
Note that this term is the numerator of the expression for the mean distance between zero crossings propod by Canny.It is a thin-plate spline-like regularization which is a standard technique to constrain a solution to be smooth (low bandwidth).
The thresholding step is easier if the respon is flat along the boundary.The oscillation of the respon along the boundary (x axis)can be minimized by penalizing
R p ¼Z R 2
j h xx ðx;y Þj 2dx dy ð22Þ
The terms will force the filter to be smooth making the
respon is less oscillatory,thus resulting in fewer fal detections.
3.2Derivation of the Optimal Detector
We combine the individual terms to obtain a single criterion
C ¼S ÁLoc À ðR o þR p Þ|fflfflfflfflfflffl{zfflfflfflfflfflffl}
R
:
ð23Þ
The filter in the family described by (4)that maximizes this
criterion,subject to the constraint 3Noi ¼1,is our optimal detector.The free parameter >0controls the smoothness of the filter;a high value makes the respon less prone to fal maxima and reduces oscillation along the ridge.However,the properties impo a trade off on the localization of
the respon.
In this work,we are also interested in performing a scale-independent design.In other words,if we dilate the
window by a factor ,using g ðx Þ¼ À1
2g ðx Þ,we want our solution to retain the shape independently of .This requires that we weight each of the terms in (23)using an appropriate power of the dilation factor.This issue is discusd later for each feature model parately.
For the ea of notation,we collect the component
functions of (4)into a function vector g of length ðM ðM þ3Þ
2
Þ,who components are ½g i ðx;y Þ¼
@k Àn @x k Àn @n @y n
g ðx;y Þwith i ¼
ðk À1Þðk þ2Þ
2þn k ¼0...M;n ¼0::k:
Hence,an arbitrary function in the family is reprented in a compact form as
h ðx Þ¼a T g ðx Þ;
ð24Þ
where a is the vector containing the i;k s in (4);it has the
same length as the function vector.Now,we express the terms of the criterion in a matrix form as S ¼a T s ,Loc ¼a T q ,Noi ¼a T P a ,and R ¼a T R a ,where
s ½ i ¼h f 0ðx Þ;g ðÀx Þ½ i i
ð25Þq ½ i ¼D
f 0ðx Þ;ð
g ðÀx Þ½ i Þyy
E ð26ÞP ½ i;j ¼h g ½ i ;g ½ j i
ð27ÞR ½ i;j ¼D
ðg ½ i Þyy ;ðg ½ j Þyy )þD ðg ½ i Þxx ;ðg ½ j Þxx E :
ð28Þ
g yy ðx;y Þand g xx ðx;y Þdenote @2g ðx;y Þ=@y 2and @2g ðx;y Þ=@x 2,
respectively.P and R are matrices of size M ðM þ3Þ2ÂM ðM þ3Þ
2
,while the vectors q and s are of length M ðM þ3Þ
2
.Here,P is ensured to be nonsingular.In the above expressions,the inner product of two functions is defined as
h f 1;f 2i ¼Z
R 2
f 1ðx;y Þf 2ðx;y Þdx dy:
Thus,the criterion (23)can be expresd in the matrix form as
C ¼a T Q À R ½ a ;
ð29Þ
where
Q ¼s q T :
ð30ÞSince all the terms in the criterion are quadratic,the solution for the optimal parameters can be found analytically by using Lagrange’s multiplier method.To maximize the
1010IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.26,NO.8,AUGUST 2004
3.This constraint is just a normalization factor.Setting Noi to another constant will give detectors of the same shape,but with a different energy.
criterion subject to the constraint,we t the gradient of Cþ Noi to zero:
2QÀ Rþ P
½ a¼0:ð31ÞRearranging the terms,we get
PÀ1QÀ R
½ a¼À að32Þwhich implies that is an eigenvalue of the matrix ðÀPÀ1½QÀ R Þ.The total number of eigenvalues is given
by the dimension of a.The corresponding eigenvectors a
i
need to be scaled so that the constraint a T
i Pa
i
¼1is
satisfied.The optimal solution is therefore given by
a¼max a T
i
QÀ R
½ a
i
;i¼0...MðMþ3Þ=2
n o
:ð33ÞThus,the design of the optimal feature detector boils down to an eigen-decomposition followed by an appropriate weighting of the eigenvectors so as to satisfy the constraint.
3.3Feature Detection by Local Optimization
Due to(5),the optimal angle Ãin(1)is obtained as the solution of
@ @ ðfðxÞÃhðR ÃxÞÞ¼
X M
k¼1
X k
i¼0
f k;iðx;yÞ
@
@
ðb k;ið ÞÞj ¼ Ã
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
c k;ið ÃÞ
¼0:
ð34Þ
It is easy to e from(7)that each of the terms in b k;ið Þare of degree k in cosð Þand sinð Þ;c k;ið Þis of degree k as well. Hence,(34)is a polynomial of order M(in cosð Þand sinð Þ) and,thus,the estimation of the optimal angle involves the solution of an M th order polynomial in two variables.
If hðx;yÞhas only odd/even order partial derivatives(this is the ca for many detectors),then b k;ið Þwill be a polynomial with only odd/even degree terms(of cosð Þand sinð Þ)prent.Conquently,(34)can be reduced4to a form where only terms of degree M are prent.In this ca,(34) can be further simplified(by dividing both the sides by ðcosð ÞÞM)to a polynomial in only one variable—tanð Þ.We then have an analytic solution if M<¼3[15].This ca is illustrated in Section4.1.3.When M¼2,the solution can also be computed as an eigen-decomposition of the Hessian matrix,which is better known(but,also,boils down to the above mentioned solution).This ca is described in Section
4.2.2.When the solution of(34)is not trackable analytically,it can be solved numerically using an an iterative root finder such as the Newton-Raphson method.
4T WO-D IMENSIONAL F EATURE D ETECTORS
We now design operators optimized for the detection of different2D features.We cho the window function to be a Gaussian5gðx; Þ,where is the standard deviation.When it is clear from the context,we will suppress the dependence on to simplify the notation.
4.1Edge Detection
As model for the edge,we choo the ideal step function
f0ðx;yÞ¼
1if y!0
0el:
&
ð35ÞSince it is an odd function of y,the even order derivatives do not contribute to the signal energy;we therefore ignore6 them in(4).
4.1.1Ca1:M¼1
To illustrate the derivation of the optimal filter,we explain all the steps in detail in this simple ca.Substituting the function vector g¼g x;g y
ÂÃ
in the corresponding expressions,we get
s¼À
ffiffiffi
p
0;1
½
q¼À
2
ffiffiffi p
0;1
½
P¼
2
江西婺源篁岭
10
01
免费论文!
R¼君影草
9
4
10
01
!
:
Thus,
Q¼q T s¼2
00
01
!
:
The matrices Q and P are independent of ,while R is inverly proportional to 4.So,we weigh R by 4to have a scale-invariant solution.Hence,
PÀ1QÀ 4R
ÂÃ
¼
À18 0
04À18
!
:ð36ÞThe eigenvalues of PÀ1QÀ 4R
ÂÃ
are 1¼À18 and 2¼4À18 ,respectively.The corresponding scaled eigen-vectors(so,as to satisfy the constraint)are
0;À
ffiffiffi
2
r
"#
and
À
ffiffiffi
2
r
;0
"#
;
respectively.When substituted in the criterion,they yield4À18 andÀ18 ,respectively.Thus,the optimal solution is
a¼0;À
ffiffiffi
2
r
"#
非理性决策(as >0),which corresponds to Canny’s edge detector (cf.Fig.1c).
JACOB AND UNSER:DESIGN OF STEERABLE FILTERS FOR FEATURE DETECTION USING CANNY-LIKE CRITERIA1011
4.If there is a term of degree MÀ2n,we can multiply it by ðcosð Þ2þsinð Þ2Þn to make it of degree M.
5.It is the only function that is isotropic and parable.
6.If we were to include them in the solution,their optimal coefficients would turn out to be zero anyway.
4.1.2Higher Order Cas
For higher M ,we obtain a family of solutions that are increasingly smooth when goes up.A few examples of higher order templates are given in Table 1with the filter impul respons shown in Fig.3.By comparing Figs.3b and 3c,we obrve that,as increas,the filter becomes smoother at the cost of directionality.The higher order templates are more elongated thus having higher SNR and localization (cf.Table 1);they should therefore result in better detections,at-least for idealized edges.The depen-dence of SNR on 2implies that this figure can also be improved by increasing the variance of the Gaussian.However,the ability to resolve two adjacent parallel edges decreas as increas.
4.1.3Implementation
Here,we develop the implementation procedure mentioned in Section 3.3for the special ca of third order edge detection.A general third order edge template (for different values of )is given by
h ðx Þ¼ 1;0g x þ 3;0g xxx þ 3;2g xyy :
ð37Þ
The rotated version 7of this template h is given by h ¼ 1;0ðg x cos ð Þþg y sin ð ÞÞþ
3;0
g xxx cos 3ð Þþ3g xxy cos 2ð Þsin ð Þþ
3g xyy cos ð Þsin 2ð Þþg yyy sin 3
ð Þ
þ 3;2
g xyy cos 3ð ÞþðÀ2g xxy þg yyy Þcos 2ð Þsin ð Þþ
ðÀ2g xyy þg xxx Þcos ð Þsin 2ð Þþg xxy sin 3ð Þ
:
Convolving the rotated template by f and simplifying,we get
ðf Ãh Þðr Þ¼q 1ðr Þcos ð Þ3þq 2ðr Þcos ð Þ2sin ð Þþ
q 3ðr Þcos ð Þsin ð Þ2þq 4ðr Þsin ð Þ3;
ð38Þ
where
q 1ðr Þ¼ 3;0f 3;0ðr Þþ 3;2f 3;2ðr Þþ 1;0f 1;0ðr Þ
ð39Þq 2ðr Þ¼ð3 3;0À2 3;2Þf 3;1ðr Þþ 3;2f 3;3ðr Þþ 1;0f 1;1ðr Þð40Þq 3ðr Þ¼ð3 3;0À2 3;2Þf 3;2ðr Þþ 3;2f 3;0ðr Þþ 1;0f 1;0ðr Þð41Þq 4ðr Þ¼ 3;0f 3;3ðr Þþ 3;2f 3;1ðr Þþ 1;0f 1;1ðr Þ:
ð42ÞWe multiplied the single degree terms in cos ð Þand sin ð Þwith ðcos 2ð Þþsin 2ð ÞÞso that we get a polynomial with
1012IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.26,NO.
8,AUGUST 2004
7.The expression for a general rotated template is given by (5)and (7).However,for simple templates,it may be easier to derive it directly in the Fourier domain as in (60).
TABLE 1
Edge Detectors for Different
Parameters
Fig.3.Edge Detectors for different parameters.The detectors become more orientation lective as M increas.(a)Canny’s edge detector.(b)M ¼3; ¼0:09.(c)M ¼3; ¼0:2.(d)M =5; ¼0:15.
