Sub-pixel Edge Fitting Using B-Spline Fr´ ed´ eric Bouchara 1 , Marc Bertrand 1 , Sofiane Ramdani 2 , and Mahmoud Haydar 1 1 Universit´ e du Sud Toulon-Var,UMR CNRS 6168 LSIS, B.P. 20132, 83957 La Garde Cedex, France {bouchara,bertrand}@univ-tln.fr 2 Universit´ e de Montpellier I, EA 2991 EDM, France [email protected]Abstract. In this paper we propose an algorithm for the sub-pixel edge detection using a B-spline model. In contrast to the usual methods which are generally sensitive to local perturbations, our approach is based on a global computation of the edge using a Maximum Likelihood rule. In the proposed algorithm the likelihood of the observations is explicitly computed, it ensures the filtering of the noisiest data. Experiments are given and show the adequacy and effectiveness of this algorithm. 1 Introduction Extracting object boundaries accurately is one of the most important and chal- lenging problems in image processing. An important approach of edge extraction is concerned with improving the detection accuracy and different sub-pixel edge detectors can be found in the literature. A popular approach used to compute the subpixel location of the edge is based on the moments of the image. Among the methods belonging to this category one can find algorithms using the gray level moments [1], the spatial moments [2,3,4,5] or the Zernike moments [6,7]. The interpolation of the image is another method used to compute the sub- pixel coordinates of the edge. In [8], Nomura et al. propose a method in which the first derivative perpendicular to the orientation of the edge is approximated by a normal function. The location of the edge is estimated to subpixel accu- racy by computing the maximum of this function. Other studies are based on linear [9], quadratic interpolation [10], B-spline [11] or other kind of non-linear functions [12]. Others have extended the sub-pixel localization technique to corners and ver- tex [13,14] or circular edges [15]. However, in all of these approaches, the estimation is local and does not in- clude a model of the noise. Hence, in these processes, the local perturbations strongly disturb the final result. An usual filtering approach consists in intro- ducing an a priori information in the estimation process. In edge detection, this can be achieved by using a deformable model such as the snake model proposed by Kass et al. [16]. In more recent works, the deformable contour is modelled A. Gagalowicz and W. Philips (Eds.): MIRAGE 2007, LNCS 4418, pp. 353–364, 2007. c Springer-Verlag Berlin Heidelberg 2007
Transcript
Sub-pixel Edge Fitting Using B-Spline
Frederic Bouchara1, Marc Bertrand1, Sofiane Ramdani2,and Mahmoud Haydar1
1 Universite du Sud Toulon-Var,UMR CNRS 6168 LSIS,B.P. 20132, 83957 La Garde Cedex, France
{bouchara,bertrand}@univ-tln.fr2 Universite de Montpellier I, EA 2991 EDM, France
Abstract. In this paper we propose an algorithm for the sub-pixel edgedetection using a B-spline model. In contrast to the usual methods whichare generally sensitive to local perturbations, our approach is based ona global computation of the edge using a Maximum Likelihood rule. Inthe proposed algorithm the likelihood of the observations is explicitlycomputed, it ensures the filtering of the noisiest data. Experiments aregiven and show the adequacy and effectiveness of this algorithm.
1 Introduction
Extracting object boundaries accurately is one of the most important and chal-lenging problems in image processing. An important approach of edge extractionis concerned with improving the detection accuracy and different sub-pixel edgedetectors can be found in the literature.
A popular approach used to compute the subpixel location of the edge is basedon the moments of the image. Among the methods belonging to this categoryone can find algorithms using the gray level moments [1], the spatial moments[2,3,4,5] or the Zernike moments [6,7].
The interpolation of the image is another method used to compute the sub-pixel coordinates of the edge. In [8], Nomura et al. propose a method in whichthe first derivative perpendicular to the orientation of the edge is approximatedby a normal function. The location of the edge is estimated to subpixel accu-racy by computing the maximum of this function. Other studies are based onlinear [9], quadratic interpolation [10], B-spline [11] or other kind of non-linearfunctions [12].
Others have extended the sub-pixel localization technique to corners and ver-tex [13,14] or circular edges [15].
However, in all of these approaches, the estimation is local and does not in-clude a model of the noise. Hence, in these processes, the local perturbationsstrongly disturb the final result. An usual filtering approach consists in intro-ducing an a priori information in the estimation process. In edge detection, thiscan be achieved by using a deformable model such as the snake model proposedby Kass et al. [16]. In more recent works, the deformable contour is modelled
using piecewise polynomial functions (B-spline snakes) [17,18]. Such a formula-tion of an active contour allows local control, compact representation, and it ismainly characterized by few parameters.
In this paper, we propose an algorithm for the estimation of the sub-pixel edgesusing a global approach based on a B-spline model. Our approach is similar tothe model proposed in [19]. However, in our method the statistical properties ofthe observations are computed and used in a Maximum Likelihood estimationwhich insures an efficient filtering of the noisy data.
This paper is organized as follows. In section 2, we briefly recall the classicalformulation of the B-spline model. In section 3 we present the proposed extensionto the sub-pixel case. Finally section 4 is devoted to the experimental results ofthis algorithm.
2 B-Spline Model
In this section, we present a brief theoretical review of B-spline contour formu-lation (for more details see [20]).
Let {t0, t1, . . . , tk−1} be the set of so-called knots. By definition, spline func-tions are polynomial inside each interval [ti−1, ti] and exhibit a certain degree ofcontinuity at the knots. The set {Bi,n(t), i = 0, . . . , k − n − 1} of the so calledB-splines, constitutes a basis for the linear space of all the splines on [tn, tk−n].Thus, a spline curve f(t) of degree n is given by :
f(t) =k−n−1∑
i=0
piBi,n(t) (1)
where pi are the weights applied to the respective basis functions Bi,n . TheB-Spline functions are defined by the Cox-deBoor recursion formulas:
Bi,0(t) ={
1 if ti ≤ t ≤ ti+10 else (2)
andBi,n(t) =
t − titi+n − ti
Bi,n−1(t) +ti+n+1 − t
ti+n+1 − ti+1Bi+1,n−1(t) (3)
The Bi,n(t) are nonnegative, Bi,n(t) ≥ 0 and verify the partition of unity prop-erty :
∑i Bi,n(t) = 1 for all t.
In the sequel of the paper, cubic B-spline will be used and without lost ofgenerality we will drop the index n. Relation (1) can be expressed compactly inmatrix notation as:
f(t) = BT (t).θ (4)
B(t) is the vector of B-spline functions: B(t) = (B0(t), B2(t), . . . BN−1(t)) andθ is the vector of weights: θ = (p0, ., pN−1)T
The R2 version of (1) describes an open parametric snake on the plane: v(t) =(x(t), y(t)
). The pi = (pxi, pyi) are now 2D vectors and are called the control
points.
Sub-pixel Edge Fitting Using B-Spline 355
To describe a closed curve, which is the case of interest for boundary repre-sentation, a periodic extension of the basis functions and of the knot sequenceis generally used.
3 Subpixel B-Spline Fitting
3.1 Observational Model
In this sub-section we shall derive the expression of the likelihood of the observa-tion. In order to get tractable computations, we have developed our model in thecase of a white additive Gaussian noise b. We denote σb its standard deviation.
Let ve ={ve(i.h) =
(xe(i.h), ye(i.h)
), i ∈ {1, ..., M}
}be a set of edge pixels
with integer coordinates computed by a classical algorithm (as the classical activecontour algorithm). For the sake of simplicity, we will denote in the following vei
the element ve(i.h), this rule will be applied for all the variables relative to thepixel i.
For each pixel of this set we suppose available the observation vector Oi =(Xi, Hi). The vector Hi = (Hxi, Hyi) is the local gradient of the image andXi corresponds to an estimation of the sub-pixel position of the edge alongHi.The value Xi is estimated in the local coordinate system of the pixel thanksto a quadratic interpolation (see [21] and the appendix for more explanations).These variables are supposed to be the observation version of the ”true” variables(Yi, Gi). The cartesian coordinates of the observation Oi will be noted (xoi, yoi)in the sequel.
Let now vs(t) = (xs(t) = B(t).θx, ys(t) = B(t).θy) be the B-spline model ofthe edge, where θ = (θx, θy) is the vector of k control points (with M >> k).
The likelihood P (Oi/θ) can be expressed as the product of two elementaryprobability density function (pdf):
P (Oi/θ) = P (Xi/Hi, θ)P (Hi/θ) (5)
In order to derive the first factor of this expression, we make the assump-tion that Xi only depends on vs(t(i)) =
(xs(t(i)) = xsi, ys(t(i)) = ysi
), the
corresponding edge point in the direction Hi of which the polar coordinates are(Yi, Hi). In first approximation we have modelled the pdf P (Xi/Hi, Yi) with aGaussian law. According to our tests, this approximation remains accurate evenfor significant noise levels. Then we get:
P (Xi/Hi, θ) = P (Xi/Hi, Yi) (6)
=1√
2πσXi
exp
(− (Xi − Yi)2
2.σ2Xi
)
=1√
2πσXi
exp
(− (xoi − xsi)2 + (yoi − ysi)2
2.σ2Xi
)
356 F. Bouchara et al.
The computation of σXi, based on a modified version of the method proposedin [22], is described in the appendix.
Let us now compute the expression of P (Hi/θ). The components of Hi areclassically computed by convoluting the original image with derivative kernelsnoted Kx and Ky. Using the assumption of a Gaussian additive noise, it isstraightforward to show that P (Hi/Gi) is also Gaussian. The cross-correlationand auto-correlation functions of the vertical and horizontal components of Hare given by the following well known relation:
where E[w/z] represents the conditional expected value of w assuming z.Let Ji be the vector defined by Ji = (Gyi, −Gxi) and let Ri be the unit vector
collinear to vs in vsi. Hi and Ji are clearly related by a normal distribution.To express the law P (Hi/θ) we use the well known property of the edge to be
orthogonal to the local gradient. Then, we get:
P (Hi/θ) = P (Hi/Ri(θ)) =∫
P (Hi/Ji)P (Ji)P (Ri(θ))
(11)
where the integration is achieved for all the values of Ji collinear to Ri that isfor Ji = a.Ri. Equation (11) can be expressed by:
P (Hi/θ) =
(2πσ2
H
)−1
P (Ri)
∫ +∞
−∞exp
(− (Hyi − a.Rxi)2
2.σ2H
)×
exp
(− (Hxi + a.Ryi)2
2.σ2H
)P (Ji)da (12)
In the previous equation, P (Ji) is the a priori probability of Ji. The gradi-ent has not a privileged direction and the uniform regions are supposed to bemore frequent than the regions with a high gradient magnitude. Thus, P (Ji) ismodelled with the following centered normal law:
P (Ji) =1√πΣ
exp
(− ‖Ji‖2
Σ2
)=
1√πΣ
exp
(− a2
Σ2
)(13)
Sub-pixel Edge Fitting Using B-Spline 357
Fig. 1. Test image
Using this assumption we can compute the equation (12):
P (Hi/θ) = K. exp
(−
H2yi + H2
xi
2σ2H + Σ2
)exp
(− (HxiRxi + HyiRyi)2Σ2
2σ2H(2σ2
H + Σ2)
)(14)
with K a normalization constant involving P (Ri), Σ and σH .We assume Σ large compared to σH and we write:
P (Hi/θ) ≈ K. exp
(−
H2yi + H2
xi
Σ2
)exp
(− (HxiRxi + HyiRyi)2
2σ2H
)(15)
In the above, vector Ri is given by:
Ri =Ti
‖Ti‖where Ti =
(∂xs(t(i))
∂t,∂ys(t(i))
∂t
)T
T = ‖Ti‖ is assumed to be constant along the edge.The components of vector Ti are also B-spline functions of which the spline
basis functions have the following expression:
∂Bn,i(t)∂t
=n
ti+n − tiBi,n−1(t) − n
ti+n+1 − ti+1Bi+1,n−1(t) (16)
The global likelihood is simply estimated by assuming that the Oi’s are con-ditionally independent. Hence, the global likelihood is factorized as: P (O/vs) =∏
i P (Oi/vs).
3.2 Numerical Implementation
To compute θ we simply apply the Maximum Likelihood rule, that is:
θ = arg . maxθ
∏
i
P (Xi/Hi, θ)P (Hi/θ) (17)
358 F. Bouchara et al.
(a) (b)
(c) (d)
Fig. 2. Comparative results between the proposed algorithm (solid), the true edge (dot-ted) and the classical spline interpolation (dashed). The measured subpixel coordinatesare noted with white crosses (x). (a) 10 percent of noise, (b) 20 percent of noise, (c)40 percent of noise and (d) 50 percent of noise.
This relation classically yields to the minimization of an energy function which,in our case, is the following quadratic function:
W is a (3M × 3M) diagonal matrix with Wi,i = Wi+M,i+M = 1σ2
Xiand
Wi+2M,i+2M = 1T 2σ2
H,
B is a (3M × 2N) matrix: B =
⎛
⎝B 00 BBx By
⎞
⎠
The components of matrix B are obtained from the Bi(t) functions: Bij =Bj(t(i)). In the current implementation of the algorithm we have simply used auniform distribution of the i′s points, that is, t(i) = i/M .
Sub-pixel Edge Fitting Using B-Spline 359
The components of Bx and By are computed from (15) and (16): Bxij =NHxi (Bi,2(i/M) − Bi+1,2(i/M)) and Byij = NHyi (Bj,2(i/M) − Bj+1,2(i/M))
As classically, the solution θ = argminθ E(θ) is given by the weighted leastsquare relation: θ = (BT WB)−1BT Wd.
(a) (b)
(c) (d)
Fig. 3. Comparative results between the proposed algorithm (solid), the true edge(dotted) and the classical spline interpolation (dashed). The measured subpixel coor-dinates are noted with white crosses (x). (a) p0 = 0.2 γ = 0.3, (b) p0 = 0.2 γ = 0.6,(c) p0 = 0.4 γ = 0.3 and (d) p0 = 0.4 γ = 0.6.
4 Experimental Results
In order to validate the proposed algorithm, we have achieved several tests onthe synthetic 256 × 256 image, normalized between 0 and 1, shown on figure1. We have numerically simulated the optical integration of the intensity overthe pixels in low-pass filtering and sub-sampling a 2048 × 2048 version of thisimage. The reference ’true’ edge can be hence easily computed. For all the testsachieved in this section we have computed the gradient by using the derivativeof a Gaussian kernel with a standard deviation equal to 1. The number of knotsN is set to M/4.
360 F. Bouchara et al.
We present the results for a small area localized by a small square in figure 1.First, we have tested this algorithm for a white additive Gaussian noise with
standard deviations equal to 0.1, 0.2, 0.4 and 0.5 which represent respectively10, 20, 40 and 50 percent of the maximum amplitude. In figure 2, we comparethe results obtained with this algorithm with the reference ’true’ edge and theclassical spline interpolation without using the likelihood model of section 3.The white crosses (×) represent the measured subpixel coordinates. As usualwith this kind of approach, the most noisy measures are weakly involved in theestimation process en hence filtered.
The proposed model has been developed in the case of a Gaussian noise.However, it remains efficient for other kinds of noise. In figure 3 we present theresults obtained with a salt & pepper noise of which the probability densityfunction is defined by:
P (x) = p0δ(x − γ) + (1 − 2.p0)δ(x) + p0δ(x + γ)
where δ(.) is the Dirac function.The standard deviation of such a noise is given by: σn = γ
√2p0.
We have achieved the tests for the case of a low rate noise (p0 = 20) and ahigh rate noise (p0 = 40) with two different values of the amplitude γ.
5 Conclusion
In this paper we have presented an algorithm for the estimation of the edge withsubpixel accuracy. In the proposed algorithm, the edge is modelled thanks tothe B-spline approach. The parameters of the model are computed by using aML estimation based on a Gaussian model of the noise. The likelihood of thedata is computed from the observation model which includes both orientationand position information.
Comparative experiments of this algorithm and the classical spline interpo-lation have been carried out for two different kinds of noise: a Gaussian noiseand a Salt & Pepper noise. These numerical experiments have shown that ouralgorithm outperforms the classical approach for these two kinds of noise.
References
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Sub-pixel Edge Fitting Using B-Spline 361
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A Computation of σX
The method used to interpolate the subpixel coordinate is basically an im-provement of a well-known edge detection method sometimes called NMS (Non-Maxima Suppression). The NMS method consists of the suppression of the localnon-maxima of the magnitude of the gradient of image intensity in the direction
362 F. Bouchara et al.
of this gradient [23]. The sub-pixel approximation proposed in [21] adds a newstep to the NMS process: when the point (x, y) is a local maximum then theposition of the edge point in the direction of the gradient is given by the max-imum of a quadratic interpolation on the values of the squared gradient normsat (x, y) and the neighboring points (figure 4).
Fig. 4. Estimation of the subpixel location Xi by using quadratic interpolation of thesquare gradient norm
Let i be an edge pixel and let a, b, c, d, e, f , u and v be its vicinity. Let Na,Nb ... and Ni be the corresponding squared gradient norms.
Let (Hxs, Hys) be the two coordinates of the gradient vector at S and let:
α =Hys
Hxs(19)
We consider here the case α < 1. In the direction defined by α the squaredgradient norms Nk and Nl in k and l are interpolated using two quadratic func-tions involving respectively Na, Nb, Nc and Nd, Ne, Nf . The subpixel coordinateXs of the edge is obtained by computing the maximum of a third quadraticcurve defined by the points k, l and i. After some computations we obtain:
Xs =(Ne − Nb) + 1
2 (Na − Nc + Nd − Nf)α + . . .
2(Nb + Ne − 2Ni) + (Nc − Na + Nd − Nf)α + . . .
. . . 12 (2Nb − Na − Nc + Nd − 2Ne + Nf )α2
. . . (Na − 2Nb + Nc + Nd − 2Ne + Nf )α2 (20)
Which can be written:
Xs =(M1 ⊗ N)(0,0)
(M2 ⊗ N)(0,0)=
A
B(21)
Sub-pixel Edge Fitting Using B-Spline 363
Where ⊗ represents the convolution operator. Matrices M1 and M2 have thefollowing expressions:
M1 =
⎛
⎝−α − α2 0 −α + α2
−2 + 2α2 0 2 − 2α2
α − α2 0 α + α2
⎞
⎠ M2 =
⎛
⎝2α + α2 0 −2α + α2
4 − 4α2 −4 4 − 4α2
−2α + 2α2 0 2α + 2α2
⎞
⎠
To compute the standard deviation of Xs = AB we first define this ratio of two
random variables as follows:
A
B=
Ac + E[A]Bc + E[B]
(22)
In the above, Ac and Bc are the centered parts of A and B. We then applya second order power series approximation on the denominator variable aroundE[B]. After some computations we get:
E
[A
B
]≈ E[A]
E[B]− CAB
E[B]2+
E[A]σ2B
E[B]3(23)
E
[(A
B
)2]
≈ E[A]2
E[B]2+
σ2A
E[B]2+ 3
E[A]2σ2B
E[B]4− 4
E[A]CAB
E[B]3
+3σ2
Aσ2B + 2C2
AB
E[B]4(24)
Where σ2A = E[A2
c ], σ2B = E[B2
c ] and CAB = E[AcBc].To compute the equations (23) and (24) we need the expressions of E[A],
E[B], E[A2], E[B2] and E[AB].From (21) and using a classical result we can write:
Note that in the above, the expectations are conditional assuming Hs andhence α is deterministic.
The expected value E[Ns] on a given pixel s is simply obtained from theexpression of Ns by using equation (7):
E[Ns] = E[H2xs + H2
ys] = E[H2xs] + E[H2
ys]
= Cxx(0, 0) + Cyy(0, 0) + E[Hxs]2 + E[Hys]2 (26)
To compute the joint second moments of A and B we first express the productAB as follows (the demonstration is given for E[AB] but the generalization toE[A2] and E[B2] is straightforward):
AB =∑
s
∑
t
M1(−s)M2(−t)N(s)N(t) (27)
364 F. Bouchara et al.
which yields to:
E[AB] =∑
s
∑
t
M1(−s)M2(−t)E[N(s)N(t)] (28)
Using the same method as for (26) we can compute the joint second momentE[NsNt]. After some computations, one can express E[NsNt] with the followingcompact relation:
