Procesamiento

Experimental science is devoted to fitting a model that depends on adjustable pa- rameters to a given set of observations. The common approach is to select or de- sign a merit function that measures the agreement between the data and the model with a particular choice of parameters. The model parameters are then adjusted to achieve a minimum in the merit function, yielding a set of best-fit parameters. The adjustment process is basically a problem of minimization in many dimen- sions. Finding the set of parameters that takes the function to a minimum or a maximum value is considered an optimization problem.

Definition7.3 (global optimization). In general, an optimization problem requires finding a set ofP∈S, whereSis a bounded set onRn_{, such that a certain qual-} ity criterion f :S → R, typically called the objective function, is minimized or equivalently maximized. Without loss of generality, it is sufficient to consider only minimization tasks, since maximizing f() is equivalent to minimizing−f(). The problem then is to find a pointPmin ∈Ssuch that f(Pmin) is a global minimum

onS. More specifically, it is required to find anPmin∈Ssuch that

∀P∈S: fPmin

≤ f(P). (7.10)

The tasks of maximization and minimization are trivially related to each other as one being the inverse of the other. An extremum (maximum or minimum) can be either global, truly the best solution, or local, the best around a neighborhood. Finding a global extremum is, in general, a very difficult problem. Moreover, in fitting data usually the merit function is not unimodal, with a single minimum, which makes the problem harder. On the other side, there are important issues that are beyond the mere finding of best fit parameters. Data are generally not ex- act! Data are subject to measurement errors. Thus, typical data never fit exactly the model that is being used, even when the model is correct. It is customary to assume that the measurements are independent random variables. Each measure- ment (f(xi,yi),xi,yi) have a mean and a standard deviation. Fitting such a model

G. Olague and B. Hern´andez 145 to the data is carried out through the well-known technique of least squares. The approach is to define anχ2_{merit function and determine the best fit parameters}

by its minimization. Because of nonlinearities, the minimization should proceed iteratively. Given an initial trial solution, sufficiently close to the minimum for the parameters; the process improves the trial solution iteratively untilχ2 _{stops, or}

effectively stop decreasing. Our approach is to apply a global optimization tech- nique using the least squares method as a local process in order to improve the search of the global optimum. Moreover, as a by-product of the minimization, the covariances of the parameters are obtained.

7.4.1. Modeling L-corners as an optimization problem

The above analysis suggests that a global optimization technique can be used to solve the problem of guessing the initial parameters. Thus, corner’s localization is obtained by fitting our parametric model to the image intensities. Estimates for the model parametersP = (p1,. . .,pn) ∈ R2 are found by minimizing the

squared differences between the (nonlinear) model function and the considered gray values: Q=χ2_=F(_P)= m i=1 m j=1 Iui,vj −ML xi,yj,P2. (7.11)

The intensities and the function values of the model in the considered image area areI(ui,vj) andML(xi,yj,P), respectively. Previous approaches used by Rohr [29] applied the method of Powell utilizing only function values or used the method of Levenberg-Marquardt (Press et al. [27]) incorporating partial derivatives of the model function in order to reduce the computation time. However, a drawback presented on these approaches is that the identification result relies on the initial parameter values and as usual with nonlinear cost functions, in general, we cannot guarantee to find the global minimum. This problem is studied in this work using an evolutionary algorithm due to the success achieved on this kind of problems. In summary,nis the number of parameters to minimize in our model.m=2w+ 1 defines the size of the input data.P=(σ1,μ1,ϑ1,σ2,μ2,ϑ2,A,B) are the parameters

ofMLthat describe the behavior of our L-corner.ML(xi,yj,P) is the corner model evaluated at thePparameters on the model coordinate system.I(ui,vj) are the intensity values of an image in a gray scale, which is a square subimage of size

m×mpixels within the entire image.F(P) is theχ2 _{estimator. Equation (7.11)}

includes two coordinate systems: the image coordinate system (u,v) and the model coordinate system (x,y), which are different.

7.4.2. Genetic algorithms for function optimization

Evolutionary algorithmsare considered a rich paradigm for global optimization. Previous methodologies as thedownhill simplex methodandsimulated annealing

146 Autonomous model-based corner detection are well-known techniques for multidimensional optimization [27]. This section is devoted to our affine evolutionary algorithm for global optimization. Currently, evolutionary algorithms for numerical optimization use real-code parameters for which a set of special transformations has been developed. In real coding imple- mentation, each chromosome is encoded as a vector of real numbers of the same length. Several crossover operators have been introduced under the name of arith- metical operators. The arithmetical operators are built by borrowing the concept of linear combination of vectors from the area of convex sets theory. Generally, crossover produces an offspring, which is calculated from the weighted average of two vectorsy1andy2as follows:

y1=λ1y1+λ2y2,

y2=λ2y1+λ1y2;

(7.12) if the multipliers are restricted to

λ1+λ2=1, λ1>0, λ2>0, (7.13)

the weighted form is known as convex combination. If the nonnegativity condition on the multipliers is dropped, the combination is known as affine combination. Finally, if the multipliers are simply required to be in real space, the combination is known as a linear combination [12]. Another operator is known under the name of dynamic mutation, also called nonuniform mutation, introduced by Janikow and Michalewicz [20]. Dynamic mutation is designed for fine-tuning capabilities aimed at achieving high precision. Given a parenty, if the elementykis selected for mutation, the resulting offspring isy=[y1,. . .,yk,. . .,yn], whereykis randomly selected from the following two possibilities:

y_k=yk+Δ t,yU k −yk (7.14) or y_k=yk−Δ t,yk−ykL , (7.15) where Δ(t,y)=yr 1− t T b . (7.16)

The functionΔ(t,y) returns a value in the range [0,y] such that the value ap- proaches 0 astincreases. This property causes the operator to search the space uniformly initially, whentis small, and very locally at later stages.tis the gener- ation number,bis a parameter determining the degree of nonuniformity, andr

is a random number between [0, 1]. It is possible for the operator to generate an offspring which is not valid. In such a case, we can reduce the value of the random numberr.

G. Olague and B. Hern´andez 147

7.4.3. A novel evolutionary representation

The operations of crossover and mutation can be encapsulated into a single com- plex transformation as follows. In order to handle affine geometry algebraically, we have to characterize the lineiby an invariant equation, and we will suppose that this equation isy0=0. Since the points ofiare now regarded as ideal points,

no point with y0 =0 is actual, and this means that we can represent the actual

points of the affine plane by pairs of nonhomogeneous coordinatesY =(Y1,Y2),

where Y1= y1 y0 , Y2= y2 y0 . (7.17)

The allowable representationsRAof the affine plane are those representationsR ofS2 in which the lineihas the equation y0 = 0; and this leads at once to the

following theorem.

Theorem 7.4. If RA is any allowable representation of the affine plane, then the whole class (RA)of allowable representations consists of all those representations, which can be derived fromRAby applying a transformation of the form

Y1=b11Y1+b12Y2+C1,

Y2=b21Y1+b22Y2+C2,

(7.18)

where the coefficients are arbitrary real numbers subject to the condition|brs| =0.

UsingTheorem 7.4, it is possible to transform thenvariables of two solutions into a new pair of solutions, according to the following transformation:

Y11 Y 12 · · · Y 1n Y21 Y 22 · · · Y 2n = ⎡ ⎢ ⎣ b11 b12 b21 b22 # $% & crossover C1 C2 #$%& mutation ⎤ ⎥ ⎦ n ⎛ ⎜ ⎝ Y11 Y12 · · · Y1n Y21 Y22 · · · Y2n 1 1 · · · 1 ⎞ ⎟ ⎠. (7.19) Equation (7.19) can be expanded to the whole population. The advantages of this representation are as follows.

(1) Standardized treatment of all transformations.

(2) Complex transformations are composed from single transformations by means of matrix multiplication.

(3) Ann-dimensional point can be transformed by applying a set ofntrans- formations.

(4) Simple inversion of the transformation by matrix inversion.

(5) Extremely fast, hardware-supported matrix operations in high-power graphic workstations.

148 Autonomous model-based corner detection (a) (b) (c) (d) (e) (f) Ib Ia 20 40 80 (a) (b) (c)

Figure7.6. On the right structures (a), (b), and (c) of the test synthetic image with Gaussian noise scaled byλ=20, 40, 80.