On novel developments of controlled evolution of level sets in the field of inverse shape problems



[1] Novel developments of the so-called controlled evolution of level sets [Ramananjaona et al., 2001b], which is devoted to shape identification of homogeneous scattering obstacles buried in a known space from time-harmonic wave field data, are considered herein. The emphasis is twofold: regularization of the geometry of the sought shape—enforced via a speed of motion of the level set in (pseudo)time and space resulting from the minimization of a properly penalized objective functional; and improvement of the convergence of the scheme itself by a choice of a time step adapted to the evolution of the level set and to the sought decrease of the objective functional. Key elements of the theoretical analysis are given, and several numerical examples illustrate pros and cons.

1. Introduction

[2] Level set methods are increasingly used to compute and analyze the motion of a contour. Investigated in-depth by Osher and Senthian [1988] for the modeling of the propagation of fronts, they are nowadays considered for various applications as is illustrated, e.g., by Osher and Fedkiw [2001] and references therein. One of the major advantages of these methods appears to be their capability of merging or splitting obstacles without topological constraints.

[3] The so-called controlled evolution of level sets whose novel developments are considered herein is based on a combination of two methods:

  1. The level set representation of shapes [e.g., Osher and Senthian, 1988], which was first proposed by Santosa [1996] for an inversion procedure, enables us to seek the closed contours of a scatterer as the zero-level of a level set. This level set is evolved in space and (pseudo)time in a prescribed search domain so as a suitably chosen objective function is minimized. The evolution is governed by a Hamilton–Jacobi equation which links, through a velocity motion, its partial derivative with respect to time to the norm of its gradient in space.
  2. The Speed or Velocity Method, whose theoretical features are now well established for the solution of design and sensitivity problems [Sokolowski and Zolésio, 1992], enables us to choose a velocity such as the time derivative of the objective functional is negative or null, and consequently to decrease this functional with time (pending limitations that might arise at the discretization stage).

[4] The present paper is focused on an electromagnetic inverse scattering problem where one has to retrieve a homogeneous, closed obstacle (a possibly multiply connected lossy dielectric cylinder with same z-orientation and unknown cross-sectional shape in the orthogonal (x-y) plane) which is buried in a known lossy dielectric half-space. This must be done from scattered fields that are observed above this half-space when the said obstacles are probed by prescribed wave fields. More specifically, the polarization of the wave field is transverse electric (TE), i.e., the only nonnull component of the magnetic (H) field (incident as well as scattered) is parallel to the z axis. Pioneered by Litman et al. [1998] for the reconstruction of penetrable obstacles in free space in the dual Transverse Magnetic polarization (the only nonnull component of the electric (E) field is parallel to the cylinders' axes), the controlled evolution of level sets has been extended by Ramananjaona et al. [2001a] to the retrieval of buried obstacles in both TM and TE polarizations and it has been applied to the reconstruction of obstacles in free space from experimental microwave data [Ramananjaona et al., 2001a]. Here, the purpose of our investigation is twofold.

[5] First, regularization procedures that are involving either the length of the cross-sectional contour or the area of the cross-section itself are proposed in order to avoid unrealistically jagged boundary curves while still preserving the ability to produce multiply connected objects. Such a regularization is enforced via the objective functional and consequently is shown to only entail changes in the velocity motion, with the occurrence of either curvature-driven or advection terms—in effect as it was suggested earlier by Santosa [1996] from inspection of existing material on front propagation.

[6] Second, a numerical scheme for the choice of the time step Δt (needed at the discretization stage of the Hamilton–Jacobi equation involved in the evolution of the level set) is put forth by combining theoretical analysis and numerical experimentation. This time step is indeed one of the few key parameters at the implementation stage of the solution algorithm—as expected, too small a time step leads to a slow decrease of the objective functional, and too high a time step leads to divergence of the algorithm. Up until now [Ramananjaona et al., 2001a, 2001b], Δt is chosen by pure numerical experimentation. Here, two adaptative schemes are proposed: one is based on a CFL criterion [Sethian, 1999], where Δt is a function of the spatial discretization and of the maximum amplitude of the speed of motion of the level set; the other assumes a maximum time step chosen so as a properly introduced L norm of the level set is decreased from one iteration to the next.

[7] The plan of the paper is as follows. In section 2 geometry and the TE formulation of the wave field are sketched. In section 3 key features of the inversion method are reminded. In section 4 regularization of the inversion via geometrical constraints is discussed. In section 5 adaptative time steps are introduced. In section 6 a number of illustrative numerical examples are given. A short outline of forthcoming studies follows in section 7.

2. TE Formulations of the Wave Field

[8] The geometrical and electric configuration of interest is sketched in Figure 1. A homogeneous dielectric cylindrical obstacle of axis parallel to the z axis is embedded within a homogeneous dielectric lower half-space (numbered 2). Its possibly multiconnected cross-section Ω is of smooth enough boundary contour and remains contained inside a prescribed box ��. The lower half-space is separated by a planar interface x-z from the upper half-space (numbered 1).

Figure 1.

Geometrical and electrical configuration of study. All following figures display the results within the search domain �� only, the global orientation—interface at the top—being kept.

[9] All constitutive materials are linear, isotropic and nonmagnetic (permeability μ0). Their known permittivities read as εm = ε0 εrm, m = 1, 2, Ω, at implied circular frequency ω—omitting the time-dependence exp (−jωt)—where ε0 is the permittivity of the upper half-space assumed to be air (i.e., εr1 = 1), and where both εr2 and εrΩ may take complex values with positive or null imaginary parts (e.g., via a conductivity contribution). Corresponding propagation constants km are such that km2 = ω2 μ0εm with positive or null imaginary parts. The electrical contrast function χ(r) defined in �� is also introduced. χ(r) is constant-valued and equal to 0 everywhere in �� except for any r in Ω where χ(r) is equal to η = equation image.

[10] Ideal time-harmonic magnetic current line sources Sj (j = 1, …, Ns) parallel to the z axis are placed strictly above the interface x-z. Ideal sensors are also placed above this interface and collect the single z component of the scattered magnetic field Hscat at Nr locations. For simplicity one henceforth assumes the existence of a certain probing line equation image at constant height yr along which the sensors are equally distributed, while the sources are also taken equally distributed at constant height ys.

[11] u, uinc and uscat are denoted as the single z components of the total, incident and scattered magnetic field (the incident field is due to the chosen source radiating at ω in the unperturbed environment and accounts for the layering). The data field collected in the measurement domain ℳ is denoted as ζ. Subscript j emphasizes whether necessary that the corresponding source is Sj.

[12] Application of the Green theorem to the Helmholtz wave equations satisfied by u, and accounting for the appropriate TE transmission conditions and Sommerfeld radiation conditions at infinity in both half-spaces, yields the rigorous state and data in equations (1) and (2) (refer, e.g., to Lambert [1998]):

equation image
equation image

The two-component ∇′ operator in the above is taken with respect to the primed variables and the Green functions Gmn(r, r′) model the radiation of a source at r′ in the half-space n and the observation of the field at r in the half-space m, where m = 1, 2, n = 1, 2.

3. Key Features of the Inversion Method

[13] The method has been pioneered by Litman et al. [1998] and investigated since then by the authors [e.g., Ramananjaona et al., 2001b], noteworthy investigations being led elsewhere [e.g., Dorn et al., 2000; Ito et al., 2001]. As already said, it combines a level set representation of shapes [Sethian, 1999] the first use of which in an inversion procedure goes back to Santosa [1996] and the Speed Method which has been much studied in design and sensitivity problems [Sokolowski and Zolésio, 1992]. Here one has no place to detail its involved machinery. However, it is still necessary to sketch how the method actually goes in order to appreciate which key interrogations remain and which novel material is brought forth.

[14] With reference to the previous section, an obstacle with interior domain Ω is assumed to be made of a known material, but is of unknown shape, location and connectivity—there may be isolated parts, and holes within them. Ω is wholly included in a prescribed domain �� (a clear yet not unrealistic prior). It is probed by time-harmonic sources Sj and corresponding scattered fields ζ are collected in ℳ, a set disjoint from ��. One assumes that an initial domain Ω0 with boundary Γ0 at time t = 0 is prescribed and is associated to a certain level set function ϕ0—its level 0 is Γ0, it is negative in the interior of Ω0, and positive in the exterior—such as a signed distance function. At time t ≥ 0, a new domain Ωt of boundary Γt, corresponding to the zero level of the set ϕt, is derived from an Hamilton–Jacobi equation

equation image

where V is a certain velocity field and n the unit normal to the contour Γt.

[15] The choice of V is key to the effectiveness of the method. The Speed method enables us to select V so as a well chosen objective functional equation image is decreased (which is equivalent to equation image). In our case equation image is given by

equation image

Its derivative (as demonstrated by Ramananjaona et al. [2001b] by a Min Max analysis developed from the Green-based integral formulations of the field) follows as

equation image
equation image

p is an adjoint field whose source term is made as usual from the conjugated discrepancy of the data error in equation image and which satisfies equations similar to those satisfied by u. g is the normal component of the shape gradient at Γt. V(x, t) is the normal component of a heuristic choice of V(x, t) at any point x in the search domain �� (including the boundary Γt) which enforces the decrease of equation image for all t according to

equation image

4. Regularization: From Penalization to Level-Set Evolution

[16] What we concentrate upon in this section is the topic of regularization. The regularization is enforced via the objective functional J, and as a consequence via V, in order to avoid unrealistically jagged shape boundaries Γt while still preserving the ability to produce multiply connected obstacles Ωt; let us however notice that ad hoc techniques already exist, like shortening a jagged contour Γt via a secondary evolution [Dorn et al., 2000], while one has observed that reinitializing at certain t's the level-set function ϕt as a signed distance function (calculated from contour Γt at such t's) had some smoothing effects on the contours, but significantly slackened the inversion procedure.

[17] A three-term objective functional J is introduced. It is made of a data error equation image, previously defined in (4), and of two additional penalty terms: JΓ, which is proportional to the perimeter of the reconstructed object Ωt, and with which one is aiming at short cross-sectional contours Γt; JΩ, which is proportional to its cross-sectional area, and with which one is aiming at minimizing this area. J thus reads as

equation image


equation image

[18] In the above a and b are real positive constants (their choice is deciding, but in practice one has to resort to numerical experimentation), and a0 and b0 are squared norms associated with the initial guess of the obstacle domain. Classical calculations of the time derivative of these penalty terms are demonstrated in continuum mechanics theory [Germain and Muller, 1980; Delfour and Zolésio, 2001]. Carrying them out yields this derivative as

equation image

As was done previously (see equations (6) and (7)), the normal component of the velocity field V can be defined everywhere in �� as

equation image

in order to ensure nonpositive dJ/dt, the curvature κt being the one of each level set and being calculated as ∇ · (∇ϕ/∣∇ϕ∣). This curvature-dependent velocity has been widely used to regularize the computation of motion of fronts via the level set method [Harabetian and Osher, 1998], as well in the field of image processing [e.g., Mumford and Shah, 1989; Malladi et al., 1995], but appears to have only been recently proposed in order to tackle inverse scattering problems [Santosa, 1996; Feng et al., 2000, 2002].

5. Discretized Hamilton–Jacobi Equation

[19] For numerical calculations, equation (3) is discretized into

equation image

equation imageijt) is a numerical hamiltonian corresponding to the term ∣∇ϕ(x, t)∣ which is describing the propagation and advection of the front with respect to a certain velocity. equation imageijt) is a numerical hamiltonian corresponding to the term κt(x)∣∇ϕ(x, t)∣ which is describing a curvature-driven evolution. Both terms are computed using discretization space steps Δx and Δy in the directions x and y, respectively. Numerical implementation of the above is discussed in-depth by Sethian [1999].

[20] The time discretization is a main hurdle of an effective reconstruction procedure based on an evolution equation. Even though a legitimate time step Δt would be a constant one satisfying a CFL condition connected to the Hamilton–Jacobi equation of the motion of the level set, other choices are worthwhile studying.

5.1. CFL Time Step

[21] Because of the nonlinearity of the Hamilton–Jacobi equation (3), one cannot put forth an explicit CFL condition. So, instead of keeping the time step as a constant, it might be useful in terms of stability to adapt its value both to the space discretization step and to the amplitude of the velocity. Operating in an analogous fashion to what is usually done with discretized evolution equations, a CFL condition which characterizes a relation between these two quantities is written as

equation image

with h = min(Δx, Δy). This condition implies that the front of each level does not move beyond one space cell from its previous location—refer to Sethian [1999] for details.

5.2. Maximal Time Step

[22] Speeding up the shape reconstruction (i.e., quickly converging toward the least cost in a least number of iterations) and at the same time preventing a blow-up of the level set might be achieved by keeping the global minimum (or maximum) of the level set as a constant. (A similar idea was introduced by Dorn et al. [2000], a level-set evolution normalized with a parameter chosen such that the L(��) norm of the level-set is constant being implemented.) This analysis restricts the study to unconstrained cases (b = 0 and a = 0, because the curvature-driven term must satisfy a CFL condition) and exhibits a large time step selection. For any time step, one may draw an analogy between the search of a local minimizer using a optimization procedure such as a steepest descent method and the evolution of a steady state with respect to an ordinary differential equation [Higham, 1999].

[23] In this frame of thoughts, our focus is to converge globally toward a fixed point in a minimal number of iterations, without taking care of the regular evolution of the level set and possibly getting out of local minima. One is able to introduce an upper bound in the discretized Hamilton–Jacobi evolution of the level set as

equation image

Replacing ϕijt+1 by its corresponding expression yields

equation image

In order to obtain a suitable time step, one is now looking for τ such that

equation image

The value obtained for τ may then be used as a time step for the motion of the level set. If no solution τ is found for equation (16), we try to find a solution equation image, or equation image, and so on. However, large time step values used at some iterations do not involve an evolution which satisfies the minimization condition of the objective functional, and a constant velocity is something which is disagreeing with the nonlinearity of the Hamilton–Jacobi equation. This might yield a oscillating (but stable) procedure, or even a nonconvergent search—i.e., convergence is not guaranteed.

[24] Introducing a weighting function w(J) which is depending upon the value of the objective functional J and whose role is to decrease the time step Δt according to the decay of this value of J tends to alleviate this problem. One simply writes:

equation image

where function w(J) should be such that

equation image

During the numerical experimentations the second feature is replaced by equation imagew(J) ≃ 10−2 and a specific weighting function is chosen as

equation image

[25] To conclude this section, one should emphasize that other time steps might be employed. For example, a time step which would be only decreasing in proportion to the value of the objective functional could be used. However, doing so has been observed as not enabling us to leave local minima. Other norms, such as Lp(��), might be used in order to determine a time step value; but in such cases one would have to cope with the problem of finding an upper bound for the norm of the level set.

6. Numerical Illustrations

[26] The purpose of our investigation is illustrated with a small number of numerical examples. The configuration of study is as follows: the upper half-space is air and the lower half-space is of relative permittivity εr2 = 2 and conductivity σ2 = 0 S/m. Two obstacles of relative permittivity εrΩ = 3 and conductivity σΩ = 0 S/m are embedded within this lower half-space: one is squared of side 0.29 m and is centered at (−0.18 m, −0.68 m); the other is rectangular with sides 0.18 m × 0.29 m and is centered at (0.28 m, −0.25 m). The whole set is illuminated by 15 sources regularly spaced between −5 m and 5 m at height 0.5 m above the interface, and the scattered field is collected on 47 receivers regularly spaced between −10 m and 10 m at height 0.125 m above the interface. The operating frequency is 200 MHz (the wavelength in the embedding half-space is thus about 1.06 m). As a general rule, the initial guess of the obstacle domain is a single disk centered in the prescribed search domain ��, the latter being taken as a square of side 1 m centered at (0 m, −0.5 m).

[27] The synthetic data ζ are provided by solving the direct problem with a 44 × 44-pixel grid whereas the inverse problem is handled with a 30 × 30-pixel grid in order to avoid the inverse crime. The corresponding model error computed as the L2(ℳ) norm of the difference between the scattered fields simulated with those two grids is valued at about 9.510−3. Notice that using the Born approximation (i.e., by letting the scattered field in equation (2) be induced by sources proportional to the incident field and not to the total one) would lead to a discrepancy between the Born-approximated field and the exact one of about 0.4104 at 200 MHz, with much higher values often observed for intermediate shapes (such as the disks chosen as first guesses in the absence of other means of initialization).

[28] Results displayed here in terms of level sets and object domains are either those observed at the lowest value of the objective functional J reached during the first 500 iterations, or earlier, after a small number of iterations. A number of evolutions of J as a function of the number of iterations are given also.

6.1. Illustration of the Regularization

[29] The iterative procedure of solving the inverse scattering problems de facto requires the solution of two direct problems at each step. This discretized direct problem and its corresponding adjoint problem can be handled either exactly, by a Gauss-type solver in N3 operations, or less accurately, by a conjugate-gradient solver in N iterations, where N is the number of unknowns. In this subsection we will show the influence of this computational error onto the reconstructions and how the regularization allows us to overcome the problem. The level-set procedure is initialized with a single centered obstacle of radius 0.25 m (Figure 2a) and the time step is kept constant to Δt = 3 10−3.

Figure 2.

(a) Initialization (iteration 0), (b) Gauss solver of the direct and adjoint problems (results at iteration 489), (c) same as in (b) with a CG solver without regularization (iteration 405), (d) same as in (c) with regularization (iteration 312). Frequency is 200 MHz.

[30] In Figure 2b the level set and its corresponding domain obtained by using a Gauss-type solver without regularization are displayed. The best result (as a reminder, the one associated to the least value of J observed during the first 500 iterations) shows a good localization of the two obstacles. When using a conjugate-gradient solver (Figure 2c) without regularization one can observe the emergence of jagged contours. It should be noted that even with such a contour the obstacles are relatively well located still. This phenomenon is due to too low a required precision (10−6) of the conjugate-gradient solver; increasing its precision to 10−8 leads to the disappearance of the oscillations. Such a behavior will allow to test the efficiency of the regularization at low cost.

[31] As shown in Figure 2d, the addition of a regularization constraint onto the contour (a = 1 and b = 0) yields much better results with a smoother contour and a better location of the obstacles. The corresponding objective functions equation image as a function of the number of iterations are depicted in Figure 3. (Suitable values of regularizing parameters have been determined by numerical experimentation with different values of a and b, no theoretical criterion for optimality being available.)

Figure 3.

Evolution of the objective functional equation image as a function of the number of iterations (refer to Figure 2) for three different solution algorithms of the direct and adjoint problems: exact Gauss solver (dashed curve), CG solver without regularization (dotted curve), CG solver with regularization (solid curve). Frequency is 200 MHz.

6.2. On the Time Step Choice

[32] In this subsection the fields are computed using a Gauss-type solver and no regularization is enforced. The level-set procedure is initialized by a single centered obstacle of radius 0.15 m. In Figure 4 three different time step strategies are considered, the results being shown at an early stop (iteration 50) and at the end of the procedure (for the lowest objective functional).

Figure 4.

Top: Obstacle cross-sections retrieved early on (at iteration 50); bottom: best solutions (lowest objective functional). Three different time steps Δt, constant value, CFL, and maximal time step are used, from left to right. Frequency is 200 MHz. The variations of the objective functionals are shown in Figure 5 and Figure 6.

[33] First let us compare at iteration 50, the result in Figure 4a obtained with a predetermined, constant time step Δt = 0.003 (the objective functional is about 2.2 10−2), the one in Figure 4b obtained with an adaptative CFL scheme (the objective functional is reduced to about 910−3), and the result in Figure 4c obtained with a maximal adaptative time step (the objective functional is much smaller, at about 4.4 10−3). We observe that the adaptative time step enables a much faster convergence, the maximal one giving us the lowest cost.

[34] Then, letting the reconstruction go up to 500 iterations (Figures 5 and 6), we observe that the quality of the results does not depend on the type of time step chosen, minimal values of the objective functional being equal to about 1.85 10−3 (constant time step, Figure 4d), 3.3 10−3 (adaptative CFL, Figure 4e), and 1.421 10−3 (maximal time step, Figure 4f) at iterations 421, 433 and 455, respectively. The interest of the choice of the maximal time step is especially manifest in the first iterations of the procedure, when the objective functional is decreased very quickly, this being associated with a large amplitude of the normal velocity. However, when the solution starts to stabilize itself, choosing the maximal time step or the CFL one provides a similarly oscillating behavior at values that remain below, even much below, or around the model error of 9.510−3 previously given. So, a stopping criterion is strongly in demand in this phase of oscillations, since pursuing the calculations at quite high a computational cost (i.e., during a large number of iterations) does not bring us more pertinent information.

Figure 5.

Variation of the objective functional equation image as a function of the number of iterations for a constant time step Δt (dashed curve) and a CFL one (solid curve).

Figure 6.

Refer to Figure 4. Variation of the objective functional equation image as a function of the number of iterations for a constant time step Δt (dashed curve) and a maximal one (solid curve).

7. Conclusion

[35] The level set representation of shapes, combined with the speed method, enables many kind of evolutions, given the velocity field, such as deformations, splitting or merging, and performs a worthwhile role in the field of inverse scattering problems as already underlined in previous publications [Ramananjaona et al., 2001a, 2001b].

[36] We have observed the efficiency of using geometrical penalization terms in the cost functional (and their corresponding terms in the expression of the velocity) as a regularization tool when the interior fields are computed with too low a precision, the same observation being true also for relatively noisy scattered field data. This regularization may be very useful in reconstructing 3-D buried objects, where the inversion of a matrix with a Gauss-type solver becomes nearly impossible in the scattered field computation and requires a conjugate-gradient solver which needs quite less computational memory.

[37] The CFL time step discretization actually gives us a good “first choice” of the time step in order to process the motion of the level set via the Hamilton–Jacobi equation. But a time step which is satisfying a condition of stability such as the one considered here and linked to the L(��) norm of the level set may enable a faster convergence of the cost functional, especially when the evolution of the level set is performed in the whole discretization domain ��. The choice of the weighting function w(J) in order to enforce a decreasing time step is peculiar, but it yields a convergent search in many configurations, although many other choices are available.

[38] Finally the results shown here confirm that a clever stopping criterion should be implemented in the method. Indeed, there is no need to let the inversion procedure be associated with a cost functional that oscillates for a long time around a plateau of low magnitude (usually smaller, even much smaller than the model error), while only a few pixels vary in the object representation at the same time. The first idea being presently considered is a statistical study of the evolution of the image, for example stopping the evolution when the average variation of the image (the pixel distribution associated with the level set) tends to zero within a certain number of iterations. Another possibility would be to evaluate an a priori value of the model error, which corresponds to the theoretical accessible minimum.