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Registration of Deformed Images using Energy Minimization Constraints

Abstract

In some applications, as in image registration of serial sections in biology or medicine, it is often necessary to find a rigid body transformation that matches features under natural deformations in two images. Usual approaches deal with natural deformation as a problem of error minimization of displacement of markers. In this paper we present a registration technique which effectively models the deformation by using the physical model of an elastically deformable thin plate. The proposed method makes use of Thin Plate Splines as a registration function enabling a clear separation between the effects of the sensor positioning and the effects of deformations. The application of the method is shown using simulated data, and its strengths and flaws are analyzed

Image registration; Thin Plate Spline; energy minimization; deformation modeling; image matching; rigid body transformation


Registration of Deformed Images using Energy Minimization Constraints

José Eduardo C. Castanho

UNESP - FET - DEE,

Caixa Postal 473. 17001-970. Bauru, SP,Brazil.

phone: (014) 230-2111 ext. 194, fax: (014) 230-5070

castanho@azul.bauru.unesp.br

Clésio L. Tozzi

UNICAMP - FEEC - DCA

Caixa Postal 6101 13083-970 Campinas, SP,Brazil

phone: (019) 239-8385, fax: (019) 239-1395

clesio@dca.fee.unicamp.br

Abstract: In some applications, as in image registration of serial sections in biology or medicine, it is often necessary to find a rigid body transformation that matches features under natural deformations in two images. Usual approaches deal with natural deformation as a problem of error minimization of displacement of markers. In this paper we present a registration technique which effectively models the deformation by using the physical model of an elastically deformable thin plate. The proposed method makes use of Thin Plate Splines as a registration function enabling a clear separation between the effects of the sensor positioning and the effects of deformations. The application of the method is shown using simulated data, and its strengths and flaws are analyzed.

Keywords:deformation modeling, Image registration, Thin Plate Spline, energy minimization, image matching, rigid body transformation.

1 Introduction

In this paper we discuss the problem of registering images of serial sections. In this case, registration consists of finding the parameters of a rigid body transformation based on a set of correspondent points in both images. However, serial sections poses additional technical difficulties for registration since the images are of different slices and may show natural deformations. Usually the deformations are treated as an error in the location of markers and statistical methods of error minimization are employed. This approach does not take into account the nature of deformations. A more interesting way of modeling deformations is the use of physically based models of elastically deformable bodies. The approach we present herein models the registration mapping as being the superposition of the effects of natural deformations and the effects of a rigid body transformation which express the image acquisition misalignment. We use Thin Plate Spline (TPS) as registration function in order to correctly model the transformation. The obtained result is optimum in the sense that total energy of bending (deformation) is minimized and agree with models of elastically deformable plates. TPS's are functions that model an infinite metal thin plate under deformation and that can be considered a linear combination of a plane and a general spline surface. So a clear separation between the effects of the sensor positioning (linear part of the transformation) and the effects of the implicit image deformation is provided. Since we are looking for the parameters of a rigid body transformation, a method is provided to extract them from the TPS functions. The final result takes deformation into account but its effects are clearly separable and can be eliminated from the final rigid body registration function. So the proposed registration method is more coherent with the nature of the problem being approached than those that deal with deformation as an error in the data. In the next sections we discuss image registration models and we present the implementation and conceptual aspects of using TPS for image registration. In the sequence, the method for extracting the rigid body transformation embedded in the TPS's is presented, and the obtained results are evaluated. Finally, we present a discussion and the conclusions concerning the proposed method.

2 A Review of Image Registration Models

An usual step in most image processing and computer vision applications is image registration. A number of image registration methods has been presented in the literature. A detailed survey of registration methods can be found in [5], and a revision of those methods more concerned with the medical area can be found in [13]. In a simple way, the general goal in image registration is, given two images of the same scene which present differences due to the acquisition process (as for example, temporal differences, from viewpoint, etc.), to get one mapping function between both images such as the differences are minimized. The parameters of the functions u = fx (x, y) and v = fy (x, y) that relates the reference image coordinates (x,y) to the target image coordinates (u,v), are in general determined in accordance with a convenient criterion of optimization (being common to minimize Cartesian distances).

An application of image registration, usual in biology and medicine, as well as other areas, is to align images of serial sections to get a 3D model. In these applications the main goal is to recover the alignment of the sections which is lost during the slicing process. The transformation necessary to recover the alignment is easily identified as a rigid body type, that is, a concatenation of a rotation, a scaling, and a translation. The mapping function is then represented by the equations (1) and (2).

(1)

(2)

Where ax and ay are translations in x and y directions respectively, a is the scaling factor, and j is the rotation angle.

The most difficult problems for registering these kind of images arise from the fact that they may show natural differences of shape or features. Consequently, the alignment lost in the acquisition process may be mixed with the natural deformations and it becomes difficult to determine the rigid body transformation parameters using intrinsic image features. Determining the rigid transformation that makes the registration in the presence of local deformation implies in isolating both effects. One possibility is to choose features or to use fiducial marks that are not affected by the deformation. It happens that this procedure is often of difficult realization. This procedure may also need specific knowledge of the material being worked with and it is not of easy generalization. So deformations must be incorporated to the model that describes the registration transformation and its effects must be isolated.

2.1 Modeling Deformation as an Error

When deformations are not significant least square methods may yield good results [24],[18], [15]. In cases in which the differences are local methods based on the statistical norm, as the Generalized Hough Transform [2], [10], may produce better results [8]. In the first approach the deformation is included in the model as a statistical error of displacement of the features that is minimized in the process of searching for the transformation parameters. The effects of deformation are then spread all over the image even if they are local. As a result the image registration tends to be a function of an average displacement of the used control points. In the second case it is supposed that the effects of deformation are overcome by making the registration a function of the statistical mode. In this case, the whole image can be registered taking as reference a region (or feature) that has not been affected by the deformation. Regions with deformation tend to have less influence in the registration process. The main flaw of the methods based on error minimization is to treat deformation as being an implicit error in the data. As deformation is not really an error this misinterpretation yields wrong results. Therefore it is necessary to determine a model in which the deformation could be effectively incorporated and its behavior adequately described making possible to separate the effects of deformation from those of a global rigid transformation.

2.2 Elastic Models

A more natural way of modeling deformations is to apply the analogy with elastically deformable bodies. The elasticity theory has been employed in several applications for modeling of natural deformations. Its application relies on using the physical laws that govern object deformation which are under the action of external forces as well as intrinsic properties. Through the simulation of physical properties such as stress and rigidity it is possible to model the static shapes shown by a great number of deformable bodies as rubbers, springs, paper, clothes, and other flexible bodies. The addition of physical properties as mass or dumping enables the simulation of the dynamics of these objects. These simulations usually take on the application of numerical methods to solve partial derivatives equations that govern the evolution of the object and its movement through the space [30].

A rigid body under deformation forces has the tendency to accommodate in its state of minimum energy for a given configuration of applied forces. So the elastic modeling usually leads to the solution of a differential equation for the condition that minimizes the deformation. A classical example is the interpolating cubic spline which is the abstraction of the shape of a thin elastic beam, whose minimum bending energy can be described by a differential equation [20]. The elasticity theory leads to the generalization of the idea of the spline for representation of curves, surfaces, and solids.

This concept is not new and has been used in several works where it is necessary to match two images or two models with deformation [29], [25], [1]. In computer graphics, for example, Terzopoulos et al. [30], and Metaxas and Terzopoulos [25] have used models of elastically deformable bodies for representation and animation of bodies with deformation. In computational vision Kass et al [19] have present a method for matching elastic contours of an object. Terzopoulos [29] applies the concept of elastic deformation to fit a smooth surface to a set of 3D points acquired in a stereo vision system. His method employs results of variational calculus to determine the surface with minimum bending energy that fits the data.

Several image registration methods have associated the idea of elastic models with the modeling of deformations in images. Generally, such methods iteratively approximate correspondent features in both images based on similarity measures. The image or object is modeled as an elastic body and the measure of similarity between points or features in both images act as external forces that shapes the body. These forces are counterbalanced by rigidity and smoothing constraints. The process consists in determining the state of minimum energy whose resulting deformation transformation defines the registration. Elastic methods are usually employed to solve problems of medical imaging where it is necessary to match structures. While sometimes referred as registration methods these applications aim to matches two images or structures removing any difference between them.

The elastic approach for image matching was first employed by Burr [6], [7]. His method is based on a proximity measure between features to produce a displacement field which will be used to gradually and locally approximate one image toward the other. The displacement in each point of the image is determined by a weighted averaging of all displacements. At each new iteration of the process the local displacements have more influence on the weighting. Bajcsy and Kovacic [1] have proposed the application of elastic models for matching 3D images and images from atlases. The process starts with a gross global alignment using conventional techniques. After that the images are locally approximated to match each other. A similar technique was presented in by Mehran Moshfeghi and Surendra Ranganath's work [27], where it is proposed the use of elastic model for matching 3D images of different modalities.

An elegant example of elastic model for image matching is the Thin Plate Spline. The application of TPS to match images with deformations was first proposed by Goshtasby [16]. In his work he affirms that TPS is superior to polynomial approach in image warping since the influence of local deformations are less likely to spread over all the image. Bookstein [4] shows an interesting application of the TPS to evaluation of deformations in biology context. By using TPS as mapping functions, Bookstein matches contours in different images. The shape changing in the matched contours is then analyzed decomposing the TPS bending energy matrix in its eigenvectors. These can be interpreted as components of shape deformation at different geometrical scales. In the next section we review the characteristics and the conceptual aspects of TPS as well as its application to the problem of image matching.

3 Image Matching With Thin Plate Splines

TPS's are functions which present some interesting characteristics to solve this kind of problem. They model an infinite metal thin plate surface, in which load points yield deformations. The infinite surface is assured to interpolate the load points with minimum bending energy, and that is smooth (C1 exists and it is continuous). TPS's are compositions of radial basis functions and the effect of the functions diminishes as we go far from the reference points. So although they are global functions the influence of a local deformation tends to spread all over the image, but with less emphasis the farther it is from the point of deformation. The farther toward the infinite the function assumes an almost linear behavior. This is in contrast to the behavior of global methods based on least squares polynomial fitting, that have the effects of local distortions weighted, or equalized, all over the image [32], [16]. Moreover the functions can be determined analytically instead of numerically as is usually the case when other elastic models are employed.

Determining the functions u = fx (x, y), and v = fy (x, y) using Thin Plate Splines can be interpreted as being a problem of interpolation of a smooth surface over a set of scattered points. So given a set of points (xk , yk) in the source image and a set of corresponding points (uk , vk) in the target image, determining the functions fx (x, y), v = fy (x, y) is equivalent to determine two smooth surfaces: one passing through the set of points (xk , yk , uk) and another through the set (xk , yk , vk). If the problem of interpolation is modeled with an affine transformation, then the interpolated surface will be a plane. In the case that the target image has deformations, the interpolating surface will be a linear combination of plane and an irregular surface (representing the deformation).

The TPS function is obtained by minimizing an energy penalty functional. For a thin plate under a small bending, the bending energy at a point is proportional to the norm .

The function f(x,y) that minimizes the integral

(3)

over a class of interpolation functions is given by the solution of the biharmonic equation D2 f (x, y) = 0 that is known [11] to be z (x, y) = r2 where is the radial distance.

Minimizing equation 3 is equivalent to find the minimum bending energy of the plate all over the space. More details of the theoretical foundations of Thin Plate Splines can be found in [11], [12] or in [21],[22],[23].

In order to determine the mapping functions in x and y, constrained to the given n load points, it is necessary to solve the linear system composed by the equations 4, 5, 6, and 7.

(4)

(5)

(6)

(7)

The first equation in the system assures that the interpolating surface is smooth while the equations 5, 6, and 7 assure an almost linear behavior of the TPS when evaluated far from the control points.

Given n control points (xi , yi ) with f (xi , yi ) = zi, the following set of equations (8) results from equation 4:

(8)

The linear equation system is solved for fx (x, y) = u, determining the mapping in x, and for fy (x, y) = v determining the mapping in y, to find the parameters a0, a1, a2, and ki, with i=1, ..., n, and n is the number of control points.

a) b) c)

Figure1: Image matching (c) of two slices (a) and (b) using Thin Plate Splines. The dashed contour in figure (c) is the contour (a) after registration.

Figure 1 shows the contour matching process of two slices of a green pepper using TPS. The crosses in the contours are the position of manually chosen control points used to get the TPS function.

4 Extracting Rigid Body Transformation from TPS

In the TPS formulation, as presented in the last section, the linear component of transformation is given by the coefficients a0 , a1, a2 (see Equation 4) which defines an affine transformation. This formulation can be interpreted as the superposition of a deformed surface over the plane a0 + a1x + a2y, or in other words, the transformation can be split on a linear component and another given by the effects of deformation modeled by the surface spline.

Ftotal = f Affine+ f Spline

In the case of registration of cross section images, however, this model needs further constraints. Inside this context we are interested in searching for the rigid body transformation that aligns the sections. So a better model to the problem is a rigid body transformation with the superposition of the effects of a deformation.

Ftotal = f RigidBody+ f Spline

Within this framework we propose a method for determining the rigid body transformation in registration of serial sections, which is based on the assumption that the scaling of images is isotropic, that is, the scale factor is the same in x and y.

This restriction can be imposed on the registration functions fx and fy , as formulated in Eq. 4, by making the terms a1, and a2, in both functions, two orthogonal vectors. Then the mapping functions become:

(9)

(10)

where a is the scaling factor and j is the rotation angle of the transformation.

To satisfy this constraint it is necessary to solve fx and fy simultaneously, in contrast with original TPS that are interpolated separately. In equations 9, and 10 the parameters to be determined are: (a sin j), and (a cos j), ax ,ay ,bi , and ci with i = 1, ..., n. They can be found solving the following linear system:

(11)

where

and Kx = Ky is the design matrix (12) of the spline given by:

(12)

with r2ij = (xj - xi )2 + (yj - yi )2, and i, j = 1, ..., n.

4.1 Extracting the Rotation and the Translation Parameters from TPS's

In the proposed method there is a refinement of the transformation model. The linear part now represents a rigid body transformation with scaling. The non-linear component represents a displacement applied to each point and that models the deformation which matches both images. However, from the point of view of registration of serial section images the scaling factor must also be considered as a deformation since we are not considering changes like those in the gain of lens or in the distance of acquisition. Therefore, it is necessary to set apart the effect of rotation from the effect of scaling embedded in the terms (a .sin), (a .cos) suppressing the scaling factor from the mapping function. The separation can be easily accomplished through the computation of the norm of the vectors of the rotation/scaling transformation given in matrix form as in 13.

(13)

where a is the scaling factor, or using the relation

(14)

to get the rotation angle and with backward substitution in (a .sin) or in (a .cos) to get the scale. We must note, however, that the simple elimination of the scaling factor will yield a wrong result. Supposing we are working with contours the following steps are proposed to separate the effects of rotation from those of scale:

  1. Get the center of mass of the reference contour.

  2. Apply the gotten linear transformation, including translation, scaling and rotation, but excluding the effects of deformation on the reference contour.

  3. Determine the center of mass of the transformed contour.

  4. Determine the translation of registration using the difference between the coordinates of center of mass of contours before and after applying the transformation;

  5. Determine the rotation using expression 14.

Although the above sequence is based on the use of contours, a similar approach can be developed for other situations. We must note that translation parameter is computed using the center of mass of the transformed contour without the effects of deformation and the same contour without any transformation. This is different from computing the translation parameter using the center of mass of each original contour, reference and target, which is sometimes used as a registration method.

a) b) c)

Figure 2: The original and the deformed contours generated for tests.

5 Results

To evaluate the proposed registration method we present in this section some experiments using two polygons representing contours. The first polygon was generated by digitizing the extern contour of the image of a green pepper slice. A total of 31 points were used to describe the contour. The second polygon was derived from the first through simulation of deformations. Deriving one polygon from the other avoids problems with the correspondence of control points. The deformed polygon was rotated and translated to be used as the target contour for registration with the original contour. The resulting parameters of transformation obtained with the registration method are compared to the applied transformation enabling a controlled evaluation of the method.

Figure 2 shows the contours generated for the tests without the effects of rotation and translation. In all figures, the continuous line represents the deformed contour and the dashed line the original contour. In figure 2-a the deformed contour was produced by the displacement of some vertices in the original contour. In figure 2-b the deformed contour is now scaled with a factor of 1.1. In figure 2-c a greater deformation was applied to the contour and it was also scaled with the factor 1.1.

After generating the contours with deformation they have been translated and rotated with the following values:

rotation:
x translation:
y translation: 60
200
50

Finally, the original contour is registered with each deformed and transformed contour. The numerical results are presented in the table 1 and in the figure 3 we can observe the contours after registration for the three cases.

Table 1:
The numerical results of registration tests.

a) b) c)

Figure 3:a) Contour of fig. 2-a after registration. b) Result of registering the deformed contour of fig. 2-b (continuous line) with the original contour (dashed line). c) Registration of the deformed contour of fig.2-c with the original contour (dashed line)

Some considerations must be given with regard to the interpretation and analysis of the accuracy of experiments. First, it is necessary to say that the examples were conceived with the aim of reproducing real situations as close as possible and the polygons were produced based on a digitized image of a green pepper slice (see Figure 1). However, they are still simulations and also do not represent all possibilities of employing the method. It is also necessary to emphasize that the experiments were conducted using contours with artificially produced deformations, which do not necessarily obey the criterion of registration, that is, deformations with minimum energy of deformation. The results are still very significant, however. In the first test the small deformation has not influenced the results too much, and the error was less than one pixel in the translation and one degree in the rotation. In the second test with scaling present the result was also very accurate, showing the small influence of scale in the process. This was expected since the scaling, being a linear transformation, does not modify the bending energy of the TPS. In the third experiment with the more pronounced deformation the result has a small deterioration in its accuracy. It is very good and visually pleasant, however. This last result shows that a more global deformation (the whole upper side of the contour was deformed) has increased influence in the transformation. As consequence the translation and scaling factors were affected by the deformation.

This formulation works well under the assumption that the" scaling'' is almost homogeneous in all directions, that is, when the deformation is regularly distributed all over the body. In fact, the proposed model assumes that the body scaling factors are the same in the x and y directions. The solution of the linear system will give then the optimal solution for the problem, adjusting the values of the angle of rotation, scaling factor and deformation that satisfies the given constraints. However, when the body has deformations which differ greatly on a given direction the model will fail to provide the right solution for the registration problem. The problem arises because scaling factor, rotation and deformation are free to be adjusted. Consequently, when we have a case in which there is only scaling in one direction, say x direction, the three parameters of the transformation will be conveniently adjusted although there is only one real change in the scale in the x direction. We could suppose that such variation could be adequately modeled by the spline component, but that is not the case since it will yield a non-minimal energy of bending. This case is illustrated by the following example of registration of two rectangles.


Figure 4: a) Two rectangles, R1 on the left bottom, and R2 on the upper right corner, that must be registered. b) Result of registering R1 (dashed line) with R2 (full line) using the proposed approach.

Test 4: Registration of images with anisotropic scaling.

To demonstrate the preceding discussion we will register two rectangles R1 and R2. The rectangle R2 was derived from R1 by scaling it in the x-direction by a factor of .5, then rotating it by 60, and finally applying a translation of 20 pixels in x and y. Figure 4-a shows both rectangles before registration. The rectangle R1 was registered against R2 using the process described in the earlier section 4. Numerically the results from the registration were:

X translation
Y translation
Rotation
Scale 19.49
19.62
49.14
0.93

As we can see by the numerical values as well as by observing figure 4-b, the resulting registration is incorrect. It doesn't mean the method has a wrong conception but that it was incorrectly applied. As a counter example we show another case.

Test 5: Registration of images with isotropic scale. In this example we will verify the behavior of the method in a case of isometric scaling. Now, the rectangle R2 was created from R1 by scaling it by a factor of 0.5 in both x and y directions, by rotating it by 60, and translating it by 20 pixels in x and y directions. Figure 5-a shows both rectangles before registration and figure 5-b after registration.


Figure 5: a) Two rectangles, R1 on the left bottom, and R2 on the upper right corner, that must be registered. b) Result of registering R1 (dashed line) after registration with R2 (full line).

The numerical results are:

rotation
scale
tx
ty 60.0
0.50
20.0
20.0

As we can observe in both numerical and visual results, the registration was correctly solved by the method. This last example shows that when the method is applied in a case whose scale factor can be adjusted conveniently, it will work fine, otherwise it will fail.

6 Discussion

Some implementation aspects of the method must also be commented. Concerning the computational aspect Flusser [14] notes in his work that the cost for solving the linear system necessary to find the TPS parameters is high when the number of control points exceeds 50 points, making the approach useless. More recently, Barrodale has shown that advances in numerical techniques have minimized the importance of this problem [3]. Anyway, in many applications the needed number of control points may be much less than 50.

The adoption of this technique assumes that the application, or objects in the images, obey the criterion of minimal bending energy or at least shows an approximated behavior. On the other hand, the minimum energy criterion is present in several other physical processes and also in the modeling of the human visual process [29], [17]. From this view point, modeling the natural phenomena of deformation, as is the case in discussion, by using the criterion of minimum energy is very well supported. This suggests the adoption of the method as an approximation in cases which there exists an unknown deformation type.

In the presented method it is necessary to establish a set of correspondence points in both images and to assure that the points in the set show high reliability with regard to their location, otherwise the obtained registration will present an inaccuracy (although could be difficult to quantify this error). Furthermore it is necessary that the amount of control points in the image be enough to characterize adequately the deformations and variations of interest. The method does not permit an approach similar to that presented in [26] or [6] in which by using a measure of proximity new points of correspondence are determined at each new iteration, because transformation parameters are got analytically instead of iteratively. However it is possible to use the information of bending energy to validate the control points when computing the TPS parameters. This can be done as follows:

  1. An initial registration is obtained with a set of points that can not be assured to be reliable;

  2. After the initial registration a new configuration of points is generated, where measures of similarity (e.g. proximity) can be used to evaluate the reliability of the set;

  3. Based on the new configuration of the set of points a new registration is got.

This procedure can be repeated until no meaningful modification is produced in the results. Obviously, depending on the reliability of the initial configuration a greater or lesser number of iterations will be necessary. The process of registration can be evaluated by the amount of bending energy needed for each configuration of control points. If a reliable initial configuration of points can not be obtained, the above procedure can be changed to incorporate an initial step of registration based on other method (e.g. the Generalized Hough Transform) resulting in an initial gross registration. From this point, and based on the detection of significant points along the contour it can be established a set of points making use of similarity measures. Properties that can be used as similarity measures are curvature, proximity, edge slope, etc. These values can be used separately or combined to provide more reliability.

With regard to the method for choosing the control points some alternatives are possible. For example, when dealing with contours one can identify meaningful points as point of maximum curvature or inflection points. Points of maximum curvature and points of inflection are good candidates to the purpose because they are naturally tied to the characterization of shape contour and consequently of the object to which it pertain. However, if a contour has few points with these characteristics then they can be in insufficient number for determining the transformation with the desired accuracy. The solution is to find all the points which present some reliable characteristic and to interpolate a certain number of points between those points along the contour, following a given criterion. An approach similar to that presented by Davatzikos [9] can be used to get the control points, depending on the type of the image.

The interpolant characteristics of TPS also suggests its application to the image interpolation, with the advantage over other methods that make use of polynomials. Similar proposal is made by Ruprecht [28] with the use of other radial basis functions. The application of the same technique for registration and interpolation of the same images can eventually bring additional computational gain in the process.

Some extensions to this work are right at hand. The concept of modeling deformation by energy minimization superposed to a rigid transformation underlying the method would be very desirable to the 3D registration case. Concerning the practical implementation of the method the determination of a reliable technique for determination of the position of control points could be very interesting and valuable. This could be much more attractive if accomplished using the energy minimization information. A possible approach would be to impose further constraints to the energy minimization model in order to assure a correct location of the control points. Another interesting path of research to be pursued is the investigation of the use of TPS as an approximation function instead of an interpolating one [31]. This could lead to a method that would minimize the misplacement of control points.

7 Conclusions

We have presented in this paper a technique for image registration based on the use of TPS as a mapping function. The presented method shows simple modifications, from the point of view of the implementation, when compared with the original formulation of TPS. However, the added constraints represent a significant change in the interpretation of the conceptual model mainly in the context of image registration. In our approach the registration is done modeling a rigid body transformation with the superposition of effects of deformation and scaling. On the other hand, in the direct application of the TPS the registration is modeled by a linear combination of a general affine transformation and a deformation component. The resulting mapping function obeys, in both cases, the criterion of minimum energy of bending (deformation). The presented model can be applied to register images of serial sections more consistently with the nature of the problem than other methods presented in the literature, in which the effects of deformation are treated as a problem of minimization of the error that is supposed to be implicit in the data.

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Publication Dates

  • Publication in this collection
    13 Oct 1998
  • Date of issue
    Apr 1997
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