Optimization in Differentiable Manifolds in Order to Determine the Method of Construction of Prehistoric Wall-Paintings

10/09/2012 ∙ by Dimitris Arabadjis, et al. ∙ National Technical University of Athens 0

In this paper a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall-paintings. The approach includes a) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, b) choice of prototype curves families, c) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, d) clustering of the contour parts and the prototypes, to determine a minimal number of potential guides, e) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The introduced methodology simultaneously deals with two problems: a) the arbitrariness in data-points orientation and b) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4-manifolds. Application of this methodology to celebrated wall-paintings excavated at Tyrins, Greece and the Greek island of Thera, manifests it is highly probable that these wall-paintings had been drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings' lines with an exceptionally low average error, less than 0.39mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities, more than 1000 years before their axiomatic formulation in Classical Ages.

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1 Introduction

In this paper a methodology of general applicability is presented for answering the question if an artist used a number of archetypes to draw a painting or if he drew it by free hand. In fact, the contour line parts of the drawn objects that potentially correspond to archetypes are initially spotted. Subsequently, the exact form of these archetypes and their appearance throughout the painting is determined. The method has been applied to celebrated wall paintings, ”Lady of Mycenae”, of the 13th century B.C. excavated at Mycenae, Greece and ”Naked Boys” drawn c. 1650 B.C. excavated at Akrotiri, Thera, Greece.

1.1 Previous Works on Fitting Data to Curve Prototypes

In the bibliography there are various approaches to the problem of optimally fitting curves to 2D-data, either using the explicit or the implicit form of the model curves. Namely, in [1]

, the optimal fit of implicit plane curves to data points is treated via a square distance minimization procedure. This minimization is performed over translation, rotation and implicit function parameters, in order to deal with rigid body motion of the model curve and with its shape variance. Previously, in

[2]

, authors had employed orthogonal distances and the related tangential quantities between data and model curve points, in order to optimally fit conic sections to the given data points, via a non-linear regression algorithm. A lower bound for the least squares non-linear regression, between an implicit model curve and the data points of a contour, is given in

[3], thus offering a measure for the optimality of the orthogonal distance curve fitting methods. In [4], authors analytically determine the least-squares optimal translation and rotation of an explicit curve model, in order to fit a given set of data points. Then the primary parameters of the model curve are obtained via a 2D iterative region descending process. The alignment of an implicit curve model to given contour points is also dealt in [5], in a manner that the fitting process is invariant under affine transformations. Namely, the authors use affine invariant Fourier descriptors for an area-parametrization of the model curve and, by matrix annihilation, they determine the harmonic implicit form for the model curve descriptive equation. In [6] authors exploit point-wise distribution of shape templates in order to define shape descriptors and develop the corresponding distance measure. This distance measure offers exact point-wise correspondence between two different shapes, allowing for optimal matching transformation of one shape to the other. In [7] authors deal with the problem of fitting explicitly described B-Spline model curves to data-points lying on a 2D-surface. The B-Spline is restricted so as to lie on the same surface with the data-points by treating the surface as a differentiable 2-manifold. Then, the fitting procedure is performed so as the on manifold geodesic distance between data and model curve points is minimized. Analogous technical elements for modifying planar objects so as to lie on a manifold are developed in [8], where the authors re-evaluate the ”Beltrami short time kernel” in the case that the image to be filtered lies on a manifold. In [9] authors deal with the problem of fitting algebraic curves to data points, via their implicit representation, by constraining their functional form so as to have closed zero level sets, thus making the curve determination more robust. The implicit functional form of the model curves (algebraic) is adopted in [10] to define itself a matching error via the polynomial function level sets. In order to deal with arbitrary orientation, authors adopted invariant computation techniques of polynomial coefficients. Finally, in [11] the representation of an object as a unification of sub-objects, that can be analytically described by implicit curve or surface forms, is used, in order to segment a complex visual form into ”meaningful” sub-shapes.

1.2 The Present Problem and the Proposed Approach

In the present work, the demand was the determination of an unambiguous relative placement between a prototype implicit curve and the contour data, in the sense that attribution of various contours data to a prototype curve is orientation invariant. Namely, given an implicit form of a prototype curve, the integral of the least distances between a curve part and each contour data points is computed immediately, by means of the flat curvature of the implicit functional form of the prototype curve, with no need of intermediate optimal placement of the data points along the prototype curve points. Additionally, for the present application, one must spot a unique prototype curve that gave rise to a considerable number of different contour realizations, where each such realization corresponds to a different part of the prototype curve. So, the determination of the model curve that optimally fits a set of partial realizations of its planar graph is a problem of increased degrees of freedom, compared with the widely treated problems of curve fitting. Namely, the problem of optimally fitting the model curve to the data runs over the space of the free parameters of the curve model and, simultaneously, over all possible line segments of it and all possible data points subsets that could have been generated by the same prototype. There are various approaches that deal with the determination of the parameters that optimally fit a model curve to a given set of points in a statistically efficient manner. In

[12], there is a compilation of the existing implicit curve fitting techniques, together with statistical analysis of their efficiency. Concerning the works referred and outlined in Sect. 1.1, they are mainly split into 2 categories: a)those who treat the problem of fitting curve models to data points and b) those who treat the problem of shape matching. Among these publication there are approaches that treat the problem of curve fitting (e.g.[1], [5], [4], [10]), or shape matching (e.g. [6]) with an orientation invariant approach, but none of these techniques dealt with the problem of simultaneous fitting the same implicit curve model to different sets of data points. Moreover, in most curve-fitting approaches, determination of the optimal values for the free parameters of the model curve is based on iterative error minimization algorithms like Gauss-Newton, Gradient Descent, etc. that do not necessarily converge to a global error minimum. Concerning shape matching methods, although they offer unsupervised shape similarity measures, minimization of these measures deforms the model shape without constraining any of its geometrical characteristics. In order to deal with the demands of unambiguously optimal fit between the determined model curve and all contour data attributed to it, we had to develop a unified optimization method of the least possible dimensionality that exhaustively acts on the set of all possible prototypes of the same parametrized curves family. The fitting process introduced here treats each class of prototype curves as a 4-manifold and performs a curvature driven exhaustive error minimization between the class of prototype curves and the available data points. By applying this methodology two times, first with respect to the drawn contours and next with respect to the prototypes (Sects. 4.3, 4.6 respectively), we determine a unique prototype curve that optimally fits all data points attributed to it.

2 Fundamental Notions and Definitions Concerning the Method of Wall-Paintings Drawing

A first crucial hypothesis for the method the artist(s) might employed for drawing the wall-paintings c. 3500 years ago
”The fresco technique asked for highly fast and precise execution of the drawing process. At the same time, the stability of the contour line of the figures depicted on the painting had always been a primary goal of the artists throughout human history. Thus, a plausible assumption about the method of drawing of these prehistoric wall paintings is that the artists employed guides or instruments to support the drawing object.” The quality of drawing of the paintings considered in this paper, suggests an additional content to this assumption: ”the artists took special care and paid attention to ensure continuity of the drawn contour line and, wherever possible, of its tangent, at the points where a change of guide occurred.”

Definition 1 (The object part)

If the aforementioned assumption is correct, then there will be subsets of the contour line of the various figures appearing in the wall-painting, which they had been drawn by a continuous stroke of the paintbrush. These subsets are called object parts.

Definition 2 (The object)

One can define the object to be a subset of the wall painting border, which represents a thematic unit, is smooth and its beginning and end points are discontinuities of the border line or of its tangent. Evidently an object is a concatenation of object parts, since it has been drawn by a usually small number of continuous strokes of the paintbrush.

2.1 A rigorous determination of the object part

Consider ideally that, in the wall painting, all objects’ contour lines are continuous, described by the piecewise twice differentiable mono-parametric vector equation

. Then, an object part is spotted by determining a contiguous subset of the object curve whose beginning and end points are one of the following: 1)the beginning or the end of the object, 2)a point where the curvature is discontinuous. Analogous pictorial feature definitions can be found in [13], where authors use curvature extremes and sign changes in order to define strokes, corners, endpoints and junctions.

The notion of an object part is crucial, since if the artist(s) indeed used stencils, then, each time he placed the stencil on the wall and drew a line, we consider he created an object part.

If is the length of an arbitrary object then we select an appropriate small percentage of , say . Subsequently, in order to spot points where there is change of stencil, we proceed as follows: Let an arbitrary point on object contour, say the i-th, where , then we determine the 3rd degree polynomials of object length that best fit object points and the polynomial of 5th degree of object length that best fits the points and satisfies the tangent continuity on , namely .

In order to decide if the i-th pixel is an ending point of an object part we check for discontinuity of the object contour curvature on it. Hence, the two approximations of the contour curvature from the left and from the right should manifest an abrupt difference. Equivalently, we demand , where a very small angular threshold. If this demand is met then the i-th point is the end of the current potential object part. The beginning of the next potential object part is the first point after i-th, say the j-th, where the demand is met. By application of this method to all available objects of the wall-painting figures, we generate an ensemble of potential object parts with contour point vectors , . We once more stress that we consider that each such object part was drawn by one paintbrush stroke. We also assume that if we achieve in corresponding these object parts into a small number of prototypes, then we will support the hypothesis that these ”one-stroke contour lines were made by means of a stencil or another equivalent instrument that guided the brush.” If we further adopt the plausible assumption that each such stencil had a specific geometric form, namely in modern mathematics a specific functional type, then we may deduce that the derivatives of all orders of remain continuous along an object part. On the contrary, at the points of change of stencil, or of replacement of the stencil, it is highly probable that the curvature will essentially change, even in the case where the artist(s) achieve continuity of and to ensure a smooth and good aesthetic result.

3 A Brief Description of the Employed Methodology for Determining Possible Geometric Guides

Symbol Meaning
implicit description of a model curve
Cartesian coordinates
primary parameters of the model curve
the flat curvature function of the model curve
the model curve, parametrized via curve length , which is resulted by a fixed pair of primary parameter values
the primary parameters isocontour, parametrized via curve length , as it is resulted by
a fixed pair of Cartesian planar coordinates
the curve length while moving on -plane in a direction normal to
the curve length while moving on -plane in a direction normal to
the internal product between the directions normal to and
The local frame defined by the level-sets of : is the curve length differential along and
the remaining curve length differentials evolve along the curves selected to define the tangent space of :
evolves along direction, evolves in a direction normal to covariance of and
and normal to covariance of and
the unit directional vectors normal and tangent to respectively
the unit directional vector spaces normal and tangent to the level-sets of respectively
the Euclidean norm of evolution on the level-sets of (i.e. of the level-sets’ curvature)
TABLE I: Basic Symbols

After application of the aforementioned methodology, we have determined an ensemble of object parts, i.e. an ensemble of parts of the painted figures’ borders with small curvature fluctuations along each such part. Now, the question arises if there is a small number of prototype curves that optimally fit these object parts and if yes, how to determine the exact functional form of these prototypes. In order to deal with this problem we have applied a novel methodology outlined in the following steps.
Step 1 - Choice of a reasonable criterion that quantifies how well an object part fits a prototype curve part
Suppose the prototype curve in that is described by the implicit functional form and let be the unit vectors normal to it. Moreover, let an arbitrary point on an object part and its vectorial distance from

. Suppose, for a moment, that the object part is a continuous curve. Then, evidently, a measure for the level of fitting the considered object part to

is the integral of along a proper part of .
Step 2 - Alternative version of the aforementioned criterion
Because is calculated along , by applying Stokes theorem on domain , bounded by the object part and the curve part of , we obtain.

(1)

But, actually, , where is the flat curvature of the implicit functional form of the curve model . Consequently, minimization of the integral of on a proper part of can be obtained by minimization of the corresponding area integral of . In Sect. 4.1 and in appendix A, we show that minimization of the integral of the flat curvature over can be obtained by minimizing the integral of on the boundary of ,
Step 3 - Primary Parameters definition
In general, the functional form of depends on a significant number of parameters. For example, conic sections implicit functional form depends on 5 parameters in general. In Sect. 4.2, it is shown that, independently of the number of parameters and the exact functional form of , two parameters are sufficient to describe all possible deformations of the flat curvature of on the -plane. We will call these parameters as the ”primary parameters” of .
Step 4 - Extending the fitting process in
Assume a given object part, OP and a parametrized class of prototypes , implicitly described by the equation . We look for the determination of the proper primary parameters values and the proper relative placement between OP and , so as the integral of the point by point distances (see Step 1) between these curves are minimized. Hence, we extend the object part - prototype fitting analysis from to , where two of the coordinates of refer to the coordinates of the curves in , while the other two coordinates refer to the primary parameters of . At the beginning, this specific subset of is assumed to be the Cartesian product of the -space with the -space. But, in the following, given that we need to calculate distances between OP and all possible deformed versions of S, we treat the space of the level sets of as a differentiable manifold. Via the curvature of the tangent space of this manifold, we look for the curve which optimally fits OP along the normal vectors as it is described in Sects, 4.3, 4.4. The aforementioned approach and the resulting criterion form an efficient procedure that simultaneously offers the primary parameters of the prototype curve and its optimal relative position with OP.
Step 5 - Determining the different types of potential stencil-guides
After application of the fitting process outlined in Step 4, each object part corresponds to a prototype curve part of a specific implicit functional form with primary parameters . Then, we apply standard clustering techniques in order to group the estimated primary parameters of the same class of prototype curves into clusters of neighboring parameters values. We assume that each such cluster corresponds to a specific stencil-guide. In other words, we assume that all object parts that correspond to the same cluster of primary parameters have been generated by one stencil - guide, the optimal cluster representative.
Step 6 - Determination of the optimal cluster representative, i.e. of the potential guides parameters
Consider a cluster and the set of object parts that correspond to it, . Then, we redefine the sub-domain between the level sets of the prototype curves and the image of on the -manifold, as it is determined via the estimated . Application of the resulting criterion, (see Sect. 4.6), offers a unique prototype curve with specific primary parameters, to which all are optimally fitted.

We stress that the aforementioned procedure accounts for all possible prototype curve versions, described by the functional form and acts on the manifold they define independently of rotation and translation transforms. So, the fitting process is independent of the initial orientation of the object parts on the -plane. Moreover, by computing the fitting error as an integral on the manifold of a class of prototype curves and by performing the error minimization exhaustively, the developed procedure offers very consistent fitting results, even in the case that the considered drawings suffer serious wear.

Fig. 1: Block Diagram outline of the developed methodology

4 Differentiable Manifold Analysis for Fitting the Outlines of a Painting to Prototype Curves

4.1 Determination of the most probable stencil part that created an object part

Let an object part consist of pixels with point vectors , . Then the contour length that corresponds to each point vector is approximated via . So, we can express the object part contour with curve parameter the length , namely .

If the painter had used a part of a specific prototype curve to draw this object part, we expect that the drawn contour and the corresponding part of the prototype curve should have close curvature values along them. So let a prototype curve (stencil) given by the equation . Curvature at length of this stencil , is also obtained via the relation
.

In order to compute the minimum distance of an object part from a properly prototype curve we adopt the flat version of the curvature and for any point of the object part we compute the value . Assume, for a moment, that we properly place the object part around the adopted prototype curve so as to minimize the distances between them. For the computation of the integral of these minimum distances along the prototype curve we define the domain between the prototype curve and the object part. Then, using Stokes’ Theorem [14] for the integral of on , we obtain

(2)
(3)

We also use Stokes’ Theorem to obtain a boundary form for the integral of on , as described below:

(4)

Since is the unit normal vector of isocontours, the area differential of can be written , where and . So,

(5)

Then, by applying Stokes’ Theorem in (4) we obtain

(6)

In order to obtain the most possible part of stencil used for drawing the considered object part, we should determine the proper that minimizes . But as it is shown in Appendix A minimization of coincides with minimization of . Thus, the part of stencil that had, most probably, been used for drawing the considered object part consist of points , , where

(7)

4.2 Definition of the primary parameters for a family of prototype curves

The analysis of previous Sect. 4.1 has offered a functional representation of the optimal fitting between an object part and a given prototype, , described by the locus . But, the prototype curve, , can be described as a realization out of all the curves that belong to the parametrized family ; where is the vector of the free parameters for this class of curves. The parameters that form vector describe spatial transformations (rotation and translation) and shape (curvature) variances between the curves of the corresponding family. Since, analysis of Sect. 4.1 makes use of the flat curvature functional over the descriptive equation of the prototype curve, we should determine the number of parameters which are sufficient to describe all curvature variances between the curves of the same family.

Namely, we should exploit the derivative , where , , and is the flat curvature that corresponds to the class of curves with descriptive equation . For the subsequent analysis we should recall the definition of the unit directional vectors and employed in Sect. 4.1, where , . So, the flat curvature is related with and via the expressions

(8)

and the partial derivatives of the directional vectors with respect to parameter read

(9)

Subsequently, using expression (8) for the curvature and (9) for and , the -derivative of reads , since (8) and (9) imply that
. Using equation (9) for and next (8) for we obtain

(10)

Next, by expanding the derivative, , of the function , one obtains and subsequently (10) results

(11)

Using equation (11) we obtain the differential of in the parameters space , since

commutes with addition. But the maximal number of linearly independent vectors

is 2. Equivalently the minimal number of parameters sufficient to describe all curvature variances among the curves of the same family is 2.

In order extract the primary parameters of a curves family parametrized via a parameters vector of arbitrary size, we exploit the influence of affine mapping to the flat curvature function. Namely, if , where ,

(12)

Next, since , are linearly independent, we suppose that any parameters (i.e. non-spatial) variation of can be obtained by variations of , . So, and since

(13)

where and .

Hence, given a parametrized implicit curve , its primary parameters can be recovered by the mapping and then any differential variation of the primary parameters evolves free parameters vector as equation (13) describes.

4.3 Analysis for optimally fitting an object part to a class of prototype curves

In Sects. 4.1 and 4.2 it has been shown that the action of the flat curvature functional over the implicit form of a class of curves is sufficient to optimally align an object part along the curve defined by a fixed parametrization . But, still, the pair of parameters that offer the best alignment of the object part has to be determined. In order to deal with the variances of parameters we let function act on and on the configuration . The vectors normal to the tangent space of the manifold defined by constant are given by . Thus, the curvature of the manifold tangent space is given by the differential

(14)

If we project this differential on we obtain implying that lies on the tangent space of constant. Moreover, the tangent space of can be obtained by extending the tangent spaces of the projections to or planes. Namely
, where , , , , , , . Hence, the curvature of the tangent space is expressed via the differential vector with flat representation

(15)

where , , .

But the differential vectors and are related with and via their covariance with respect to function

(16)

where and are the variances , that are normal to the isocontours of .

Covariance of and can be obtained by the demand that the total differential satisfies , which gives

(17)

By substituting (17) into (16) we obtain

(18)

Equivalently, and are related with and by an analogous concept of covariance between and with respect to function , thus obtaining

(19)

Having determined all quantities related with , we can return to (15) and reformulate it as a differential form using basis , which defines the volume form , the line form and its Hodge-star dual area form . Next, we define the flat curvature of the tangent space as

(20)

where , , . Then , where denotes the exterior derivative. Consequently, the integral of the flat curvature over a domain , by exploitation of the Stokes theorem, reads

(21)

Namely, the integral of the flat curvature over a sub-domain in is equal to the integral of the distances between points of its boundary along the normals of . This property will be used later in order to determine the fitting error between a family of prototype curves and the image of an object part in the curves’ family manifold.

In order to avoid integration of the flat curvature over the whole domain , we also adopt the integral of , where evidently . Consequently application of the Stokes theorem to the integral of over a domain gives

(22)

Consider now the domain so as its boundary is determined by the following three elements:

  1. the interpolated data points’ contour

    as it was obtained in Sect. 4.1,

  2. a segment of a prototype curve,

  3. a strip along the curve of parameters , , , where .

Then, at a point of the prototype curves sub-domain, let the minimal distance of the aligned data points sub-domain be . The integral of these distances over the prototype curves family sub-domain can be obtained using (21) and by means of the following formula

(23)

The problem of finding the optimal fitting of an object part data points to a proper parametrized implicit prototype curve is now expressed via the demand of finding the that minimizes . But, in Appendix A it is shown that minimization of demands minimization of the integral of equation (22), which we will call . These integrals consider domain with respect to the manifold defined by the level-sets of . So, under consideration of the basis together with the relations (18) and (19) the volume form of reads

(24)

where ,