A Morse theoretic approach to the geometrical feature terms specified in ISO 25178-2 and ISO 16610-85

Gert W Wolf

doi:10.1088/2051-672X/abfdff

1. Introduction

While in ISO 25178-2 [1, 2] terms, definitions and parameters for the determination of surface texture by areal methods are described, ISO 16610-85 [3] is devoted to the terminology and concepts for areal morphological segmentation. As has been pointed out by Blateyron [4], in the actual version of ISO 25178-2 [1] several ambiguities do exist, whose elimination is the goal of the revision under way¹ . According to him, the intended modifications concern (a) the concepts closed motif and open motif, (b) specifications related to heights and feature attributes, (c) the definition of the peak curvature and (d) the introduction of new shape parameters.

In the present paper the attempt is made to go one step further and to put the whole set of concepts related to geometrical feature terms, which can be found in ISO 25178-2 [2] and ISO 16610-85 [3], into a Morse theoretic framework, with the motivation for this endeavour resulting from three observations. The first observation can be characterised as imbalance within the standards and refers to the fact that on the one hand very accurate definitions, which include even mathematical formulas, are given, while on the other hand, most definitions related to geometrical feature terms lack a comparable rigour. The second observation can be characterised as conflict with mathematical conventions and refers to the fact that the respective definitions are based on the image of the height function and not on its domain, or in terms of the standards, they are based on the scale-limited surface instead of the reference surface. The third observation, finally, can be characterised as partial lack of explanation and refers to the fact that different concepts are used side by side without any explanation of the formal relationships between them. Apart from the elimination of the aforementioned inconsistencies and ambiguities, the application of Morse theory would have the additional advantage that a powerful mathematical tool could be exploited in order to gain new insights into surface structures.

The remainder of this paper is organised in such a way that in a first step a brief survey of the required mathematical concepts is given (section 2), while in a second step proposals for redefinitions of the geometrical feature terms are discussed (section 3).

2. Mathematical background

To start with, it should be emphasised that surfaces do not exist per se, but they are nothing else than models, and thus they may be characterised by the following three features that go back to Stachowiak [5]: (a) the mapping feature saying that a model is always a model of something, i.e. of a natural or an artificial original, (b) the reduction feature saying that a model does not capture all properties of the original, but only those that are important from the creator's point of view and (c) the pragmatic feature saying that a model is not clearly assigned to its original, but it is designed with respect to specific subjects, specific time intervals and specific mental or actual operations.

Researchers engaged in computational geometry and computer graphics were among the first to realise that natural language definitions of surface features are ambiguous. In order to overcome this deficiency and to provide models being eligible for an adequate characterisation of surfaces and surface features, they adopted concepts from multivariable calculus and topology. As, from a mathematical point of view, the surfaces occurring in computational geometry and computer graphics are basically the same as the ones occurring in surface metrology, GIScience and many other research areas, the same mathematical concepts may be applied for their formal characterisation.

With respect to the purpose of this paper, in this section the focus will be laid (a) on functions f(x, y) of two variables, although all definitions and theorems are generalisable to functions of n variables, and (b) on functions that are defined on the (x, y)-plane or a subset of it, although the theory presented can be generalised to functions defined on smooth manifolds, independent of their dimension; examples of two-dimensional smooth manifolds are, apart from the (x, y)-plane, for instance, the sphere, the torus, the double torus or the Earth's surface. To avoid misunderstandings, it should be noted that in the theory related to manifolds, the term surface is used to denote a two-dimensional manifold, which in general represents the domain on which a function f(x, y) is defined.

Next, attention will be paid to those type of functions that prove to be indispensable with respect to a Morse theoretic characterisation of feature parameters; however, only the most relevant concepts from multivariable calculus can be given in this section (a comprehensive introduction to this topic can be found, for example, in Ghorpade and Limaye [6] or Moskowitz and Paliogiannis [7]).

Definition 2.1. A function $f(x,y)$ is termed $k$ -fold continuously differentiable, or of class ${C}^{k}$ , if all partial derivatives up to order $k$ exist and are continuous.

A smooth function is a function of class ${C}^{\infty }$ .

Fundamental for the formal characterisation of functions are their local maxima, local minima and saddle points (saddles), which are defined as follows.

Definition 2.2. A point $({x}_{0},{y}_{0})$ is a local maximum of $f(x,y)$ , if $f(x,y)\leqslant f({x}_{0},{y}_{0})$ for all ( $x,y$ ) in some neighbourhood of $({x}_{0},{y}_{0})$ .

A point $({x}_{0},{y}_{0})$ is a local minimum of $f(x,y)$ , if $f(x,y)\geqslant f({x}_{0},{y}_{0})$ for all ( $x,y$ ) in some neighbourhood of $({x}_{0},{y}_{0})$ .

A point $({x}_{0},{y}_{0})$ is a saddle point (saddle) of $f(x,y)$ , if there exist regular paths ${p}_{1}$ and ${p}_{2}$ that intersect transversally at $({x}_{0},{y}_{0})$ (i.e. the tangent vectors at $({x}_{0},{y}_{0})$ are defined and not multiples of each other) in such a way that $f(x,y)$ has a local maximum at $({x}_{0},{y}_{0})$ along ${p}_{1}$ , whereas $f(x,y)$ has a local minimum at $({x}_{0},{y}_{0})$ along ${p}_{2}$ .

It should be emphasised that, according to definition 2.2, the local maxima, local minima and saddle points are elements of the domain of the function and thus located in the (x, y)-plane. Different from them are their images under f(x, y), which correspond to points lying on the graph of the function. Nevertheless, many authors denote as local maximum, local minimum or saddle point both the argument (x₀, y₀) and its image f(x₀, y₀)² . In everyday language, the images of local maxima are also referred to as peaks, summits or tops, the images of local minima as pits, depressions or hollows and the images of saddle points as passes or cols.

Figure 1 visualises the graph of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2], which has four local maxima at (−1, −1), (−1, 1), (1, 1) and (1, −1), one local minimum at (0, 0) and four saddles at (−1, 0), (0, 1), (1, 0) and (0, −1)³ .

As the local maxima, local minima and saddle points of a function can also be characterised by their first-order and second-order partial derivatives, it seems appropriate to proceed with the formal definition of the Hessian matrix and Hessian determinant.

Definition 2.3. Let $f(x,y)$ be a function with partial derivatives ${f}_{{xx}},{f}_{{xy}},{f}_{{yx}}$ and ${f}_{{yy}}$ . The matrix ${H}_{f}=\left(\begin{array}{cc}{f}_{{xx}} & {f}_{{xy}}\\ {f}_{{yx}} & {f}_{{yy}}\end{array}\right)$ is termed the Hessian matrix of $f$ . The Hessian matrix evaluated at $({x}_{0},{y}_{0})$ is denoted by ${H}_{f}({x}_{0},{y}_{0})$ and defined by $\left(\begin{array}{cc}{f}_{{xx}}({x}_{0},{y}_{0}) & {f}_{{xy}}({x}_{0},{y}_{0})\\ {f}_{{yx}}({x}_{0},{y}_{0}) & {f}_{{yy}}({x}_{0},{y}_{0})\end{array}\right)$ .

The determinant $\det ({H}_{f})$ of the Hessian matrix ${H}_{f}$ is called the Hessian determinant; when evaluated at $({x}_{0},{y}_{0})$ , it is denoted by $\det ({H}_{f}({x}_{0},{y}_{0}))$ .

When applying the previously defined concepts, the local maxima, local minima and saddle points can be characterised by the following two theorems, whereby the first one is well-known, while the second one is crucial with respect to Morse theory.

Theorem 2.1. Let $f(x,y)$ be twice continuously differentiable.

If ${f}_{x}({x}_{0},{y}_{0})={f}_{y}({x}_{0},{y}_{0})=0$ , $\det ({H}_{f}({x}_{0},{y}_{0}))\gt 0$ and ${f}_{{xx}}({x}_{0},{y}_{0})\lt 0$ , then $f(x,y)$ has a local maximum at $({x}_{0},{y}_{0})$ .

If ${f}_{x}({x}_{0},{y}_{0})={f}_{y}({x}_{0},{y}_{0})=0$ , $\det ({H}_{f}({x}_{0},{y}_{0}))\gt 0$ and ${f}_{{xx}}({x}_{0},{y}_{0})\gt 0$ , then $f(x,y)$ has a local minimum at $({x}_{0},{y}_{0})$ .

If ${f}_{x}({x}_{0},{y}_{0})\,={f}_{y}({x}_{0},{y}_{0})=0$ and $\det ({H}_{f}({x}_{0},{y}_{0}))\lt \,0$ , then $f(x,y)$ has a saddle point at $({x}_{0},{y}_{0})$ .

Theorem 2.2. Let $f(x,y)$ be twice continuously differentiable, ${f}_{x}({x}_{0},{y}_{0})={f}_{y}({x}_{0},{y}_{0})=0$ and $\det ({H}_{f}({x}_{0},{y}_{0}))\ne 0$ .

If the number of negative eigenvalues of ${H}_{f}({x}_{0},{y}_{0})$ is two, then $f(x,y)$ has a local maximum at $({x}_{0},{y}_{0})$ .

If the number of negative eigenvalues of ${H}_{f}({x}_{0},{y}_{0})$ is one, then $f(x,y)$ has a saddle point at $({x}_{0},{y}_{0})$ .

If the number of negative eigenvalues of ${H}_{f}({x}_{0},{y}_{0})$ is zero, then $f(x,y)$ has a local minimum at $({x}_{0},{y}_{0})$ .

Local maxima, local minima and saddle points are subsumed under the generic term critical points. Strictly speaking, they are so-called nondegenerate critical points, in contrast to the degenerate critical points, which comprise, for example, dog saddles, monkey saddles or crossed pig troughs [8]. The formal criterion for distinguishing whether a critical point is a nondegenerate or degenerate one is the value of the Hessian determinant obtained at the respective point.

Definition 2.4. A point $({x}_{0},{y}_{0})$ is called a critical point of $f(x,y)$ if ${f}_{x}({x}_{0},{y}_{0})={f}_{y}({x}_{0},{y}_{0})=0;$ the real number $f({x}_{0},{y}_{0})$ is termed critical value of $f(x,y)$ .

A critical point $({x}_{0},{y}_{0})$ is said to be a nondegenerate one if $\det ({H}_{f}({x}_{0},{y}_{0}))\ne 0$ .

A critical point $({x}_{0},{y}_{0})$ is said to be a degenerate one if $\det ({H}_{f}({x}_{0},{y}_{0}))=0$ .

The number of negative eigenvalues of the Hessian matrix H_f, as mentioned in theorem 2.2, is an essential factor within Morse theory and forms the basis of the following definition.

Definition 2.5. Let $f(x,y)$ be twice continuously differentiable and let $({x}_{0},{y}_{0})$ be a nondegenerate critical point of $f(x,y)$ . The index of $({x}_{0},{y}_{0})$ is the number of negative eigenvalues of the Hessian matrix ${H}_{f}({x}_{0},{y}_{0})$ .

Due to definition 2.5, a local maximum is a critical point of index two, a saddle is a critical point of index one and a local minimum is a critical point of index zero.

The combination of the two features smooth function and nondegenerate critical point leads to a new type of function, which forms the basis for a more formal characterisation of the geometrical feature terms specified in the two standards ISO 25178-2 [2] and ISO 16610-85 [3].

Definition 2.6. A Morse function is a smooth function all of whose critical points are nondegenerate.

Morse functions have a number of noteworthy properties with the most relevant ones being quoted next (for a comprehensive introduction to Morse theory confer, for example, Knudson [9], Matsumoto [10] or Zomorodian [11]).

Morse lemma.

Theorem 2.3 Let $({x}_{0},{y}_{0})$ be a nondegenerate critical point of a Morse function $f(x,y)$ . Then it is possible to choose local coordinates $(x,y)$ in such a way that $f(x,y)$ takes one of the following three standard forms:

(i) $\,f(x,y)=f({x}_{0},{y}_{0})-{x}^{2}-{y}^{2}$ ,

(ii) $\,f(x,y)=f({x}_{0},{y}_{0})+{x}^{2}-{y}^{2}$ or

(iii) $f(x,y)=f({x}_{0},{y}_{0})+{x}^{2}+{y}^{2}$ .

The function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2], whose graph is depicted in figure 1, is obviously a Morse function as it is smooth and its critical points are nondegenerate. Theorem 2.3 says, in clear and simple terms, that in the neighbourhood of a local maximum a Morse function always looks like an inverted paraboloid (case (i)), in the neighbourhood of a saddle point always like a hyperbolic paraboloid (case (ii)) and in the neighbourhood of a local minimum always like an upright paraboloid (case (iii)); the three cases are visualised in figure 2. In addition, the Morse lemma ensures that the three types of critical points shown in figure 2 are the only ones that a Morse function f(x, y) can have.

**Figure 2.** The three types of nondegenerate critical points of a Morse function f(x, y): (a) local maximum, (b) saddle point and (c) local minimum.
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The following theorems are direct consequences of the Morse lemma.

Theorem 2.4. The critical points of a Morse function are always isolated, i.e. for each critical point $({x}_{0},{y}_{0})$ it is always possible to find a neighbourhood of $({x}_{0},{y}_{0})$ in which no other critical points are situated.

Theorem 2.5. Morse functions that are defined on a smooth compact manifold have only a finite number of critical points.

Theorem 2.6. Morse functions are everywhere dense in the space of all smooth functions on a smooth manifold.

To put it differently, theorem 2.6 states that any smooth function can be converted into a Morse function by an arbitrary small perturbation splitting the degenerate critical points into a certain number of nondegenerate ones [12]. It can be assumed that this property accounts for the popularity of Morse functions with respect to practical applications.

The last theorem of interest, finally, referring to the critical points are the Morse inequalities, which express fundamental relationships between the number of the critical points of a Morse function. With regard to the purpose of this paper, however, only a special case of the Morse inequalities, which is, moreover, an equality will be given below.

Morse inequalities.

Theorem 2.7 Let $f(x,y)$ be a Morse function that is defined on a sphere. Then the number of local maxima minus the number of saddle points plus the number of local minima of $f(x,y)$ equals two.

Of particular interest with respect to a multitude of applications is the fact that, under an additional assumption, theorem 2.7 remains valid for Morse functions being defined on a simply connected domain that is bounded by a closed contour line on which there are no critical points [13], with this finding going back to Maxwell [14]. A domain as previously described can be imagined as an island in the sea, while the Morse function can be thought of as a mapping associating with each point of the island its respective altitude. Evidently, the domain of the Morse function shown in figure 1 meets the requirements as it is simply connected, i.e. it has no holes, and it is bounded by a closed contour line, viz., the one of height zero, with no critical point lying on it.

The additional assumption concerns, in terms of island in the sea, the points lying outside the island⁴ . In order to ensure that the domain is compact in a mathematical sense, all these points have to be identified as a local minimum of altitude −∞; the identification process itself is termed one-point compactification [11, 15], the pit obtained either surrounding pit [8, 16–18] or virtual pit [19–22], depending on the respective author. When looking at the graph of the function visualised in figure 1, one recognises that there are four local maxima at (−1, −1), (−1, 1), (1, 1) and (1, −1), one local minimum at (0, 0) and four saddle points at (−1, 0), (0, 1), (1, 0) and (0, −1). Taking also the virtual pit into account, then the number of local maxima minus the number of saddle points plus the number of local minima equals 4 − 4 + 2 = 2, just as it should be according to theorem 2.7.

For the sake of completeness, it should be mentioned that an analogous concept, termed surrounding peak or virtual peak, having an altitude of +∞, also exists. In terms of the sea, this concept is required when analysing the area lying below sea level. An area like this is shown in figure 3 visualising the graph of the function g (x, y) = 2x⁴ − 4x² + 2y⁴ − 4y² − 4 for x, y ∈ [−2, 2], which has one local maximum at (0, 0), four local minima at (−1, −1), (−1, 1), (1, 1) and (1, −1), and four saddles at (−1, 0), (0, 1), (1, 0) and (0, −1).

**Figure 3.** Graph of the function g(x, y) = 2x⁴ − 4x² + 2y⁴ − 4y² − 4 for x, y ∈ [−2, 2], which has one local maximum at (0, 0), four local minima at (−1, −1), (−1, 1), (1, 1) and (1, −1), and four saddle points at (−1, 0), (0, 1), (1, 0) and (0, −1).
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After the detailed discussion of the critical points, the focus will be laid on the so-called integral lines, also referred to as integral curves ⁵ , which are defined as follows [11, 24–27].

Definition 2.7. Let $f(x,y)$ be a Morse function that is defined on a smooth compact manifold $M$ . An integral line (integral curve) of $f(x,y)$ is a maximal path p on M, whose tangent vectors agree with the gradient vector field $\left(\begin{array}{c}{f}_{x}\\ {f}_{y}\end{array}\right)$ .

${\mathrm{lim}}_{t\to -\infty }p(t)$ is termed the origin of $p$ , while ${\mathrm{lim}}_{t\to +\infty }p(t)$ is termed the destination of $p$ .

Figure 4(a) visualises the graph of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2] together with some paths of steepest ascent, figure 4(b) depicts the integral lines corresponding to the paths of steepest ascent shown in figure 4(a).

It should be noted that the terminology concerning integral lines is not unique. According to definition 2.7, integral lines are situated on the manifold on which the Morse function f is defined [11, 24–27]; a minority of authors, however, denote as integral line not (only) the path p itself, but (also) its image f(p) [25, 28–31] (for this group the lines shown in figure 4(a) are also integral lines).

The previously introduced concepts can be illustrated in simple words in the following way: If a hiker, who can hardly see anything due to heavy fog, still wants to reach the summit of a mountain, the best strategy for him to achieve this goal is to follow the line of steepest ascent, i.e. when climbing up the mountain he will follow one of the lines shown in figure 4(a), depending on where he is at the moment; when describing the chosen route to his friends at home, he will, however, take a topographic map and show them the projection of the path of steepest ascent on the (x, y)-plane, i.e. he will show them the respective integral line depicted in figure 4(b).

Figure 4 reveals several essential properties of integral lines, which are specified in theorem 2.8 in a formal way [11, 25–28, 31, 32].

Theorem 2.8. Let $f(x,y)$ be a Morse function that is defined on a smooth compact manifold M. Then the following assertions hold:

(i)
Integral lines are open at both ends.
(ii)
Both the origin and the destination of an integral line are critical points.
(iii)
Two integral lines are either disjoint or the same.
(iv)
Apart from the critical points, the entire manifold is covered by integral lines.
(v)
Integral lines are perpendicular to contour lines.

Figure 4 seems to contradict assertion (ii) of theorem 2.8 because in the graphic integral lines do not only originate from the critical points, but also from the boundary of the domain (in case of the function g(x, y) = 2x⁴ − 4x² + 2y⁴ − 4y² − 4 for x, y ∈ [−2, 2], whose graph is shown in figure 3, integral lines would not only end at critical points, but also at the boundary of the domain) [28, 33]. This contradiction, however, is only a seemingly one because, due to the one-point compactification, all points lying beyond the boundary are identified as a virtual pit, which represents a critical point, too. Thus, from a topological point of view, the integral lines, which seem to originate from the boundary of the domain, originate in fact from the virtual pit (the analogy holds for integral lines, which seem to end at the boundary of the domain, but end in fact at the virtual peak).

The validity of assertion (v) of theorem 2.8 becomes obvious when having a look at figure 5, which visualises the contour lines of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2], as they would appear on a topographic map, together with some integral lines; as can be seen, the contour lines and the integral lines form rights angles at their points of intersection.

Of special interest with respect to the topics discussed later in this paper are those integral lines, which originate from local minima and end in saddle points, and those, which originate from saddle points and end in local maxima. In the literature, integral lines featuring one of these two properties are often referred to as separatrices [28, 29, 33, 34].

Figure 6(b) visualises the separatrices of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2], the corresponding paths of steepest ascent are shown in figure 6(a). The paths of steepest ascent being associated with separatrices are known under different names, such as, for example, as course line, course or watercourse in case the separatrix is leading from a local minimum to a saddle point, and as ridge line, ridge or watershed in case the separatrix is leading from a saddle point to a local maximum.

After the discussion of the zero-dimensional elements (critical points) and one-dimensional elements (integral lines) of a manifold, the focus will be laid on two-dimensional structures. Based on the concept of integral lines, it is possible to define ascending manifolds, also termed unstable manifolds, and descending manifolds, also termed stable manifolds, in the following way [24, 25, 27, 31, 32].

Definition 2.8. Let $f$ be a Morse function that is defined on a smooth compact manifold $M$ .

An ascending manifold (unstable manifold) $A(p)$ of a critical point $p$ is the set of all points $x$ belonging to integral lines ${l}_{x}$ whose origin is $p$ , or formally, $A(p)=\{x\in M| {\mathrm{lim}}_{t\to -\infty }{l}_{x}(t)=p\}$ .

A descending manifold (stable manifold) $D(p)$ of a critical point $p$ is the set of all points $x$ belonging to integral lines ${l}_{x}$ whose destination is $p$ , or formally, $D(p)=\{x\in M| {\mathrm{lim}}_{t\to +\infty }{l}_{x}(t)=p\}$ .

Figure 7(b) visualises some of the integral lines belonging to the ascending manifold of the local minimum (0, 0) with respect to the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2]. Note that due to definition 2.7 and definition 2.8 neither the local minimum itself nor the separatrices forming the boundary are part of the ascending manifold, which represents an open subset of M [24, 31]. Figure 7(a) illustrates the associated paths of steepest ascent together with the ridge lines forming the boundary of the corresponding region on the graph of the function. Regions like this are commonly known as basins, catchment areas or drainage basins. For the sake of completeness, it should be mentioned that in figure 7(b) the white area lying outside the square, which represents the ascending manifold A((0, 0)), corresponds to the ascending manifold of the virtual pit.

Figure 8(b) exhibits some integral lines belonging to the descending manifolds of the local maxima (−1, −1), (−1, 1), (1, 1) and (1, −1) with respect to the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2]. In an analogous way to ascending manifolds, neither the local maxima themselves nor the separatrices forming the boundaries are part of the descending manifolds, which represent open subsets of M, too [24, 31]. Figure 8(a) illustrates the associated paths of steepest ascent together with the course lines forming the boundaries of the corresponding regions on the graph of the function. Regions like these are commonly known as hills, mountains or mounts.

In the literature, the two concepts ascending manifold and descending manifold are generally introduced in the context of two-dimensional structures. However, both concepts do not only apply to two dimensions, but, in the case of a two-dimensional manifold, to all dimensions less than or equal to two. The following theorem describes the relationship between the index of a critical point and the type of its ascending and descending manifolds formally [11, 15, 24].

Theorem 2.9. Let $f(x,y)$ be a Morse function that is defined on a two-dimensional smooth compact manifold $M$ and let $({x}_{0},{y}_{0})$ be a critical point of index λ.

The ascending manifold $A(({x}_{0},{y}_{0}))$ is an open cell⁶ of dimension $2-\lambda$ .

The descending manifold $D(({x}_{0},{y}_{0}))$ is an open cell of dimension λ.

Table 1, which is based on theorem 2.9, gives a detailed survey of the relationship between the index of a critical point and the type of manifold associated with it [11, 15, 24, 25, 35, 36]. To recap, due to definition 2.5, a critical point of index two is a local maximum, a critical point of index one is a saddle and a critical point of index zero is a local minimum.

Table 1. Survey of the relationship between the index of a critical point (x₀, y₀) and the type of manifold associated with it.

index of (x₀, y₀)	ascending manifold A((x₀, y₀))	descending manifold D((x₀, y₀))
2	local maximum	two-dimensional open region

1	separatrix leading from a saddle to a local maximum	separatrix leading from a local minimum to a saddle

0	two-dimensional open region	local minimum

Concerning the critical points of index one it can be said that, due to the assumption that f(x, y) is a Morse function, it is ensured that (a) exactly two separatrices terminate in a saddle and exactly two separatrices originate from a saddle and (b) separatrices connect always nondegenerate critical points of different indices, thus excluding saddle-saddle-connections.

Figure 7 and figure 8 reveal several essential properties of ascending and descending manifolds, which are specified in theorem 2.10 in a formal way [11, 24, 35].

Theorem 2.10. Let $f(x,y)$ be a Morse function that is defined on a smooth compact manifold $M$ . Then the following assertions hold:

(i)
The ascending (descending) manifold of $f$ is the descending (ascending) manifold of $-f$ .
(ii)
The ascending (descending) manifolds are pairwise disjoint.
(iii)
The ascending (descending) manifolds cover the manifold $M$ .
(iv)
The boundary of an ascending (descending) manifold is the union of lower-dimensional cells.

The feature described by assertion (i), termed duality, represents a widely used concept within mathematics. For the purpose of illustration, let us consider the two functions f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 and g(x, y) = − f(x, y) = 2x⁴ − 4x² + 2y⁴ − 4y² − 4, whose graphs are shown in figure 1 and figure 3, respectively. A look at figure 3 and figure 7(b) reveals that the ascending manifolds shown in figure 7(b) correspond to the descending manifolds of the function g(x, y) = − f(x, y). In an analogous way, a look at figure 3 and figure 8(b) reveals that the descending manifolds shown in figure 8(b) correspond to the ascending manifolds of the function g(x, y) = − f(x, y).

The disjointness addressed in assertion (ii) results from the fact that both ascending manifolds and descending manifolds are open sets.

Assertion (iii) becomes obvious when having a look at figure 7 and figure 8, which demonstrate how the manifold M is covered by the ascending manifolds and descending manifolds, respectively. As already mentioned, the area lying outside the square in figure 7 corresponds to the ascending manifold of the virtual pit.

Assertion (iv) says, in clear and simple terms, that a separatrix (one-dimensional manifold) is bounded by critical points (zero-dimensional manifolds) and a region (two-dimensional manifold) is bounded by separatrices (one-dimensional manifolds) in combination with critical points (zero-dimensional manifolds).

The assertions put forward in theorem 2.10 lead to the following definition [11, 15, 24–26, 31].

Definition 2.9. Let $f$ be a Morse function that is defined on a smooth compact manifold $M$ .

The union of all ascending manifolds forms the ascending Morse complex.⁷

The union of all descending manifolds forms the descending Morse complex.

When applying the terminology introduced in definition 2.9, it can be said that figure 7(b) visualises the ascending Morse complex of the manifold M, while figure 8(b) visualises its descending Morse complex.

For the sake of completeness, it should be pointed out that, in contrast to the majority of authors, who define ascending and descending manifolds as specified by definition 2.8, a small minority of them, such as, for example, Edelsbrunner [26, 35], Gyulassy [15] or Zomorodian [11], add the critical points to the corresponding ascending and descending manifolds.

In order to bring the discussion of the mathematical concepts to a close, it should be noted that when Morse theory is used for the characterisation of surface features, the terms Morse-Smale function, Morse-Smale cell and Morse-Smale complex are usually explained, too. As these terms, however, are of no relevance with respect to the topics presented in section 3, their discussion will be refrained from. Instead, it will be demonstrated how a formalisation of the geometrical feature terms, as specified in ISO 25178-2 [2] and ISO 16610-85 [3], can be achieved by means of the topological concepts presented in this section.

3. Geometrical feature terms and Morse theory

The segmentation of a surface into ascending or descending manifolds is a problem common to many areas of research, such as, for instance, computational geometry, computer graphics, GIScience or surface metrology. A study of the literature reveals that several concepts were partially developed in parallel, with the consequence that nowadays identical conceptions have often different names and identical names are often used for different conceptions. To give an example, in surface metrology areal motifs are defined as the result of the segmentation of a given surface into mutually exclusive hills or basins [19, 37–39], thus resembling to some extent the ascending and descending manifolds discussed in section 2.

The purpose of the present section is to transfer the previously described theoretical concepts to surface metrology, thereby addressing in particular the following two topics: (a) the investigation of the interrelationships between Morse theory and the definitions of the geometrical feature terms given in ISO 25178-2 [2] and ISO 16610-85 [3], and (b) the presentation of some redefinitions whose aim is the elimination of several inconsistencies and ambiguities that can actually be found in the two standards, with the motivation for this endeavour resulting from three observations.

The first observation can be characterised as imbalance within the standards. To be precise, on the one hand, there exist very accurate definitions, which include even mathematical formulas, for numerous concepts, such as the local gradient vector, the autocorrelation function or the root mean square height, to mention only a few. On the other hand, most definitions related to geometrical feature terms lack a comparable rigour, although appropriate mathematical concepts are available.

The second observation can be characterised as conflict with mathematical conventions. More precisely, as the key concept of ISO 25178-2 [2] and in the sequel of ISO 16610-85 [3], is the scale-limited surface, almost all definitions in the standard are based on it. From a mathematical point of view, however, the scale-limited surface is the graph of a height function h(x, y), which is defined on the reference surface. In order to be in accordance with mathematical practice, all geometrical features terms should therefore not be defined with respect to the scale-limited surface, but rather with respect to the reference surface.

The third observation, finally, can be characterised as partial lack of explanation. To give an example, in the two standards it is mentioned that an unwanted side effect induced by the first step of a segmentation process is an over-segmentation of the surface due to measurement noise; the over-segmentation itself is described in terms of the motifs, i.e. the dales and the hills. In the following step it is described how this over-segmentation can be eliminated by pruning the change tree; the elements of the change tree, however, are the pits, passes and peaks of the surface, and not its dales and hills. The description of the pruning process is correct, however, an explanation highlighting that its validity is due to the one-to-one correspondences between the dales and the pits and between the hills and the peaks, respectively, is missing. In contrast, when applying Morse theory this problem will not occur because the aforementioned one-to-one correspondences are direct consequences of the definitions of the ascending and descending manifolds.

3.1. Geometrical feature terms in ISO 25178-2

In this section, the geometrical feature terms specified in ISO 25178-2 [2] will be discussed in individual subsections. If a subsection's caption exhibits a number of the type (a. b. c) or (a. b. c. d), the number in parentheses refers to that section of ISO 25178-2 [2], in which the respective definition can be found. In contrast, an (add.) in a subsection's caption indicates that this term should be added to the standard, in order to simplify definitions given later on. Each subsection consists of two parts, namely, (a) the proposed redefinition and (b) comments explaining the motivation for the redefinition. The subsections themselves are ordered in such a way that, first of all, general terms are discussed and, hereafter, zero-dimensional features, one-dimensional features and, finally, two-dimensional features.

3.1.1. Reference surface (3.1.10)

Definition 3.1. A reference surface is a two-dimensional manifold.

When considering the reference surface as two-dimensional manifold, it follows, due to the explanations given at the beginning of section 2, that it meets all requirements to serve as domain of that function, which associates with each element (x, y) its respective height (cf definition 3.2).

The examples given in section (3.1.10) of ISO 25178-2 [2], viz., the plane, the cylinder and the sphere, in combination with the explanations given in section (3.2.5) of the standard cited, reveal the validity of the assumptions underlying definition 3.1. This impression is reinforced when reading in section (3.2.6) of the predecessor version ISO 25178-2:2012 [1] that the coordinate system is based on the reference surface.

The relationship between reference surface and scale-limited surface will be discussed in section 3.1.3.

3.1.2. Height function (add.)

Definition 3.2. The height function $h(x,y)$ associates with each element ( $x,y$ ) of the reference surface its respective height $z=h(x,y)$ .

Section (3.2.5) of ISO 25178-2 [2] is devoted to the two terms height and ordinate value, whereby it can be assumed that both terms denote the same entity. Although not explicitly stated, the notation used in ISO 25178-2 [2] supports the assumption that the height h(x, y) represents a function in the mathematical sense. The confusion is due to the fact that in mathematics the symbol f(x, y) is used to denote (a) the function value at the point (x, y), i.e. a specific numerical value, as well as (b) a whole function, i.e. a binary relation between two sets that associates with every element of the first set exactly one element of the second set; evidently, the two conceptions described in (a) and (b) are something quite different.

In several sections of ISO 25178-2 [2], properties of the height function are implicitly described, such as, for example: (a) in section (3.2.6) the local gradient vector is defined, thus implying that h(x, y) must be (at least) once differentiable; (b) in section (3.2.15) the Fourier transformation is explained, thus inducing that h(x, y) must be integrable; (c) in section (3.3.1) it is described how plateaus, representing degenerate peaks, can be avoided, in section (3.3.2) the same is done for pits, with both reminding on theorem 2.6, which states that any smooth function can be converted into a Morse function by an arbitrary small perturbation splitting a degenerate critical point into a certain number of nondegenerate ones.

For the reasons listed above, it will be assumed that the height function h(x, y) is a Morse function. It should be emphasised that this assumption, which also concerns the key concept scale-limited surface, is arbitrary, but proves to be useful with respect to the definitions given later on. In addition, as a scale-limited surface represents the graph of a height function h(x, y) (cf section 3.1.3), its visual appearance is completely determined by the properties of h(x, y). As already noted at the beginning of section 2, one must be aware that surfaces do not exist per se, but they are nothing else than models and thus they are not right or wrong, but appropriate or inappropriate.

Definition 3.2 may be used instead of the definition given in section (3.2.5) of ISO 25178-2 [2]; in addition, the definition of depth, which can be found in section (3.2.5.1) of the standard cited, may be omitted because negative function values are not worth mentioning.

For the sake of completeness, it should be noted that in case that the height function is not defined in analytic form, but only by its values at a finite number of points (x_i, y_i) with i = 1,...,n, classical Morse theory can no longer be applied. In this instance a modified approach, which goes back to Banchoff [40], may be used. His concept relies on a triangulation of the manifold and assumes the height function to be linear on every triangle, i.e. piecewise linear on the manifold. Due to these assumptions, the gradients are no longer continuous and hence they do not generate pairwise disjoint integral lines, as required for the definitions of ascending and descending manifolds. Nevertheless, also for piecewise linear manifolds an adequate theory exists, which has already been adopted within computational geometry and computer graphics [26, 35, 41–44].

Another option for the handling of a finite number of data points (x_i, y_i, z_i) = (x_i, y_i, h(x_i, y_i)) with i = 1,...,n is the application of discrete Morse theory, as introduced by Forman [45, 46]. While classical Morse theory establishes a connection between the topology of a manifold and the critical points of a smooth function defined on it, Forman derived an analogue theory for functions defined on manifolds that are discretised in form of simplicial or cell complexes. In addition, Forman demonstrated how discrete analogues of intrinsically smooth concepts, such as the gradient vector field, the gradient flow or the Morse complex can be defined. As discrete Morse theory provides a robust computational framework, it is gaining more and more interest especially within computational geometry and computer graphics [9, 15, 24, 29, 34, 47–55]. Within surface metrology, the two aforementioned approaches seem to have great potential with respect to the formal analysis of freeform surfaces (a comprehensive introduction to this topic can be found in Jiang and Scott [56]).

3.1.3. Scale-limited surface (3.1.9)

Definition 3.3. A scale-limited surface is the image of the reference surface under the height function $h(x,y)$ .

Due to section (3.1.10) of ISO 25178-2 [2], the reference surface is associated to the scale-limited surface according to a criterion; this assertion says, in mathematical terms, that the reference surface is the image of the scale-limited surface. On the other hand, one can read in section (3.2.5) of ISO 25178-2 [2] that the height is the distance from the reference surface to the scale-limited surface, i.e. the assignment is made in the quite opposite direction, thus implying that the scale-limited surface is the image of the reference surface. The combination of the last two remarks reveals the contradiction, which is actually inherent in the two definitions given in section (3.1.10) and section (3.2.5) of ISO 25178-2 [2].

Definition 3.3 is apparently an option to resolve this contradiction, as it is, moreover, in line with the other conceptions described in ISO 25178-2 [2].

3.1.4. Evaluation area (3.1.11)

Definition 3.4. see text below

The definition of the term evaluation area as specified in section (3.1.11) of ISO 25178-2 [2] is unsatisfactory due to the following reasons:

First of all, the evaluation area is defined in terms of the scale-limited surface, i.e. in terms of the graph of the height function h(x, y). It must be emphasised that this is against mathematical conventions according to which restrictions concerning the image of a function have always to be made in terms of the function's domain.

Secondly, the explanations given are ambiguous for the following reason: in section (3.1.11) of the standard cited one can read that the symbol $\tilde{A}$ is used for the domain (of integration or definition), whereas the symbol A is used for the numerical value of the evaluation area. This formulation, however, makes no sense unless the height function h(x, y) is constant (according to ISO 25178-2 [2], a single value is assigned to a whole set). In contrast, a replacement of the actual formulation numerical value of the evaluation area by image of the reference surface under the height function h(x, y) can easily resolve this problem.

A comfortable way to solve all problems encountered with the specification of the term evaluation area is to comprehend the reference surface not as the whole (x, y)-plane, the whole cylinder or the whole sphere, respectively, but only as that portion, which is of relevance for the specific problem under consideration, with this approach being in accordance with mathematical conventions.

It is for the reason stated that no formal definition is given for the moment because the basic question remains open whether a definition is necessary or whether the term is superfluous.

3.1.5. Local maximum (add.)

Definition 3.5. A point $({x}_{{\max }},{y}_{{\max }})$ is a local maximum of the height function $h(x,y)$ , if $h(x,y)\,\,\leqslant h({x}_{{\max }},{y}_{{\max }})$ for all ( $x,y$ ) in some neighbourhood of $({x}_{{\max }},{y}_{{\max }})$ .

The purpose of this definition, like that of the definitions given in section 3.1.7 and in section 3.1.9, is to enable a distinction between the points lying in the reference surface, i.e. in the domain of the function h(x, y), and their images lying on the scale-limited surface, i.e. on the graph of the function h(x, y). This distinction, which is standard in mathematics, was the basis for all definitions given in section 2.

The only difference between the above definition and definition 2.2 is that instead of an arbitrary function f(x, y) the height function h(x, y) is used.

3.1.6. Peak (3.3.1)

Definition 3.6. A peak is the image of a local maximum $({x}_{{\max }},{y}_{{\max }})$ under the height function $h(x,y)$ .

Definition 3.6 can be reformulated in terms of the coordinates of the local maximum in the following way. If z_max denotes the function value of $({x}_{{\max }},{y}_{{\max }})$ under h(x, y), i.e. ${z}_{{\max }}=h({x}_{{\max }},{y}_{{\max }})$ , then a peak is a point with coordinates $({x}_{{\max }},{y}_{{\max }},\,{z}_{{\max }})\,=({x}_{{\max }},{y}_{{\max }},\,h({x}_{{\max }},{y}_{{\max }}))$ . The value z_max is denoted as peak height in section (3.3.5) of ISO 16610-85 [3].

In view of theory and the calculations involved, however, not the peak itself, which is situated on the graph of the height function h(x, y), is of importance, but the corresponding local maximum being located in the reference surface (the proof is provided by a look in any mathematical textbook).

3.1.7. Local minimum (add.)

Definition 3.7. A point $({x}_{{\min }},{y}_{{\min }})$ is a local minimum of the height function $h(x,y)$ , if $h(x,y)\,\geqslant h({x}_{{\min }},{y}_{{\min }})$ for all (x, y) in some neighbourhood of $({x}_{{\min }},{y}_{{\min }})$ .

The explanations given in section 3.1.5 for local maxima apply accordingly for local minima.

3.1.8. Pit (3.3.2)

Definition 3.8. A pit is the image of a local minimum $({x}_{{\min }},{y}_{{\min }})$ under the height function $h(x,y)$ .

The explanations given in section 3.1.6 for peaks apply accordingly for pits. In terms of the coordinates of the local minimum, a pit is a point with coordinates $({x}_{{\min }},{y}_{{\min }},\,{z}_{{\min }})=({x}_{{\min }},{y}_{{\min }},\,h({x}_{{\min }},{y}_{{\min }}))$ . The value ${z}_{{\min }}=h({x}_{{\min }},{y}_{{\min }})$ is denoted as pit height in section (3.3.6) of ISO 16610-85 [3].

3.1.9. Saddle point (saddle) (3.3.3 and 3.3.3.1)

Definition 3.9. A point $({x}_{{sadd}},{y}_{{sadd}})$ is a saddle point (saddle) of the height function $h(x,y)$ , if there exist regular paths ${p}_{1}$ and ${p}_{2}$ that intersect transversally at $({x}_{{sadd}},{y}_{{sadd}})$ (i.e. the tangent vectors at $({x}_{{sadd}},{y}_{{sadd}})$ are defined and not multiples of each other) in such a way that $h(x,y)$ has a local maximum at $({x}_{{sadd}},{y}_{{sadd}})$ along ${p}_{1}$ , whereas $h(x,y)$ has a local minimum at $({x}_{{sadd}},{y}_{{sadd}})$ along ${p}_{2}$ .

In contrast to the four previously given definitions, viz., that of a local maximum, a peak, a local minimum and a pit, the issues concerning saddle points are much more complex due to the following ambiguities being inherent in ISO 25178-2 [2].

First of all, the usage of the terms saddle point and saddle in ISO 25178-2 [2] is contrary to that in mathematics. To be precise, according to section (3.3.3) and section (3.3.3.1) of the standard cited, saddle points are located on the scale-limited surface, i.e. on the graph of the height function h(x, y). In mathematics, however, saddle points are never elements of the image of a function, but always elements of its domain, in this particular case of the reference surface (cf definition 2.2 given in section 2). Unlike the definitions given in ISO 25178-2 [2], definition 3.9 takes this into account.

Secondly, in section (3.3.3) of ISO 25178-2 [2] the term saddle is defined and in section (3.3.3.1) the term saddle point. As in mathematics both terms denote the same, this differentiation seems to be inappropriate from a mathematical point of view. For this reason, in the present work no distinction will be made between the two terms, but they will be rather used synonymously throughout this paper (cf definition 2.2 given in section 2).

Thirdly, the existence of degenerate saddle points cannot be excluded by the definitions given in ISO 25178-2 [2], which is particularly interesting because in section (3.3.1) and section (3.3.2) of the standard it is explained how degenerate peaks and degenerate pits can be avoided. To reiterate, degenerate critical points can occur unless the height function h(x, y) is assumed to be a Morse function (cf definition 2.6 and section 3.1.2). For the purpose of illustration, some examples of degenerate saddle points are shown in figure 9. To be precise, figure 9(a) visualises the graph of the function f₁(x, y) = x³ − 3xy² for x, y ∈ [−3, 3], which has a monkey saddle at (0, 0), figure 9(b) shows the graph of the function f₂(x, y) = x³ y − xy³ for x, y ∈ [−2, 2], which has a dog saddle at (0, 0), and figure 9(c) depicts the graph of the function f₃(x, y) = x⁵ − 10x³ y² + 5xy⁴ for x, y ∈ [−1, 1], which has a starfish saddle at (0, 0). To summarise, according to the definitions given in ISO 25178-2 [2], which rely on course lines and ridge lines, degenerate saddles like the ones shown in figure 9 are allowed. However, this represents an inconsistency because degenerate peaks and degenerate pits are obviously not desired due to the explanations given in section (3.3.1) and section (3.3.2) of the standard cited.

**Figure 9.** Three types of degenerate saddle points: (a) monkey saddle, (b) dog saddle and (c) starfish saddle.
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3.1.10. Pass (add.)

Definition 3.10. A pass is the image of a saddle point $({x}_{{sadd}},{y}_{{sadd}})$ under the height function $h(x,y)$ .

The explanations given in section 3.1.6 for peaks and in section 3.1.8 for pits apply accordingly for passes. In terms of the coordinates of the saddle point, a pass is a point with coordinates (x_sadd, y_sadd, z_sadd) = (x_sadd, y_sadd, h(x_sadd, y_sadd)). The value z_sadd = h(x_sadd, y_sadd) is denoted as saddle height in section (3.3.4) of ISO 16610-85 [3], although the term pass height would be more appropriate due to the explanations given in section 3.1.9.

3.1.11. Virtual peak (add.)

Definition 3.11. A virtual peak is a peak of height $+\infty$ , which represents the image of all points lying outside the domain of the height function.

The definition of the term virtual peak has to be added to ISO 25178-2 [2] because a virtual peak may occur as element of the change tree, which is defined in section (3.3.8) of the standard cited (cf section 3.1.25).

It is noteworthy that in ISO 16610-85 [3] only the definition of a virtual pit is given and not that of a virtual peak, although both concepts are essential due to theoretical reasons. To give an example, if the definition of a virtual peak is omitted in the two standards, the topological structure of the graph of the function g(x, y) = 2x⁴ − 4x² + 2y⁴ − 4y² − 4 for x, y ∈ [−2, 2], which is shown in figure 3, could not be visualised by a change tree. Due to topological reasons, either a virtual peak or a virtual pit has to be added in order to obtain a compact manifold as domain of the height function (cf theorem 2.7 and the accompanying explanations).

For the sake of completeness, it should be noted that the term virtual pit condition, which is defined in section (3.3.10) of ISO 16610-85 [3], is obsolete because, on the basis of the above, there is no choice whether to make this assumption or not.

3.1.12. Virtual pit (add.)

Definition 3.12. A virtual pit is a pit of height $-\infty$ , which represents the image of all points lying outside the domain of the height function.

The explanations given in section 3.1.11 for virtual peaks apply accordingly for virtual pits.

3.1.13. Separatrix (add.)

Definition 3.13. A separatrix is an integral line, which originates either from a local minimum and ends in a saddle point or which originates from a saddle point and ends in a local maximum.

As in the case of zero-dimensional features, it is also in the one-dimensional case important to ensure a distinction between the elements lying in the reference surface, representing the domain of the height function h(x, y), and their corresponding images lying on the scale-limited surface, representing the graph of the height function h(x, y). In the present case, this is guaranteed by definition 3.13 in combination with definition 3.14 and definition 3.15.

For a detailed explication of the terms integral line and separatrix confer definition 2.7 and the accompanying explanations given in section 2.

3.1.14. Course line (3.3.1.3)

Definition 3.14. A course line is the image of a separatrix, which originates from a local minimum and ends in a saddle point, under the height function $h(x,y)$ .

The explanations given in sections 3.1.6, 3.1.8 and 3.1.10 for zero-dimensional features apply accordingly for one-dimensional ones.

Although, from a theoretical point of view, a course line can also be defined as the image of a descending manifold of a saddle (cf table 1), it will be refrained from doing it because definition 3.14 is much more intuitive.

3.1.15. Ridge line (3.3.2.3)

Definition 3.15. A ridge line is the image of a separatrix, which originates from a saddle point and ends in a local maximum, under the height function $h(x,y)$ .

The explanations given in sections 3.1.6, 3.1.8 and 3.1.10 for zero-dimensional features apply accordingly for one-dimensional ones.

Although, from a theoretical point of view, a ridge line can also be defined as the image of an ascending manifold of a saddle (cf table 1), it will be refrained from doing it because definition 3.15 is much more intuitive.

3.1.16. Ascending manifold (add.)

Definition 3.16. An ascending manifold $A(({x}_{{\min }},{y}_{{\min }}))$ of a local minimum $({x}_{{\min }},{y}_{{\min }})$ is the set of all points $x$ belonging to integral lines whose origin is $({x}_{{\min }},{y}_{{\min }})$ .

As in the case of zero-dimensional and one-dimensional features, it is also in the two-dimensional case important to ensure a distinction between the elements lying in the reference surface, representing the domain of the height function h(x, y), and their corresponding images lying on the scale-limited surface, representing the graph of the height function h(x, y). In the present case, this is guaranteed by definition 3.16 and definition 3.18 in combination with definition 3.17 and definition 3.19.

For a detailed explication of the terms ascending manifold and descending manifold confer definition 2.8 and the accompanying explanations given in section 2.

3.1.17. Dale (3.3.2.1)

Definition 3.17. A dale is the image of the ascending manifold of a local minimum under the height function $h(x,y)$ .

In section (3.3.2.1) of ISO 25178-2 [2], a dale is defined as the region around a pit such that all maximal downward paths end at the pit. Although this definition seems intuitively correct, the following explanations will reveal the opposite.

Figure 10 visualises the graph of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2]. The image of the ascending manifold of the local minimum (0, 0) is the area within the solid lines, which represent the ridge lines forming its boundary (cf figure 7(a)). The assumption that this region coincides with that area, which is generally regarded as the dale being associated with the pit (0, 0, 4) is supported by the corresponding figure shown in ISO 25178-2 [2]. In addition, figure 10 depicts a path, outlined by a solid red line, leading from the peak (1, −1, 8) via the points (1.1, −0.9, 7.84), (1.125, −0.85, 7.7), (1.175, −0.8, 7.5), (1.2, −0.7, 7.1), (1.1, −0.5, 6.8) to the pass (1, 0, 6) and further on via (0.5, 0.25, 5.1) to the pit (0, 0, 4). In order to give an impression of the exact location of this path, three different viewpoints were chosen, namely, (a) diagonally above the graph of the function (figure 10(a)), (b) in front of the graph of the function (figure 10(b)) and (c) directly above the graph of the function (figure 10(c)). Obviously, the path under discussion is not only a downward path as the steadily decreasing z-coordinates of the points lying on it indicate, but rather a maximal downward path because it cannot be extended in any direction without losing this property.

Subject to the definition given in section (3.3.2.1) of ISO 25178-2 [2], the path depicted in figure 10 as well as an infinite number of similarly located ones are also elements of the dale being associated with the pit (0, 0, 4), although this contradicts the intuitive idea of what is considered a dale. In contrast, when using definition 3.17 in order to characterise the term dale in a formal way, a discrepancy like the previously described one cannot occur. The underlying reason is that the two definitions are based on different assumptions; while the definition given in section (3.3.2.1) of ISO 25178-2 [2] rests upon the conception of maximal downward paths, definition 3.17 relies on the conception of maximal downward paths of steepest descent, with this property being inherent in the definition of an integral line (cf definition 2.7), which in turn is the basis for the formal specification of all one-dimensional and two-dimensional geometrical feature terms being discussed within this paper.

3.1.18. Descending manifold (add.)

Definition 3.18. A descending manifold $D(({x}_{{\max }},{y}_{{\max }}))$ of a local maximum $({x}_{{\max }},{y}_{{\max }})$ is the set of all points $x$ belonging to integral lines whose destination is $({x}_{{\max }},{y}_{{\max }})$ .

The explanations given in section 3.1.16 for ascending manifolds apply accordingly for descending manifolds.

3.1.19. Hill (3.3.1.1)

Definition 3.19. A hill is the image of the descending manifold of a local maximum under the height function $h(x,y)$ .

The explanations given in section 3.1.17 for dales apply accordingly to hills, as there exists an analogous ambiguity in the definition given in section (3.3.1.1) of ISO 25178-2 [2], in which a hill is defined as the region around a peak such that all maximal upward paths end at the peak.

Figure 11 visualises the graph of the function f(x, y) = −2x⁴ + 4x² − 2y⁴ + 4y² + 4 for x, y ∈ [−2, 2]. The images of the descending manifolds of the local maxima (−1, −1), (−1, 1), (1, 1) and (1, −1) are the areas within the solid lines surrounding the four peaks; the solid lines themselves represent the course lines forming the respective boundaries (cf figure 8(a)). The assumption that these regions coincide with those areas, which are generally regarded as the hills being associated with the peaks (−1, −1, 8), (−1, 1, 8), (1, 1, 8) and (1, −1, 8) is supported by the corresponding figure shown in ISO 25178-2 [2]. In addition, figure 11 depicts a path, outlined by a solid red line, leading from the pit (0, 0, 4) via (0.5, 0.25, 5.1) to the pass (1, 0, 6) and further on via the points (1.1, −0.5, 6.8), (1.2, −0.7, 7.1), (1.175, −0.8, 7.5), (1.125, −0.85, 7.7), (1.1, −0.9, 7.84) to the peak (1, −1, 8). A comparison of this path with the one shown in figure 10 reveals that the only difference between them is the direction assigned. In order to give an impression of the exact location of the path, three different viewpoints were chosen, namely, (a) diagonally above the graph of the function (figure 11(a)), (b) in front of the graph of the function (figure 11(b)) and (c) directly above the graph of the function (figure 11(c)). Obviously, the path under discussion is not only an upward path as the steadily increasing z-coordinates of the points lying on it indicate, but rather a maximal upward path because it cannot be extended in any direction without losing this property.

Subject to the definition given in section (3.3.1.1) of ISO 25178-2 [2], the path depicted in figure 11 as well as an infinite number of similarly located ones are also elements of the hill being associated with the peak (1, −1, 8), although this contradicts the intuitive idea of what is considered a hill. In contrast, when using definition 3.19 in order to characterise the term hill in a formal way, a discrepancy like the previously described one cannot occur. The underlying reason is that the two definitions are based on different assumptions; while the definition given in section (3.3.1.1) of ISO 25178-2 [2] rests upon the conception of maximal upward paths, definition 3.19 relies on the conception of maximal upward paths of steepest ascent, with this property being inherent in the definition of an integral line (cf definition 2.7), which in turn is the basis for the formal specification of all one-dimensional and two-dimensional geometrical feature terms presented within this paper.

3.1.20. Point feature (3.3.5.3)

Definition 3.20. A point feature is a peak or a pass or a pit.

A point feature can alternatively be defined as the image of a critical point under the height function h(x, y).

Following the explanations given in section 3.1.9 and section 3.1.10, the term pass is used instead of saddle point, with the latter being used in section (3.3.5.3) of ISO 25178-2 [2].

Due to the fact that the definitions of a peak, a pass and a pit are based on Morse theory, also the definition of the term point feature is much more rigorous now. The same argument holds for the definitions given in sections 3.1.21 and 3.1.22.

3.1.21. Line feature (3.3.5.2)

Definition 3.21. A line feature is a course line or a ridge line.

A line feature can alternatively be defined as the image of a separatrix under the height function $h(x,y)$ .

The explanations given in section 3.1.20 with respect to the accuracy of the definitions of point features apply accordingly for line features.

3.1.22. Areal feature (3.3.5.1) and motif (3.3.4)

Definition 3.22. An areal feature is a dale or a hill.

An areal feature can alternatively be defined as the image of an ascending manifold of a local minimum or of a descending manifold of a local maximum under the height function $h(x,y)$ .

The explanations given in section 3.1.20 with respect to the accuracy of the definitions of point features apply accordingly for areal features.

From a theoretical point of view, a distinction between the two terms areal feature and motif is unnecessary as both of them denote the same entity.

3.1.23. Topographic feature (3.3.5)

Definition 3.23. A topographic feature is a point feature or a line feature or an areal feature.

The explanations given in section 3.1.20 with respect to the accuracy of the definitions of point features apply for all topographic features.

3.1.24. Contour line (contour, isoline) (3.3.6)

Definition 3.24. A contour line (contour, isoline) ${C}_{h}$ with height $h$ of a function $f(x,y)$ is the set of all elements ( $x,y$ ) with $f(x,y)=h$ , or formally, ${C}_{h}=\{(x,y)| f(x,y)=h\}$ .

In simple terms, a contour line, also termed contour or isoline, is the locus of all points (x, y) of the function's domain that are solution to the equation f(x, y) = h. A more formal definition of this concept can be found in Heine et al [57] or Pascucci and Cole-McLaughlin [58], who define a contour line as a special case of a more general topological concept termed level set.

The striking difference between definition 3.24 and the definition given in section (3.3.6) of ISO 25178-2 [2] is that according to the latter, a contour line is an element of the image of the height function h(x, y), i.e. of the scale-limited surface, and not of its domain, i.e. of the reference surface, which is, however, against mathematical convention. As contour lines form, moreover, the basis of contour trees, which will be dealt with in section 3.1.25, it seems appropriate to modify their definition according to mathematical usage.

3.1.25. Change tree (contour tree) (3.3.8)

Definition 3.25. see text below

The change tree, also termed contour tree, represents the basic data structure with respect to the calculation of feature parameters being specified in ISO 25178-2 [2]. It goes back to Boyell and Rushton [59], who were the first to propose a tree-like data structure for the representation of adjacency relationships between contour lines. In the years to follow, a vast amount of literature concerning this topic, including some comprehensive surveys [8, 18, 24, 57, 58, 60, 61], was published. In surface metrology the change tree was introduced by Scott [19, 37–39], whose approach is based on the assumption that the human apprehension of surfaces is not performed via numerical values, but via patterns of features.

A closer inspection of the definition of the change tree given in section (3.3.8) of ISO 25178-2 [2], however, reveals two major problems, which will be discussed next.

The first problem concerns the lack of rigour in the definition of the change tree, despite the fact that suitable mathematical concepts are available [8, 18, 57, 58, 60, 61]. This lack of accuracy refers to the definition itself as well as the accompanying explanations. To be precise, the definition of the change tree given in ISO 25178-2 [2] resembles much more the instruction manual for a drawing than a formal mathematical definition, although the latter can be found in numerous publications [8, 57, 58, 60]. Unsatisfactory from a mathematical point of view are also the accompanying explanations, because the elements of a change tree, which represents a special case of a more general topological structure termed graph, are not lines and end of lines, as can be read in the standard, but edges end vertices (nodes), which are the correct designations (for an introduction to graph theory see, for example, Bondy and Murty [62]). To give another example, the difference between peaks (pits) and passes is described by end of lines and joining lines in the standard and not in terms of the degree of the respective vertex as it should be.

The second problem is related to the data structure itself. In section (5.3) of ISO 25178-2 [2] four criteria are listed, which may be used for the definition of thresholds with respect to a proper segmentation. These four criteria comprise the dale local depth (hill local height), the perimeter of a dale (hill), the area of a dale (hill) and the volume of a dale (hill).

An inspection of the literature, however, reveals that actually only the dale local depth (hill local height) is implemented as termination rule for the simplification process of a surface. The reason for this discrepancy is a weakness of the data structure itself, which entails that no other information than height is stored in the change tree [8, 18]. A solution of this problem can be achieved by choosing one of two possible strategies, viz., (a) the additional storage of local spatial measures within the change tree or (b) the usage of weighted surface networks, a different topological data structure, instead of change trees.

Before comparing the two strategies, it should be emphasised that all effects induced by both of them are merely of technical nature, i.e. they are confined to modifications in the software performing the pruning process. Concerning (a), Carr [60, 63, 64] demonstrated how geometrical properties of features, such as, for example, the contour length, the cross-sectional area, the surface area or the volume, to mention only a few, can also be taken into account and be implemented in modified algorithms. But even if Carr's approach is chosen, another major drawback of change trees continues to exist, namely, that contour trees contain no geometrical information related to the gradient flow [15, 35, 44]. Concerning (b), weighted surface networks, which represent a different topological data structure [16, 65–69], are closely related to Morse theory. To be precise, when focusing only on the critical points and separatrices of a Morse function, while ignoring its ascending and descending manifolds, the structure thus obtained represents the associated 1-skeleton (critical net)⁸ [24, 28, 70], which is equivalent to the corresponding weighted surface network [24]. To put it in a nutshell, when balancing the pros and cons of the two approaches, the second approach seems to be the more promising and thus the preferable one (for a detailed discussion of different topological data structures confer Wolf [8, 18]).

The previous explanations should also help in answering the question how definition 3.25 should look like; should it include (a) a more formal definition of the change tree, (b) a definition of the change tree, which is modified in such a form that also local spatial measures are taken into account, or (c) the formal definition of a weighted surface network.

The effects of a possible change in the data structure from change trees to weighted surface networks with respect to the pruning process will be discussed in section 3.1.29.

3.1.26. Adjacent pass (add.)

Definition 3.26. An adjacent pass of a peak is a pass that is connected to the peak by a ridge line.

An adjacent pass of a pit is a pass that is connected to the pit by a course line.

Introducing the term adjacent pass into the standard seems advisable because in this case other definitions, such as, for example, the ones given in section (3.3.8.1), in section (3.3.8.2) and in section (3.3.8.3) of ISO 25178-2 [2], can be formulated in a much more convenient way. The reason for the latter is that in any visualisation of the topological structure of a scale-limited surface by a graph, regardless of whether in form of a change tree or a weighted surface network, specific pairs of critical points are connected by edges, with this relationship being denoted as adjacent in graph theory.

Figure 1 reveals that the pit (0, 0, 4) has four adjacent passes, viz., (−1, 0, 6), (0, 1, 6), (1, 0, 6) and (0, −1, 6), whereas the peak (1, −1, 8) has two adjacent passes, viz., (0, −1, 6) and (1, 0, 6).

3.1.27. Hill local height (3.3.8.2)

Definition 3.27. The hill local height is the difference between the height of a peak and the height of its highest adjacent pass.

In section (3.3.8.2) of ISO 25178-2 [2] the hill local height is defined as the difference between the height of a peak and the height of the nearest connected saddle point on the change tree. While this definition is applicable to change trees only, definition 3.27 is much more general and can also be applied to other topological data structures, such as, for example, weighted surface networks. The importance of the term hill local height will become obvious in section 3.1.29 in connection with the discussion of the pruning process.

3.1.28. Dale local depth (3.3.8.3)

Definition 3.28. The dale local depth is the absolute value of the difference between the height of a pit and the height of its lowest adjacent pass.

The explanations given in section 3.1.27 for the hill local height apply accordingly for the dale local depth. In the latter case, however, the absolute value of the difference must be taken in order to avoid negative results.

3.1.29. Pruning (3.3.8.1), Wolf pruning (3.3.8.4) and height discrimination (3.3.9)

Definition 3.29. Let $f$ be a Morse function that is defined on a smooth compact manifold $M$ . The pairwise elimination of either a peak together with its highest adjacent pass or of a pit together with its lowest adjacent pass is termed pruning.

In the technical sciences, the reduction of measurement noise corresponds to the simplification of a scale-limited surface, or, to put it differently, it corresponds to the process of deriving from an original scale-limited surface a second scale-limited surface of decreased complexity, but with its structural properties being retained.

The theorem of Matsumoto [10] can be considered as the core theorem with respect to the simplification of the topological structure of surfaces being characterised by their critical points. For the two-dimensional case it says, in clear and simple terms, that from a topological point of view the only valid simplification is the pairwise elimination of a peak together with its highest adjacent pass or of a pit together with its lowest adjacent pass⁹ (a detailed discussion of this topic can be found in Wolf [8, 18]). Incidentally, the height differences, which are determined by these two pairs of points, are the hill local height and the dale local depth, as specified in definition 3.27 and definition 3.28, respectively.

The importance of Matsumoto's theorem results from the fact that its assertions are based exclusively on the properties of a Morse function and not on those of a specific data structure, such as a change tree, a weighted surface network or a Morse-Smale complex.

The detailed discussion of Wolf pruning as well as of the term height discrimination is postponed to sections 3.2.9 and 3.2.8, respectively.

3.1.30. Segmentation (3.3.7) and segmentation function (3.3.7.1)

All definitions related to segmentation need no further refinement because their sound mathematical foundations can be found in Scott [39].

3.2. Geometrical feature terms in ISO 16610-85

In this section, the geometrical feature terms specified in ISO 16610-85 [3] will be discussed in individual subsections, provided they have not yet been dealt with in section 3.1. If a subsection's caption exhibits a number of the type (a. b. c) or (a. b. c. d), the number in parentheses refers to that section of ISO 16610-85 [3], in which the respective definition can be found. In contrast, an (add.) in a subsection's caption indicates that this term should be added to the standard, in order to simplify definitions given later on. Each subsection consists of two parts, namely, (a) the proposed redefinition and (b) comments explaining the motivation for the redefinition.

3.2.1. Maxwellian hill (3.1.1.1) and hill (3.1.1.3)

Definition 3.30. see text below

In ISO 16610-85 [3] one can find the definition of a Maxwellian hill (section (3.1.1.1)), which is defined as the region around a peak such that all maximum upward paths end at the peak, and that of a hill (section (3.1.1.3)), which is defined as the region around a single dominant peak whose boundary consists of a ring of course lines.

Obviously, both definitions may cause criticism due to the reasons listed below.

First of all, the definition of a Maxwellian hill given in section (3.1.1.1) of ISO 16610-85 [3] is identical to the definition of a hill given in section (3.3.1.1) of ISO 25178-2 [2]. As a consequence, both definitions suffer from the same drawbacks that have already been described in section 3.1.19.

Secondly, in the two standards ISO 25178-2 [2] and ISO 16610-85 [3] two different terms, viz., hill and Maxwellian hill are used to denote the same entity.

Thirdly, the definition of a hill given in section (3.1.1.3) of ISO 16610-85 [3] suffers from a lack of mathematical rigour because (a) the key term single dominant peak is nowhere defined and (b) the formulation ring of course lines is no mathematical one (for a correct definition confer section 3.1.19).

Fourthly, despite its imprecise mathematical formulation, the definition of a hill as specified in section (3.1.1.3) of ISO 16610-85 [3] corresponds to what is commonly regarded a hill (cf definition 3.19). This implies that in the two standards ISO 16610-85 [3] and ISO 25178-2 [2] the term hill is used to denote two different entities, viz., a correctly defined hill and an incorrectly defined hill, to put it straight.

In order to avoid the previously described problems, the following strategy seems promising: (a) specification of an appropriate definition of a hill in ISO 25178-2 [2] (cf section 3.1.19), (b) omission of the definition Maxwellian hill in ISO 16610-85 [3] and (c) modification of the definition of a hill in ISO 16610-85 [3] accordingly.

3.2.2. Maxwellian dale (3.1.2.1) and dale (3.1.2.3)

Definition 3.31. see text below

The explanations given in section 3.2.1 for hills and Maxwellian hills apply accordingly for dales and Maxwellian dales.

3.2.3. Event (3.2.1.1)

Definition 3.32. see text below

Following the explanation given in section 2, the domain of a Morse function is the disjoint union of zero-dimensional manifolds (critical points), one-dimensional manifolds (separatrices) and two-dimensional manifolds (the ascending manifolds of local minima and the descending manifolds of local maxima). Accordingly, the image of the Morse function is the disjoint union of the images of the critical points (peaks, pits and passes), of the separatrices (course lines and ridge lines) and of the ascending manifolds of the local minima and the descending manifolds of the local maxima (dales and hills). Due to definition 3.23, these features are termed topographic features. Following the definition given in section (3.2.1.1) of ISO 16610-85 and the examples listed there, each of these features is also termed event. To conclude, in the two standards ISO 25178-2 [2] and ISO 16610-85 [3] two different terms, viz., topographic feature and event are used to denote the same entity.

3.2.4. Local peak height (3.3.5.1)

Definition 3.33. The local peak height is the difference between the height of a peak and the height of its highest adjacent pass.

Definition 3.33 and definition 3.27 are identical, whereby the term hill local height replaces the old designation local peak height in ISO 25178-2 [2]. The reasons that motivated this change are described in detail by Blateyron [4]. It can be assumed that the same substitution will be part of the next revision of ISO 16610-85 [3].

3.2.5. Local pit height (3.3.6.1)

Definition 3.34. The local pit height is the absolute value of the difference between the height of a pit and the height of its lowest adjacent pass.

Definition 3.34 and definition 3.28 are identical, whereby the term dale local depth replaces the old designation local pit height in ISO 25178-2 [2]. The reasons that motivated this change are described in detail by Blateyron [4]. It can be assumed that the same substitution will be part of the next revision of ISO 16610-85 [3].

3.2.6. Wolf peak height (3.3.7.1)

Definition 3.35. The minimal height difference between a peak and its highest adjacent pass that should be retained by Wolf pruning is termed Wolf peak height.

For a more detailed discussion of the Wolf peak height confer section 3.2.9.

3.2.7. Wolf pit height (3.3.7.2)

Definition 3.36. The minimal height difference between a pit and its lowest adjacent pass that should be retained by Wolf pruning is termed Wolf pit height.

For a more detailed discussion of the Wolf pit height confer section 3.2.9.

3.2.8. Height discrimination (3.3.8)

Definition 3.37. The height discrimination is the minimum of the Wolf peak height and the Wolf pit height.

For a more detailed discussion of the height discrimination confer section 3.2.9.

3.2.9. Wolf pruning (3.3.7)

While pruning (cf section 3.1.29) refers to the elimination of a single peak together with its highest adjacent pass or of a single pit together with its lowest adjacent pass, Wolf pruning denotes the repeated application of a single pruning step as described before. To put it differently, the term Wolf pruning designates the repeated pairwise removal of a peak together with its highest adjacent pass or of a pit together with its lowest adjacent pass, in order to obtain a scale-limited surface, which is less complicated than the original one in the sense that it contains fewer peaks, pits and passes. In this process, the sequence in the elimination of the peaks and pits is determined by their local peak heights (cf section 3.2.4) and local pit heights (cf section 3.2.5), while the number of individual elimination steps depends on the pre-specified value of the height discrimination (cf section 3.2.8).

Wolf pruning can best be described in form of an algorithm. In contrast to the procedure given in section (4.4) of ISO 16610-85 [3], algorithm 1 is formulated in such a way that it is applicable to arbitrary topological data structures, such as change trees, weighted surface networks or Morse-Smale complexes.

Algorithm 1. (Wolf pruning)

1: procedure Wolf pruning

2: Specify the height discrimination

3: while (local peak height $\leqslant$ $\leqslant$ height discrimination) or (local pit height $\leqslant$ $\leqslant$ height discrimination) do

4: Compute the local peak height of all peaks and the local pit height of all pits

5: Identify the peak or pit, which is unequal to the virtual peak and virtual pit and whose local peak height or local pit height, respectively, is minimal

6: Eliminate the respective peak (pit) together with its highest (lowest) adjacent pass; modify the height differences between the peaks (pits) and their adjacent passes accordingly

7: end while

8: Stop

9: end procedure

According to line (5) of algorithm 1, the minimum has to be taken over all local peak heights and local pit heights; in addition, line (5) ensures that the virtual peak or virtual pit, respectively, will not be removed by the pruning process because its elimination would entail the indefiniteness of the study area. As has been demonstrated [8, 18, 69], the elimination of a single peak together with its highest adjacent pass or of a single pit together with its lowest adjacent pass may induce changes in the height differences between the peaks and pits and their adjacent passes; line (6) assures that after each elimination the height differences are modified in order to guarantee the consistency of the model.

For the sake of completeness, it should be mentioned that with respect to segmentation as defined in section (5.3) of ISO 25178-2 [2], algorithm 1 takes only the local peak height (local pit height) into account. If other criteria, such as the other ones specified in section (5.3) of ISO 25178-2 [2] (cf section 3.1.25) should be considered, too, algorithm 1 must be reformulated accordingly. A more general version of algorithm 1, meeting these criteria, was given by Wolf [18], who used instead of the local peak height and the local pit height the more general concept importance of a surface-specific point (relevance of a surface-specific point) and instead of the height discrimination the more general concept desired degree of simplicity.

4. Conclusions

The intention to compose the present paper was twofold. First of all, an attempt was made to demonstrate how the definitions of geometrical feature terms can be put into a Morse theoretic framework in order to take advantage of a powerful mathematical tool to gain new insights into surface structures. Secondly, it was attempted to highlight ambiguities and inconsistencies of those definitions of ISO 25178-2 [2] and ISO 16610-85 [3] that are related to geometrical feature terms. Associated with the latter was the endeavour not only to point out weaknesses, but also to illustrate possible solution strategies.

A final point that should be addressed is to demonstrate how the number of definitions presented in section 3 can be reduced by almost half without losing mathematical accuracy. In this way, the accusation that the presented mathematical concepts are too complex with respect to practical applications can easily be countered.

As a first example, let us consider figure 11 and think of it as a scale-limited surface; the maximal upward path depicted in red is of no importance with respect to the following explanations and can be ignored. While figure 11(a) gives a three-dimensional view of the scale-limited surface, figure 11(c) visualises it from above, thereby showing its course lines and hills (figure 10 visualises the ridge lines and dales in an analogous way). This interpretation, however, is only one possible, namely, the poorer one. A much better interpretation is to consider figure 11(c) not as scale-limited surface, but as the reference surface itself. In this case, not the course lines and hills are displayed, but the separatrices and descending manifolds (figure 10 visualises separatrices and ascending manifolds in an analogous way), whereby nothing of information is lost. The only difference is that instead of the terms peak, pit, pass, course line, ridge line, hill and dale the terms local maximum, local minimum, saddle point, separatrix leading from a local minimum to a saddle point, separatrix leading from a saddle point to a local maximum, descending manifold and ascending manifold have to be used, with this being fully in line with mathematical conventions.

To give a second example, this time by referring to an output as obtained by the MountainsMap^® software. Figure 12(a) gives a three-dimensional view of a scale-limited surface, while figure 12(b) displays it from above, thus showing its peaks, course lines and hills. However, as in the previous example, this is the poorer interpretation with respect to a formal mathematical approach. The much better one is to regard figure 12(b) as the reference surface itself, whereby the entities being visualised are the local maxima, separatrices and descending manifolds, respectively. The same argumentation holds for the results of the pruning process, which are visualised in figure 12(c) (5% pruning) and figure 12(d) (20% pruning) with the percentage referring in both cases to the height difference between the highest and lowest points of the surface.

When redefining the geometrical feature terms as previously described, nothing is lost, but a lot is gained. However, whether these suggestions will be taken up and put into practice, is not in the hands of the author, but in the hands of the members of the respective committees.

Acknowledgments

The assistance of François Blateyron (Digital Surf) for providing the respective graphics is gratefully acknowledged.