P-Simple Points and General-Simple Deletion Rules

Abstract

Reductions transform binary pictures only by changing some black points to white ones. Parallel reductions can alter a set of black points simultaneously, while a sequential reduction traverses the black points of a picture, and changes the actually visited single point if the considered deletion rule is satisfied. Two reductions are called equivalent if they produce the same result for each input picture. General-simple deletion rules yield pairs of equivalent topology-preserving parallel and sequential reductions in arbitrary binary pictures. This paper bridges P-simple points and general-simple deletion rules: we show that some deletion rules that delete P-simple points are general-simple, and each point deleted by a general-simple deletion rule is P-simple.

1 Introduction

Let $\mathcal{V}$ be a digital space (e.g., ${\mathbb Z}^2$ or ${\mathbb Z}^3$), where the elements of $\mathcal{V}$ are called points. A digital binary picture [12] on $\mathcal{V}$ is a mapping that assigns only two possible values, black and white to each point. A reduction [7] transforms a binary picture only by changing some black points to white ones, which is referred to as deletion. Reductions play a key role in some topological algorithms, e.g., thinning [7] or reductive shrinking [8].

Parallel reductions can delete a set of black points simultaneously, while a sequential reduction traverses the black points of a picture, and considers the actually visited point for possible deletion at a time. These two strategies are illustrated in Algorithms 1 and 2. Note that Algorithm 2 (i.e., the sequential approach) can specify a unique mapping of pictures to pictures with the help of the input parameter $\varPi$ (i.e., the order in which the points are selected by the foreach loop). For practical purposes we assume that all input pictures are finite (i.e., they contain finitely many black points).

Algorithms 1 and 2 classify the set of black points B in the input picture into two (disjoint) subsets: points in the constraint set C are not taken into consideration (i.e., each element in C is absolutely protected), and the deletion rule R associated with these algorithms is evaluated only for the elements of the set $X=B{\setminus }C$. Note that thinning algorithms can delete some border points [12], hence their constraint sets contains all interior points [12] and some types of endpoints (i.e., points that provide relevant geometrical information with respect to the shape of the objects) [7, 16] or accumulated isthmuses (i.e., generalization of curve/surface interior points) [3].

An investigated black point p is deletable by the deletion rule R, if $R\>(\>p,\>A,\>C\>)=$ true, where $A=B$ in parallel reductions (see Algorithm 1), and $A=SB\subseteq B$ in the sequential case (see Algorithm 2). Thus parallel reductions consider the initial set of black points B when the deletion rule is evaluated. On the contrary, the set of black points is dynamically altered when a sequential reduction is performed. We lay stress upon the fact that both the constraint set C and the set of points $X = B{\setminus }C$ are not modified by Algorithms 1 and 2.

Sequential reductions (with the same deletion rule) suffer from the drawback that different visiting orders (raster scans) of points in $X = B{\setminus }C$ may yield various results. A deletion rule R is called order-independent [21] if the sequential reductions with R produce the same output picture for all the possible visiting orders. By extending this concept: a sequential reduction is said to be order-independent if its deletion rule is order-independent.

The concept of simple point is fundamental in topological algorithms. A (black or white) point is called a simple point if its alteration (i.e., changing its ‘color’) preserves topology [12]. A reduction is said to be topology-preserving if the set of black points in the output picture can be obtained from the set of black points in the input picture by sequential deletions of simple points [12].

Unfortunately, simultaneous deletion of a (sub)set of simple points may alter the topology. That is why Bertrand has proposed the notion of P-simple point [1].

Two reductions are called equivalent if they produce the same result for each input picture [18]. With the help of the notion of general-simple deletion rule, the author showed that general-simple deletion rules yield pairs of equivalent and topology-preserving parallel and (order-independent) sequential reductions [18].

This paper bridges P-simple points and general-simple deletion rules. In Sect. 3, we prove that some deletion rules that delete P-simple points are general-simple, and Sect. 4 shows that each point deleted by a general-simple deletion rule is P-simple.

2 Basic Notions and Results

In this paper we consider the traditional paradigm of digital topology as reviewed by Kong and Rosenfeld [12].

Let $(k,\bar{k})$ be a pair of adjacency relations on digital space $\mathcal{V}$. A $(k,\bar{k})$ digital picture is a quadruple $(\mathcal{V}, k, \bar{k}, B)$, where each point in $B\subseteq \mathcal{V}$ is called a black point, each point in $\mathcal{V}{\setminus }B$ is said to be a white point, k-adjacency/connectivity is used for black points, and $\bar{k}$-adjacency/connectivity is assigned to white points.

A black point p is called an interior point if each point that is $\bar{k}$-adjacent to p is black. A black point is said to be a border point if it is not an interior point.

The concept of simple points is well established in digital topology: A point is called a simple point in a picture if its alteration preserves topology [12].

There are several useful characterizations of simple points in 2D [5, 9–12], 3D [5, 10, 12, 13, 22], 4D [5, 11], and higher dimensional [15] pictures. Since all examples presented in this paper assume 2D digital pictures sampled on ${\mathbb Z}^2$ (that is dual to the regular square grid), we recall the following characterization of simple points on (8, 4) pictures.

Theorem 1

[9] A (black or white) point $p\in {\mathbb Z}^2$ is simple in an (8, 4) picture if and only if p is matched by at least one of the base templates depicted in Fig. 1 or their rotated and reflected versions.

Fig. 1.

Base matching templates for characterizing (8, 4)-simple (black or white) points. Notations: each black element matches a black point; each white element matches a white point; each element depicted in gray matches either a black or a white point; each central element marked a ‘$\bigstar$’ is coincident with the investigated point. Note that interior points are non-simple points.

In order to construct topology-preserving parallel reductions and provide a verification method, Bertrand has proposed the notion of P-simple point [1]. Let us consider a set of points $B\subseteq \mathcal{V}$ and a set of points $P\subseteq B$. Then a point $p\in P$ is P-simple in B if for each set of points $Q\subseteq P{\setminus }\{p\}$, point p is simple in $B{\setminus }Q$.

Bertrand showed that a parallel reduction that deletes only P-simple points is topology-preserving [1], and he gave a local characterization of P-simple points on (26, 6) 3D pictures [2]. Note that Bertrand and Couprie linked critical kernels to minimal non-simple sets and P-simple points [4].

Figure 2 classifies black points of an (8, 4) picture into (non-simple) interior points, non-simple border points, P-simple points, and additional simple points (that are not P-simple points).

Fig. 2.

Classifying black points in an (8, 4) picture. Notations: (non-simple) interior points are marked ‘i’, non-simple border points are marked ‘n’, P-simple points are marked ‘p’, simple (border) points that are not P-simple points are marked ‘s’.

The author established a new sufficient condition for arbitrary pictures with the help of general-simple deletion rules [18]. Let us recall and rephrase the corresponding concepts and results.

Definition 1

Deletion rule R (with respect to the constraint set $C\subseteq B$) is general if $R ( p, B, C ) =$ true for a point $p\in B$, then $R ( q, B, C ) = R ( q, B{\setminus }\{p\}, C )$ for each point $q\in B{\setminus }\{p\}$.

Definition 2

Deletion rule R is general-simple if R is general and it deletes only simple points.

Theorem 2

A deletion rule R yields an order-independent sequential reduction if and only if R is general.

Theorem 3

General deletion rules specify pairs of equivalent parallel and order-independent sequential reductions.

Theorem 4

A parallel reduction is topology-preserving if its deletion rule is general-simple.

Let us summarize Theorems 2–4: if a deletion rule deletes only simple points, and the ‘deletability’ of any point does not depend on the ‘color’ of any ‘deletable’ point, then that deletion rule specifies a pair of topology-preserving equivalent parallel and (order-independent) sequential reductions.

3 From P-Simple Points to General-Simple Deletion Rules

In this section a special deletion rule is constructed that deletes P-simple points, and we verify that our deletion rule is general-simple. Throughout this section, we assume that set B contains all black points in the input picture (see Algorithms 1 and 2).

Let us define constraint set $\mathcal{C}$ as

$$\begin{aligned} \mathcal{C} = \{\ p\ |\ p\ \mathrm{is\ not\ a\ simple\ point\ in}\ B\ \}. \end{aligned}$$

(1)

Let Y be a set of black points in a picture such that $Y\subseteq B$ and $Y\supseteq \mathcal{C}$. Then consider the following set of points

$$\begin{aligned} \mathcal{P}_{\mathcal{C},Y} = \{\ p\ |\ p\in Y\ \mathrm{and}\ p\ \mathrm{is\ a\ simple\ point\ in}\ Y\ \}. \end{aligned}$$

(2)

The deletion rule $\mathcal{PS}$ (with respect to the constraint set $\mathcal{C}$) is defined as

$$\begin{aligned} \mathcal{PS}\>(\>p,\>Y,\>\mathcal{C}\>) = \left\{ \begin{array}{ll} \mathbf{true} &{} \ \text {if}~p~\mathrm{is}~P\text {-simple in}~Y \\ \mathbf{false} &{} \ \text {otherwise} \end{array} \right. \!, \end{aligned}$$

(3)

where $P = \mathcal{P}_{\mathcal{C},Y}$.

Since a P-simple point is necessarily a simple point, $\mathcal{PS}$ deletes only simple points. We show that deletion rule $\mathcal{PS}$ is general-simple.

Let us introduce a relation marked ‘$\Bumpeq$’ between two sets of black points. We say that $B_1 \Bumpeq B_2$ if $B_2$ can be derived from $B_1$ by sequential alteration of simple points. It can readily be seen that ‘$\Bumpeq$’ is an equivalence relation. We can write that $B\Bumpeq B{\setminus }\{p\}$ if $p\in B$ is a simple black point, and $B\Bumpeq B\cup \{p\}$ if $p\not \in B$ is a simple white point.

Lemma 1

Assume that $\mathcal{PS} ( p, B, \mathcal{C} ) =$ true. Then $\mathcal{P}_{\mathcal{C},B{\setminus }\{p\}} = \mathcal{P}_{\mathcal{C},B}{\setminus }\{p\}.$

Proof.

It is obvious that $\mathcal{P}_{\mathcal{C},B{\setminus }\{p\}}\ \subseteq \ \mathcal{P}_{\mathcal{C},B}{\setminus }\{p\}$. Then we prove that $\mathcal{P}_{\mathcal{C},B} \subseteq \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}}\cup \{p\}$.

Let q be a point, such that $q\in \mathcal{P}_{\mathcal{C},B}{\setminus }\{p\}$. In consequence, q is a simple point in B. Since $\mathcal{PS} ( p, B, \mathcal{C} ) =$ true for a point $p\in B{\setminus }\mathcal{C}$, p is a simple point in B. As point p is $\mathcal{P}_{\mathcal{C},B}$-simple in B, p is a simple point in $B{\setminus }\{q\}$. Hence we can write

$$\begin{aligned} \begin{aligned} B&\ \Bumpeq \ B{\setminus }\{q\}, \\ B&\ \Bumpeq \ B{\setminus }\{p\}, \\ B{\setminus }\{q\}&\ \Bumpeq \ ( B{\setminus }\{q\} ) \setminus \{p\}. \end{aligned} \end{aligned}$$

(4)

Since ‘$\Bumpeq$’ is an equivalence relation,

$$\begin{aligned} B{\setminus }\{p\}\ \Bumpeq \ B\ \Bumpeq \ B{\setminus }\{q\}\ \Bumpeq \ ( B{\setminus }\{q\} ) \setminus \{p\} = ( B{\setminus }\{p\} ){\setminus }\{q\}. \end{aligned}$$

(5)

Consequently, $B{\setminus }\{p\}\ \Bumpeq \ ( B{\setminus }\{p\} ){\setminus }\{q\}$, thus q is a simple point in $B{\setminus }\{p\}$. Hence $q \in \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}} \cup \{p\}$.    $\square$

Lemma 1 states that deletion of a point by $\mathcal{PS}$ does not alter the simpleness of the (black) simple points that are not in the constraint set $\mathcal{C}$. Thus we can state that

$$\begin{aligned} \mathcal{P}_{\mathcal{C},B} = B{\setminus }\mathcal{C}, \end{aligned}$$

(6)

and since $\mathcal{PS} ( p, B, \mathcal{C}) =$ true, then

$$\begin{aligned} \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}} = ( B{\setminus }\{p\} ) \setminus \mathcal{C}. \end{aligned}$$

(7)

We are now ready to state the key theorem.

Theorem 5

Deletion rule $\mathcal{PS}$ (with respect to constraint set $\mathcal{C}$) is general.

Proof.

It is to be proved that if $\mathcal{PS} ( p, B, \mathcal{C}) =$ true, then $\mathcal{PS} ( q, B, \mathcal{C} ) = \mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} )$ for any two points $p\in B$ and $q\in B{\setminus }\{p\}$.

Since $\mathcal{PS} ( p, B, \mathcal{C}) =$ true, $p\in \mathcal{P}_{\mathcal{C},B}$ (see Eqs. 1–3, 6). If $q\in \mathcal{C}$, then $\mathcal{PS} ( q, B, \mathcal{C} ) = \mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} ) =$ false (see Eq. 3). Hence, by Eqs. 6 and 7, it is sufficient to consider a point q such that $q\in \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}}$.

• First we prove that if $\mathcal{PS} ( q, B, \mathcal{C} ) =$ true, then $\mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C}) =$ true. We give an indirect proof. Let us assume that $\mathcal{PS} ( q, B, \mathcal{C} ) =$ true and $\mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} ) =$ false for a point $q\in \mathcal{P}_{\mathcal{C},B}$. Thus there is a set $Q \subseteq \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}} \setminus \{q\}$ such that q is not a simple point in $( B{\setminus }\{p\} ){\setminus }Q = B{\setminus }( Q \cup \{p\} )$. Hence we can write

  $$\begin{aligned} B{\setminus }( Q \cup \{p\} )\ \not \Bumpeq \ ( B{\setminus }( Q \cup \{p\} ) ){\setminus }\{q\}. \end{aligned}$$

  (8)

  Since, by Lemma 1, $Q \cup \{p\} \subseteq \mathcal{P}_{\mathcal{C},B}{\setminus }\{q\}$ holds, and $\mathcal{PS} ( q, B, \mathcal{C} ) =$ true, q is simple in $B{\setminus }( Q \cup \{p\} )$. Hence

  $$\begin{aligned} B{\setminus }( Q \cup \{p\} )\ \Bumpeq \ ( B{\setminus }( Q \cup \{p\} ) ){\setminus }\{q\}. \end{aligned}$$

  (9)

  Thus Eq. 9 contradicts Eq. 8.

• Then we prove that if $\mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} ) =$ true, then $\mathcal{PS}(q,B,\mathcal{C}) =$ true. Proving indirectly, let us assume that $\mathcal{PS}(q,B{\setminus }\{p\},\mathcal{C}) =$ true and $\mathcal{PS}(q,B,\mathcal{C}) =$ false. Consequently, there is a set $Q \subseteq \mathcal{P}_{\mathcal{C},B}{\setminus }\{q\}$ such that q is not a simple point in $B{\setminus }Q$. Thus

  $$\begin{aligned} B{\setminus }Q\ \not \Bumpeq \ ( B{\setminus }Q ){\setminus }\{q\}. \end{aligned}$$

  (10)

  There are two cases to be investigated:

  • $p\in Q$ :

    Consider the set $Q' = Q{\setminus }\{p\}$. Then by Lemma 1, $Q' \subseteq \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}}$ holds. Since $\mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} ) =$ true, point q is simple in $(B{\setminus }\{p\}){\setminus }Q'$. Thus

    $$\begin{aligned} (B{\setminus }\{p\}){\setminus }Q'\ \Bumpeq \ ( (B{\setminus }\{p\}){\setminus }Q' ){\setminus }\{q\}. \end{aligned}$$

    (11)

    Since $Q = Q' \cup \{p\}$, Eq. 11 can be rewritten as

    $$\begin{aligned} B{\setminus }Q\ \Bumpeq \ ( B{\setminus }Q ){\setminus }\{q\}. \end{aligned}$$

    (12)

    Thus Eq. 12 contradicts Eq. 10.

  • $p\not \in Q$ :

    In this case $Q \subseteq \mathcal{P}_{\mathcal{C},B}$ and $( Q \cup \{q\} ) \subseteq \mathcal{P}_{\mathcal{C},B}$.

    Since we assumed that $\mathcal{PS} ( p, B, \mathcal{C}) =$ true, point p is simple in sets $B{\setminus }Q$ and $B{\setminus }( Q\cup \{q\})$. Hence the following two equations hold

    $$\begin{aligned} B{\setminus }Q\ \Bumpeq \ ( B{\setminus }Q ){\setminus }\{p\}\ =\ (B{\setminus }\{p\}){\setminus }Q, \end{aligned}$$

    (13)
    $$\begin{aligned} ( B{\setminus }Q ){\setminus }\{q\}\ =\ B{\setminus }( Q\cup \{q\})\ \Bumpeq \ B{\setminus }( Q\cup \{q\}){\setminus }\{p\}\ =\ B{\setminus }Q{\setminus }\{p,q\}. \end{aligned}$$

    (14)

    Since, by Lemma 1, $Q \subseteq \mathcal{P}_{\mathcal{C},B{\setminus }\{p\}}{\setminus }\{q \}$ holds, and $\mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} ) =$ true, point q is simple in $(B{\setminus }\{p\}){\setminus }Q$. Thus

    $$\begin{aligned} (B{\setminus }\{p\}){\setminus }Q\ \Bumpeq \ ( (B{\setminus }\{p\}){\setminus }Q ){\setminus }\{q\}\ =\ B{\setminus }Q{\setminus }\{p,q\}. \end{aligned}$$

    (15)

    Since ‘$\Bumpeq$’ is an equivalence relation, by Eqs. 13–15, we can write

    $$\begin{aligned} B{\setminus }Q\ \Bumpeq \ (B{\setminus }\{p\}){\setminus }Q\ \Bumpeq \ B{\setminus }Q{\setminus }\{p,q\}\ \Bumpeq \ ( B{\setminus }Q ){\setminus }\{q\}. \end{aligned}$$

    (16)

    Thus Eq. 16 contradicts Eq. 10.

We proved that if $\mathcal{PS} ( p, B, \mathcal{C}) =$ true, then $\mathcal{PS} ( q, B, \mathcal{C} ) = \mathcal{PS} ( q, B{\setminus }\{p\}, \mathcal{C} )$ for any two points $p\in B$ and $q\in B{\setminus }\{p\}$.   $\square$

As deletion rule $\mathcal{PS}$ is general by Theorem 5, and it deletes only simple points, $\mathcal{PS}$ is general-simple (see Definition 2). Thus the following theorem is an easy consequence of Theorems 2–5.

Theorem 6

The followings hold for deletion rule $\mathcal{PS}$ (that deletes P-simple points, and considers constraint set $\mathcal{C}$):

• 1. The sequential reduction (see Algorithm 2) with deletion rule $\mathcal{PS}$ is order-independent and topology-preserving.

• 2. The sequential and parallel reductions specified by $\mathcal{PS}$ are equivalent (i.e., they produce the same result for each input picture).

• 3. The parallel reduction (see Algorithm 1) with deletion rule $\mathcal{PS}$ is topology-preserving.

It can readily be seen that if we combine deletion rule $\mathcal{PS}$ with an arbitrary constraint set $C\supset \mathcal{C}$, then Lemma 1, Theorems 5 and 6 hold. Hence we can get various pairs of equivalent and topology-preserving parallel and (order-independent) sequential reductions by considering different geometrical constraints (e.g., characterizations of endpoints [7, 16] and types of isthmuses [3]).

Figure 3 gives an example of a pair of equivalent topology-preserving parallel and order-independent sequential reductions with deletion rule $\mathcal{PS}$.

Fig. 3.

Example of the pair of equivalent topology-preserving parallel and (order-independent) sequential reductions that use deletion rule $\mathcal{PS}$ with respect to constraint set $\mathcal{C}$. Elements in the constraint set (i.e., non-simple points) are marked ‘c’ in the input (8, 4) picture (left), and deleted points are marked ‘d’ in the output picture (right).

4 From General-Simple Deletion Rules to P-Simple Points

In this section we show that each general-simple deletion rule deletes P-simple points.

Theorem 7

If a deletion rule R (with respect to a constraint set $C\subseteq B$) is general-simple, then each point that can be deleted by R is P-simple in B, where $P = \{\ p\ |\ p\in B{\setminus }C\ {and}\ R ( p, B, C ) = {\varvec{true}}\ \}$.

Proof.  Let $p\in P$ be a point such that $R ( p, B, C ) =$ true, and consider a set $Q\subseteq P{\setminus }\{p\}$. It is to be proved that point p is simple in $B{\setminus }Q$.

There are two cases to be investigated:

• $Q=\emptyset$ : Since deletion rule R is general-simple, it deletes only simple points. Hence deletable point p is simple in $B=B{\setminus }\emptyset =B{\setminus }Q$.

• $Q\not =\emptyset$ : The general-simple deletion rule R is general as well (see Definitions 1 and 2), hence it is order-independent by Theorem 2. It means that ‘deletable’ point p remains ‘deletable’ after deletion of some previously visited ‘deletable’ points. Thus $R ( p, B{\setminus }Q, C ) =$ true. Since the general-simple deletion rule R deletes only simple points, point p is simple in $B\setminus Q$.    $\square$

Let us recall the 2D parallel thinning algorithm proposed by Manzanera et al. [14]. That algorithm falls into the category of fully parallel thinning [7] since it uses the same reduction in each thinning phase (i.e., iteration step). The deletion rule $\mathcal{M}$ of that algorithm was given by three classes of matching templates. The base templates $\alpha _1$, $\alpha _2$, and $\beta$ are depicted in Fig. 4. All their rotated versions are templates as well, where the rotation angles are $90^\circ$, $180^\circ$, and $270^\circ$. All elements of $\alpha _1$-type and $\alpha _2$-type are removing templates, while $\beta$ and its rotated versions are preserving templates. A black point is designated to be deleted if at least one removing template matches it, but it is not matched by any preserving template. The constraint set $\mathcal{I}$ comprises all interior points in the input picture of the actual iteration step.

Fig. 4.

The three base templates associated with the deletion rule $\mathcal{M}$. Notations: each black element matches a black point; each white element matches a white point; black elements marked ‘$\bigstar$’ are the central positions of the templates; black elements with white bullets match interior points (i.e., elements in the constraint set $\mathcal{I}$).

In [17] the author proved that the deletion rule of the thinning algorithm proposed by Manzanera et al. [14] is general-simple. Thus that algorithm is topology-preserving, and it is equivalent to an order-independent sequential algorithm that uses the same deletion rule (see Theorems 2–4). By Theorem 7, we can state that the existing thinning algorithm proposed by Manzanera et al. [14] deletes P-simple points from (8, 4) input pictures.

Figure 5 is to illustrate one iteration step of the thinning algorithm in question. It is easy to check that each deletable point by $\mathcal{M}$ is P-simple, where P is the set of deletable points.

Note that Palágyi, Németh, and Kardos proposed a pair of equivalent 4-subiteration 2D sequential and parallel thinning algorithms [19], and four pairs of equivalent 6-subiteration 3D sequential and parallel surface-thinning algorithms [20]. In addition, they showed in [20] that the deletion rule of the 3D parallel surface-thinning algorithm of Gong and Bertrand [6] is general-simple. Since the deletion rules of the 2D and 3D thinning algorithm mentioned above are all general simple, all of these algorithms delete P-simple points by Theorem 7.

Fig. 5.

Example of the pair of equivalent topology-preserving parallel and (order-independent) sequential reductions that use deletion rule $\mathcal{M}$ with respect to constraint set $\mathcal{I}$. These reductions are associated to one iteration step of the fully parallel thinning algorithm proposed by Manzanera et al. [14]. Elements in the constraint set $\mathcal{I}$ (i.e., interior points) are marked ‘c’ in the input (8, 4) picture (left), and deleted points by $\mathcal{M}$ are marked ‘d’ in the output picture (right).

5 Conclusions

This work bridges P-simple points and general-simple deletion rules that specify pairs of equivalent topology-preserving parallel and (order-independent) sequential reductions. On the one hand, we showed that deletion rules that delete P-simple points (with respect to constraint sets containing all non-simple points) are general-simple. Hence parallel reductions with these deletion rules are equivalent to topology-preserving and order-independent sequential reductions. On the other hand, we proved that each point deleted by a general-simple deletion rule is P-simple. Note that the results presented in this work are all valid for digital pictures in arbitrary dimensions.