1. Introduction
Fractal surface, as a class of fractal sets in the three-dimensional Euclidean space, is an important research object in fractal geometry [
1]. At present, fractal surface has been extensively applied in a variety of academic fields such as metal materials [
2], geology [
3], computer graphics [
4] and so on. One of the most concerned problems is to investigate how to measure the geometric complexity of a fractal surface, like the texture roughness of a metal fracture surface or a computer image surface. Fractal dimension [
5] is a common measure of the geometric complexity of a surface, which can be used to describe its fractal characteristics well. It is well known that the topological dimension of a surface is two. Nevertheless, its fractal dimension increases with larger amounts of complexity or roughness, which is usually greater than its topological dimension. For instance, the fractal dimension of the relief on the earth has been found to be 2.3 in general [
6]. Beyond that, many scholars have used iterative function systems (
) to construct fractal surfaces that are attractors of certain
in fact. More details about fractal surfaces and relevant studies of their fractal dimensions can be found in [
7,
8,
9,
10].
In recent years, exploring the fractal dimension of the graph of fractal curves has drawn the attention of numerous researchers. There are some commonly used definitions of the fractal dimension, such as the Box dimension, the Packing dimension, the Hausdorff dimension and the Assouad dimension, which are denoted as
,
,
and
throughout this paper, respectively. Of the diverse fractal dimensions, the Box dimension mainly considered in the present paper shows its advantage of relatively easy calculation. Up to now, a lot of meaningful work have been done, including fractal interpolation functions [
11,
12,
13,
14],
-Hölder continuous functions [
15,
16], self-similar curves like the Von Koch curve [
17,
18], and some specific fractal functions like the Weierstrass function [
19,
20,
21,
22,
23] and the Besicovitch function [
24,
25,
26]. For more details of latest work, we refer the interested readers to [
27,
28,
29,
30,
31,
32].
Another essential issue involved recently is to estimate the fractal dimension of the superposition of two fractal curves, namely, the sum of two continuous functions of one variable. This problem can be traced back to the research made first by Falconer [
33] who showed that the Box dimension of the sum of two continuous functions equals to the greater of the Box dimensions of them. On this basic, a group of academic workers have pushed this study forward and obtained a series of preliminary conclusions, whose related progress can be found in [
34,
35,
36,
37,
38,
39]. So in this paper, we shall focus on the fractal dimension of the superposition of two fractal surfaces and investigate whether it has the same result with that of fractal curves. Based on three-dimensional Cartesian coordinate system, a fractal surface can be looked upon as a bivariate continuous function, whose fractal dimension and fractional calculus has been established in [
40]. This work will contribute to enriching the theory with regards to the fractal dimension of fractal surfaces and can be applied to the research on fractal characteristics analysis of the superposition of two metal fracture surfaces or two computer image surfaces.
The outline of the remainder of this paper is organized as follows: In upcoming
Section 2, we cover required notations, concepts and results on fractal dimensions of the graph of bivariate continuous functions for subsequent sections. Then in
Section 3, we present our main results obtained in this paper. Firstly, we give the upper bound of the lower and upper Box dimension of the graph of the sum of two bivariate continuous functions. Secondly, we calculate the exact value of the lower and upper Box dimension of the graph of the sum of two bivariate continuous functions under certain particular circumstances. Thirdly, we explore some concrete situations when the two bivariate continuous functions have the Box dimension or not and also consider the case when one of these two functions is Lipschitz. Later in
Section 4, we provide a specific example and do numerical experiments to verify the theoretical results in
Section 3. Finally in
Section 5, we sum up our conclusions and discuss the further research in the future.
2. Preliminaries
In the present paper, all the subjects we discuss are entirely real. Given a non-empty subset
and a bivariate function
, the oscillation of
f over the rectangular region
is written as
and the graph of
on
is defined as
We denote as the function which is always equal to 0 on . Let be the usual Euclidean norm in . For any and , we call a -coordinate cube in .
Below we shall briefly introduce the definition of the Box dimension. For more details about other kinds of fractal dimensions, we consult the interested readers to [
1,
5,
33,
37,
41], for example.
Definition 1 ([
33]).
Let be a bounded subset of and let be the smallest number of δ-coordinate cubes that intersect X. Then the lower and upper Box dimension of X are defined as, respectively,
If the above two are equal, we define the Box dimension of X as the common value, that is,
Remark 1. The notation in Definition 1 can also be replaced by one of the following:
- (1)
The smallest number of sets of diameter at most δ that cover X;
- (2)
The smallest number of cubes of side δ that cover X;
- (3)
The largest number of disjoint balls of radius δ with centres in X;
- (4)
The smallest number of closed balls of radius δ that cover X.
Now we provide some fundamental results, which will be used in subsequent research. The forthcoming two lemmas can be essential approaches to estimating the Box dimension of the graph of a bivariate continuous function.
Lemma 1 ([
33]).
Let .
- (1)
If f is a Lipschitz map, that is,
for and certain . Then
- (2)
If f is a bi-Lipschitz map, that is,
for and certain . Then
Here dim denotes any one of , and .
Lemma 2 ([
33]).
Let be continuous and . Suppose that m and n, respectively, are the least integer greater than or equal to and . Then the range of can be estimated as
where .
Proof. Since
is continuous on
, the estimation of
can be transformed into the oscillation of
on the above subregions. We note that the number of cubes of side length
in the part above the rectangular region
that intersect
is no less than
and no more than
Summing over all the subregions just leads to the present lemma. □
The next proposition reveals several basic properties relating to the fractal dimensions of the graph of a bivariate continuous function.
Proposition 1. Let be continuous. Given a constant , the following three statements hold.
- (1)
- (2)
For a constant bivariate function on , we have
- (3)
Proof. The following arguments for (1)–(3) are all based on Definition 1, Lemma 1 and 2.
- (1)
-
Assume that
. On one hand, it follows from Lemma 2 that
On the other hand, it is observed that
So by Definition 1, we can get
Obviously, we can assert from Definition 1 that , which leads to the conclusion of (1).
- (2)
-
Note that
when
on
. Consequently,
Combining (1) of Proposition 1,
That is,
finishing the proof of (2).
- (3)
Let us define a mapping
by
for
. By using the simple properties of norm, one can show that
and
for
. With Lemma 1, we can claim that
is a bi-Lipschitz mapping and then the result of (3) holds.
□
Remark 2.
In Proposition 1, if the Box dimension of exists on , then
In particular, if , we have
by (2) of Proposition 1. Thus for any continuous function , must be a two-dimensional continuous function on .
Up to now, some particular bivariate continuous functions with non-integer fractal dimensions have been constructed. For instance, Yu [
42] had given the following facts.
Example 1 ([
42]).
For and , let
Example 2 ([
42]).
For , let
where for . If , then and could be any numbers satisfying
3. Main Results
In this section, we present our main results for the fractal dimensions the graph of the sum of two bivariate continuous functions. For two bivariate continuous functions , our motivation is to explore the values of and . According to Definition 1, we can notice that the estimation of is key to calculate them. Hence, we begin by investigating how to attain the range of . The upcoming result about the oscillation is basic.
Theorem 1.
Let be continuous. Then the range of can be estimated as
where have been defined in Lemma 2.
Proof. From Equation (
1), we can obtain
and
Summing over all the rectangular regions in Equation (
3) just leads to the right end of the required inequality. Then combining Equations (
2) and (
3), we estimate
and
Summing over all the rectangular regions in Equation (
4) and using absolute value inequality, one can get the left end of our required inequality as well. □
In the light of Theorem 1 and Lemma 2, seems to have a certain relationship with and . The next important theorem establishes a connection among the above three.
Theorem 2.
Let be continuous. Then the range of can be estimated as
where and .
Proof. It follows from Theorem 1 and Lemma 2 that
and
This concludes the proof of Theorem 2. □
With the help of Theorem 2, we shall prove the following several conclusions. Theorem 3 and 4 give the upper bound of and , respectively.
Theorem 3.
Let be continuous. Then
Proof. Assume that
and
. Given
, by Definition 1 there must exist a certain number
such that
for
. Then by Theorem 2, we get
for
. From Definition 1, we can conclude that
Since the above formula is true for
, we have
which completes the proof of Theorem 3. □
Theorem 4.
Let be continuous. Then
Proof.
From the definition of
, there exists a positive subsequence
such that
and meanwhile
So given
, there exists a
such that
when
. Then by the definition of
, there exists a
such that
when
. Combining Theorem 2, Equations (
5) and (
6), we can obtain
when
. Thus by Definition 1, we have
In the light of the arbitrariness of , we immediately get our required result. □
Under certain particular circumstances, the previous two formulae could take an equal sign, shown in the undermentioned two theorems.
Theorem 5.
Let be continuous. If
Proof. Let
. Without loss of generality, we can assume that
From Theorem 3, it follows that
Then combining Equations (
7) and (
8), we have
which is a contradiction to Theorem 3. Therefore, we can conclude that
This means the conclusion of Theorem 5 holds. □
Theorem 6.
Let be continuous. If
Proof. Without loss of generality, we suppose that
At this time, we know that
From Theorem 4, it follows that
Next, we prove that
as below. By the definition of
and
, given
, there exists a
such that
for
. Note that
and
, thus there exists a
such that
for
. Then by Theorem 2, we estimate
for
. Consequently, one can get
by Definition 1. Since
in the above formula is arbitrary, we have
. Combining Equation (
9), we just obtain the required result. □
Now we shall deal with some concrete examples on the fractal dimensions of the graph of the sum of two bivariate continuous functions. To this end, we first need to state the definition of function spaces as follows.
Definition 2. Spaces of bivariate continuous functions.
- (1)
Let be the space of all bivariate continuous functions whose Box dimension exists and is equal to d on as . Namely, is the space of d-dimensional bivariate continuous functions on .
- (2)
Let as the space of all bivariate continuous functions whose Box dimension does not exist on . Here are the lower and upper Box dimension of the function on as , respectively.
Below we start by the case when the two bivariate continuous functions have the different Box dimension.
Proposition 2.
Let and . If , then
Proof. Without loss of generality, suppose that
. At this time, we observe that
Then it follows from Theorem 5 and 6 that
and
That is,
completing the proof of Proposition 2. □
The upcoming two corollaries discuss a few situations when at least one of two bivariate continuous functions does not have the Box dimension on . These results can easily be obtained from Theorem 5 and 6 with their proofs omitted.
Corollary 1. Let and .
Corollary 2. Let , .
- (1)
If ,
- (2)
If ,
If the two bivariate continuous functions have the same Box dimension d, the result will become more complicate. Here we discuss two situations according to whether d equals to two or not. If , we can arrive at the following two conclusions.
Proposition 3.
Let for . If the Box dimension of exists, then
Proof. Firstly, let
where
is the function given in Example 1 and
could be any number belonging to
by choosing suitable
. Then from Proposition 1 and 2, it follows that
Secondly, let
where
. At this time,
Thirdly, let
. Then we know from Proposition 1 that
According to the above discussion, we just finish the proof of the present proposition. □
Proposition 4.
Let for . If the Box dimension of does not exist, then
Proof. Let
where
is the function given in Example 2 and
could be any numbers satisfying
From Theorem 5 and 6, we can get
and
which implies that
Then by Equation (
10), we just obtain our required result. □
If , the next result manifests that the sum of two two-dimensional bivariate continuous functions can keep the fractal dimension closed.
Theorem 7. Let . Then .
Proof. From Theorem 3, it follows that
Combining (1) of Proposition 1, we obtain
Thus
namely,
. □
In particular, if one of the two bivariate continuous functions is Lipschitz, we have the following assertion.
Theorem 8.
Let be continuous. If g is Lipschitz on , then
where dim denotes any one of , , , , and .
Proof. Let us define a map
by
Since
g is Lipschitz on
, let
For
, on one hand,
On the other hand,
Then by the above two inequalities, we can obtain
where
and
satisfying
. This means that
is a bi-Lipschitz map. With Lemma 1, we just get our required result. □
4. Examples
In this section, we give a concrete example to verify the result acquired in
Section 3.
Example 3.
If , it follows from Corollary 1 that
Now we show several graphs and numerical results for Example 3.
Figure 1 indicates the graph of
when
.
Figure 2 denotes the graph of
.
Figure 3 represents the graph of
. Let
be 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, respectively.
Table 1 presents the corresponding numerical results for the Box dimension of the graph of by using computing methods stated in [
43]. In addition,
Figure 4 supports our theoretical results gained in
Section 3 where the minor deviation may be rendered by the approximation of the computer process.
5. Conclusions
In this last section, we sum up conclusions obtained in this paper.
5.1. Conclusions and Remarks
Throughout the present paper, we mainly make research on the fractal dimensions of the graph of the superposition of two continuous surfaces f and g on with certain lower and upper Box dimensions. Our main conclusions can be summarized as the following several aspects:
- (1)
.
- (2)
.
- (3)
- (4)
- (5)
It has been proved that the superposition of two continuous surfaces cannot keep the fractal dimensions invariable unless both of them are two-dimensional.
- (6)
It has been proved that the fractal dimensions of the graph of the sum of a bivariate continuous function and a bivariate Lipschitz function equals to the fractal dimensions of the graph of the former. That is, a bivariate Lipschitz function can be absorbed by any other bivariate continuous function in the sense of fractal dimensions.
Moreover, it is worth mentioning that the previous results can be extended to any closed regain . In other words, all the results attained in the present paper still hold for two continuous surfaces f and g defined on .
5.2. Applications in Other Fields
In recent years, estimation of the fractal dimensions of the superposition of continuous surfaces has been widely applied in various fields such as metal materials and computer graphics.
In metal materials, the fracture surface topography with regards to the fatigue of metals can be studied by fractal characteristics, which can been found in [
44,
45]. Furthermore, fractal dimension is closely relevant to the parameters of areal surface of metals, which has been shown in [
2]. As is known to all, there exist a good deal of approaches to calculating fractal dimensions and the results under different resolutions and methods will be slightly distinguishing. This work principally investigates how to calculate fractal dimensions by counting boxes and how to estimate fractal dimensions of the superposition of two fractal surfaces, which can just be applied into the research on fracture surface topography regarding to the fatigue of metals.
Besides, in computer graphics, texture roughness is an important visual feature of computer images, which is of great significance to image analysis, recognition and interpretation. A lot of research work has been done on texture analysis and many methods for measuring and describing texture roughness have been proposed (see [
46,
47,
48,
49], for example). Fractal dimension is one of the mostly used tools to describe the texture roughness of image surfaces, namely, the complexity of image gray surfaces, which can be a representation of image stability. The higher the fractal dimension, the more complex the surface, and then the coarser the image. The results in this paper can also contribute to calculating the fractal dimensions of the surface of the superposition of two computer images.
5.3. Further Research
In this paper, there still exist some points worthy of improvement and further consideration in the future. Here we present them and put forward several open questions in the following:
- (1)
-
This work only deals with the cases when the two bivariate continuous functions have the different upper Box dimension and the lower Box dimension of one function is larger than the upper Box dimension of the other one. People could further explore the other situations later.
Question 1.
Suppose that , . What is when and what is when ?
- (2)
-
In the present paper, we only focus on the Box dimension of the graph of the sum of two bivariate continuous functions. So other kinds of fractal dimensions, such as the Packing dimension, the Hausdorff dimension and the Assouad dimension could be further considered for this problem.
Question 2.
Let be continuous. What can , and be, respectively?
- (3)
-
This study is only about bivariate continuous functions, which could be generalized to continuous functions of n variables in the future.
Question 3.
Let be continuous. What can the fractal dimensions of be?
- (4)
-
Apart from addition, people could further investigate the fractal dimensions of the graph of bivariate continuous functions under other operations.
Question 4.
Let be continuous. What can the fractal dimensions of be?
Author Contributions
Conceptualization, X.W.; methodology, X.W.; validation, X.W.; formal analysis, X.W.; investigation, X.W.; resources, X.W.; writing—original draft preparation, X.W.; writing—review and editing, X.W.; funding acquisition, X.W.. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
No data were used to support this study.
Acknowledgments
The authors thank Nanjing University of Science and Technology, for partially supporting this study.
Conflicts of Interest
The authors declare no conflict of interest.
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