A novel approach to improve the accuracy of the box dimension calculations: Applications to trabecular bone quality

Fractal dimension and specifically, box-counting dimension, is the main tool applied in many fields such as odontology to detect fractal patterns applied to the study of bone quality. However, the effective computation of such invariant has not been carried out accurately in literature. In this paper, we propose a novel approach to properly calculate the fractal dimension of a plane subset and illustrate it by analysing the box dimension of a trabecular bone through a computed tomography scan.


1.
Introduction. There are many systemic diseases that may affect the quality of dental bones. Different studies appeared in the literature try to measure the effect of a particular disease on the trabecular structure. To deal with, the main tool applied in this context is the fractal dimension throughout the box-counting model. Such a quantity allows to detect self-similarity patterns. Thus, if some kind of disease destroys that internal structure, its fractal dimension may vary providing a measure regarding this fact. See for instance [7,8,12,14,17] and [18].
However, the effective calculation of the box dimension has not been carried out in the most accurate manner in empirical applications. In most cases, the analysed images have been treated by a hand computation inspired by the "Archimedes' exhalation method", [1,9,10,13,15] and [16]. It is worth pointing out that the box dimension may be underestimated for great values of the scale δ (c.f. [11]).
Thus, the main goal in this paper is to introduce a novel approach to accurately tackle with the calculation of the fractal dimension avoiding the human error.
Our proposal consists of applying the mathematical approach introduced in Section 2 to collect data that can subsequently be analysed in order to establish connections between different pathologies such as periodontitis, gingivitis, bone density, or even cancer. Another advantage of our mathematical method is that it allows us to study a wide range of different types of images such as cone beam, ortopantomograph, and RVG to name a few. In addition, this method can be also used in all radiographic images.
In Section 3, we shall analyze the fractal dimension of a computed tomography scan image from a periodontitis patient. It is worth pointing out that in [1], the effects of that disease in the bone quality were studied through the box dimension although the conclusions reported no connections between periodontitis and bone density, likely due to a large margin of error in the box method.
2. How to accurately calculate the box dimension of plane subsets. In this section, we shall introduce a novel approach to accurately calculate the box dimension of plane subsets. We would like to point out that our key result appears in upcoming Theorem 2.2 with the construction of the curve α involved therein being explicitly described in later Theorem 2.7.
Definition 2.1. The box dimension of F ⊆ R 2 is given by the (lower/upper) limit where N δ (F ) is the number of δ−cubes that intersect F .
It is worth mentioning that the limit in Eq. (1) can be discretized by δ = 2 −n (c.f. [3, Remark 2.5]). Our first key result stands in the following terms.

Theorem 2.2 ([4]
). There exists a parameterization α : [0, 1] → R 2 of a curve for which the next identity holds regarding the (lower/upper) box dimension of F ⊆ R 2 : Notice that Theorem 2.2 states that the box dimension of any plane subset F can be calculated in terms of the box dimension of its pre-image, α −1 (F ) ⊆ [0, 1], which is a Euclidean 1−dimensional set, up to a constant (the Euclidean dimension). In order to use Theorem 2.2 in empirical applications regarding fractal dimension, our next objective is to properly determine how to construct such a curve α. To tackle this, the concept of a fractal structure will play a key role. We would like to point out that fractal structures allow a deep study of generalized fractal dimension models (c.f. [2]).
First of all, recall that a covering of a set X is a family Γ of subsets such that X = ∪{A : A ∈ Γ}. (i) for all A ∈ Γ n+1 , there exists some B ∈ Γ n such that A ⊆ B.
It is worth pointing out that each Euclidean set can always be endowed with a fractal structure naturally. Its definition can be stated in the following terms.
It holds that the image set of any parameterized Euclidean curve can be always endowed with an induced fractal structure as stated next.  Interestingly, the next result we provide allows us to properly construct a curve α to use Theorem 2.2 in applications.  : n ∈ N} be a starbase fractal structure on a metric space X and ∆ = {∆ n : n ∈ N} a ∆−Cantor complete starbase fractal structure on a complete metric space Y . Further, let α n : Γ n → ∆ n be a family of maps satisfying the two following conditions: • If A ∩ B = ∅ with A, B ∈ Γ n for some n ∈ N, then α n (A) ∩ α n (B) = ∅.
• If A ⊆ B with A ∈ Γ n+1 and B ∈ Γ n for some n ∈ N, then α n+1 (A) ⊆ α n (B). Then there exists a unique continuous map α : X → Y such that α(A) ⊆ α n (A) for all A ∈ Γ n and n ∈ N. In addition, if Γ is Γ−Cantor complete and the two following conditions stand: • α n is onto.
The accuracy of our approach to deal with the calculation of the box dimension of F ⊆ R 2 in terms of the calculation of the box dimension of α −1 (F ) ⊆ [0, 1] (c.f. Theorem 2.2) stands as a consequence of previous Theorem 2.7. It is worth noting that Theorem 2.7 can also be understood as a result allowing the construction of space-filling curves. Next, we illustrate this fact throughout the Hilbert's curve.  Fig. 4 for its first two levels) can be stated in the following terms: This allows to define the whole covering α(Γ 1 ) = {α(A) : A ∈ Γ 1 }. We can proceed similarly with the next levels of ∆. The polygonal in each image of Fig. 4 illustrates how the unit square is filled by α in each level of ∆. Such a recursive procedure allows us to refine the definition of α n in each stage of this construction since additional information regarding the curve is provided as deeper levels are reached. Accordingly, if A ∈ Γ n → B ∈ ∆ n via α n , then in the next level, A = ∪{C ∈ Γ n+1 : C ⊆ A}, B = ∪{D ∈ ∆ n+1 : D ⊆ B}, and each C is sent to D via α n+1 . Letting n → ∞, the Hilbert's curve α stands as the limit of the sequence of maps {α n : n ∈ N}.
The plane-filling nature of the curve provided by Theorem 2.2 allows us to guarantee the accuracy of our box dimension calculations. In fact, since α crosses all the elements in each level of ∆ (resp., α −1 crosses all the elements in Γ), then all the elements in the family {B ∈ ∆ n : B ∩ F = ∅} (resp., all the elements in the set {A ∈ Γ n : A ∩ α −1 (F ) = ∅}}) are considered to tackle the calculation of the box dimension of F (resp., of α −1 (F )). Following the results above, next we To conclude this section, we pose the following Conjecture. Under the same hypothesis as in Theorem 2.2, there exists a parametrized curve α : [0, 1] → R 2 for which the next identity holds: The implications of the conjecture above from the viewpoint of applications are clear: if such a result is true, then the Hausdorff dimension could be enabled to deal with empirical applications. Recall that such a model of fractal dimension is the most accurate since its definition stands in terms of a measure. However, it may be hard to calculate or estimate empirically. Recall also that Theorem 2.7 gives the explicit construction of α, and hence, the Hausdorff dimension of a plane subset F could be calculated by the Hausdorff dimension of α −1 (F ) ⊆ [0, 1] (up to a constant). Finally, the procedure described in [6, Section 3.1] could be used to calculate the Hausdorff dimension of 1−dimensional Euclidean subsets.
3. Calculating the box dimension of periodontal tissues. In this section, we shall apply Eq. (2) to explore the fractal nature of a dental tissue. It is known that periodontal tissues do possess a self-similar nature whose structure may vary between patients having periodontitis disease to healthy subjects. (c.f. Fig. 5). For illustration purposes, the box dimension of a trabecular bone from a periodontitis patient was analyzed according to Corollary 2.10. To deal with this, we considered a high-resolution caption from a computed tomography scan (CBCT, Cone beam).
Self-similarity patterns were detected in a range of 1 − 10 levels by a correlation coefficient equal to 0.97801. The box dimension of the trabecular bone in Fig. 5 (right) calculated according to Eq. (2) was found to be equal to 1.95251.