Box-Counting Dimension Revisited: Presenting an Efficient Method of Minimizing Quantization Error and an Assessment of the Self-Similarity of Structural Root Systems

Characterizing of visual objects is an important role in pattern recognition that can be performed through shape analysis. Several approaches have been introduced to extract relevant information of a shape. The complexity of the shape is the most widely used approach for this purpose where fractal dimension and generalized fractal dimension are methodologies used to estimate the complexity of the shapes. The box counting dimension is one of the methods that used to estimate fractal dimension. It is estimated basically to describe the self-similarity in objects. A lot of objects have the self-similarity; fingerprint is one of those objects where the generalized box counting dimension is used for recognizing of the fingerprints to be utilized for authentication process. A new fractal dimension method is proposed in this paper. It is verified by the experiment on a set of natural texture images to show its efficiency and accuracy, and a satisfactory result is found. It also offers promising performance when it is applied for fingerprint recognition.

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A Modified Box-Counting Method to Estimate the Fractal Dimensions

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.58-60.1756 ◽

2011 ◽

Vol 58-60 ◽

pp. 1756-1761 ◽

Cited By ~ 1

Author(s):

Jie Xu ◽

Giusepe Lacidogna

Keyword(s):

Fractal Dimension ◽

Fractal Dimensions ◽

The Self ◽

Whole Body ◽

Counting Method ◽

Border Effect ◽

Self Similarity ◽

Box Counting ◽

Self Similar ◽

Box Counting Method

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

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The quantization dimension of distributions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005357 ◽

2001 ◽

Vol 131 (3) ◽

pp. 507-519 ◽

Cited By ~ 30

Author(s):

KLAUS PÖTZELBERGER

Keyword(s):

Asymptotic Behaviour ◽

Measure Theory ◽

Probability Distributions ◽

Geometric Measure Theory ◽

Quantization Error ◽

Geometric Measure ◽

Quantization Dimension ◽

Box Counting ◽

Box Counting Dimension ◽

Definition Of

We show that the asymptotic behaviour of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension.

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SELF-SIMILARITY OF SATISFIABLE BOOLEAN EXPRESSIONS DECIPHERED IN TERMS OF GRAPH DIRECTED ITERATED FUNCTION SYSTEMS

Fractals ◽

10.1142/s0218348x0800406x ◽

2008 ◽

Vol 16 (04) ◽

pp. 305-315 ◽

Cited By ~ 2

Author(s):

TIAN-JIA NI ◽

ZHI-YING WEN

Keyword(s):

Normal Form ◽

Iterated Function System ◽

Iterated Function Systems ◽

Conjunctive Normal Form ◽

Self Similarity ◽

Boolean Expression ◽

Box Counting ◽

Box Counting Dimension ◽

Iterated Function ◽

Boolean Expressions

The decision problem of satisfiability of Boolean expression in k-conjunctive normal form (kSAT) is a typical NP-complete problem. In this paper, by mapping the whole Boolean expressions in k-conjunctive normal form onto a unit hypercube, we visualize the problem space of kSAT. The pattern of kSAT is shown to have self-similarity which can be deciphered in terms of graph directed iterated function system. We provide that the Hausdorff dimension of the pattern of kSAT is equal to the box-counting dimension and increases with k. This suggests that the time complexity of kSAT increases with k.

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EXPERIMENTAL SIMULATION OF THE FIRE PLUMES: EFFECTS OF AN AIR ENTRAINMENT MODE ON THE SELF-SIMILARITY ZONE

Proceedings of International Symposium on Convective Heat and Mass Transfer in Sustainable Energy ◽

10.1615/ichmt.2009.conv.480 ◽

2009 ◽

Author(s):

Ahmedou Ould Mohamed Mahmoud ◽

Jamel Bouslimi ◽

Rejeb Ben Maad

Keyword(s):

Air Entrainment ◽

The Self ◽

Experimental Simulation ◽

Self Similarity ◽

Fire Plumes

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Speed vs. comfort: Testing the efficacy of a single question for predicting transit users’ route choices

10.31234/osf.io/drj2a ◽

2019 ◽

Author(s):

Joseph John Pyne Simons ◽

Ilya Farber

Keyword(s):

Data Collection ◽

Efficient Method ◽

Choice Task ◽

The Self ◽

Self Report ◽

Model Parameters ◽

Single Item ◽

Alternative Approach ◽

Single Question ◽

Insight Into

Not all transit users have the same preferences when making route decisions. Understanding the factors driving this heterogeneity enables better tailoring of policies, interventions, and messaging. However, existing methods for assessing these factors require extensive data collection. Here we present an alternative approach - an easily-administered single item measure of overall preference for speed versus comfort. Scores on the self-report item predict decisions in a choice task and account for a proportion of the differences in model parameters between people (n=298). This single item can easily be included on existing travel surveys, and provides an efficient method to both anticipate the choices of users and gain more general insight into their preferences.

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CAN AGN FEEDBACK BREAK THE SELF-SIMILARITY OF GALAXIES, GROUPS, AND CLUSTERS?

The Astrophysical Journal ◽

10.1088/2041-8205/783/1/l10 ◽

2014 ◽

Vol 783 (1) ◽

pp. L10 ◽

Cited By ~ 37

Author(s):

M. Gaspari ◽

F. Brighenti ◽

P. Temi ◽

S. Ettori

Keyword(s):

The Self ◽

Self Similarity

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Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽

10.3390/gels7020046 ◽

2021 ◽

Vol 7 (2) ◽

pp. 46

Author(s):

Pedram Nasr ◽

Hannah Leung ◽

France-Isabelle Auzanneau ◽

Michael A. Rogers

Keyword(s):

Fractal Dimension ◽

Crystallization Temperature ◽

Cayley Tree ◽

Fractal Dimensions ◽

Self Similarity ◽

Avrami Model ◽

Cluster Aggregation ◽

Box Counting ◽

Self Assembled ◽

Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

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