Box counting method

Box counting methods. These are simple statistical approaches to describe the mixing of a small... 

This method has been extended to creating rectangular boxes instead of polygons. In this box counting method, >l can be determined as follows ... 

Box counting method is commonly used to obtain mass fractal dimension from an aggregate s projected area. 

One ways of characterization the morphology and aggregation of materials is to determine their sizes. The lignin particles sizes have been studied previously with a variety of methods such as viscometry, sedimentation, and diffusion measurements in the solution state [163], the box counting method applied on a lignin surface [164], and simulations of the ceU wall structure have brought up the idea about fractal lignin [165]. 

The box counting method involves covering an aggregate image with square boxes of different sizes and counting the number of boxes Abox required to do this for each different box length Lbox- If the boxes are made to cover the entire projection of the... 

Growth kinetics data obey the simple empirical equation d = cf" for point cathode and circular anode system d is the radius of the circular envelope grown in time t, and m and log c are the slope and intercept, respectively. Scanned pictures of electro-deposits are characterized in terms of fractal dimension by box counting method. Cathode potential changes with time were also monitored during electro-deposition and dissolution processes. Next amplitude plots indicated that the oscillations were periodic in the binary system while for pure Pb and Zn, it was like random noise. Das et al. [47] developed patterns from pure copper sulphate and zinc sulphate solutions as shown in Fig. 13.21... 

Quantitatively, a self-afRne fractal is defined by the fact that a change Ax XAx (and possibly Ay —> XAy) transforms Az into X Az, where H lies between 0 and 1. The case H — 1 corresponds to a self-similar fractal. Self-affine fractal structures are no longer characterised by just one (mass or boundary) fractal dimension they require two. The first is local and can be determined by the box-counting method, for example it describes the local scale invariance and its value lies between 1 and 2. The second is global and its value is a simple whole number describing the asymptotic behaviour of the fractal. In the case of a mountain, this global dimension is simply 2. When viewed from a satellite, even the Himalayas blend into the surface of the Earth. 

The apparent dendritic-like edges of a seahorse are indicative of its fractal nature. Indeed, a detailed analysis using the box-counting method suggested previously [34] yields A = 5/ , where A is the area belonging to a given pattern enclosed within a radius R centered at the center of the pattern, S = 2.65 0.15 is the shape factor, and d = 1.7 0.1 is the fractal dimension. 

The principle of the box counting method mainly involves an iteration operation to an initial square, whose area is supposed to be 1 and which covers the entire graph. The initial square is divided into four sub-squares and so on. After the n times operations, the number of sub-squares, which contain the discrete points of the profile graph are counted and the length L of the profile is approximately obtained. Then the fractal dimension is calculated as D=l+log L/(n.log2). 

Fractal dimension. This is derived by the box counting method, which uses the surface ar of I as a function of the size of the cubic tessellation. 

One simple way of determining the fractal dimension from an image of a material is to use a box-counting method. The fractal is covered by a grid of boxes of size x and the number of boxes containing mass (Af) counted. Finer and finer grids are used, and N is plotted as a function of X, resulting in a power law behavior that can yield the fractal dimension. 

Both porosity and fractal dimension of the generated clusters show good agreement with the predicted values resulting from (10.22) to (10.24). Slight differences arise from the accomplished random agglomerate creation process. Furthermore, the determination of the fractal dimension by the box counting method [41] has a limited accuracy for relatively low particle numbers. 

---