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2 (e.g. on RGB, HSV, HSB space) – the thresholding. The results of fractal analysis (using the box counting method) are box counting fractal dimension and box counting fractal measure. This article describes a new method of determining fractal dimension and fractal measure using integral (wavelet – Haar) transformation. The main advantage of the new method is the ability to analyse grey scale (or colour) images without lengthy thresholding and calculating the box counting fractal dimension D and fractal measure K for each thresholded image in the set. More detailed results of the analysis using this new method can be found on our web pages http://www.fch.vutbr.cz/lectures/imagesci. The interpretation of the results obtained from the black & white image analysis (using the new method) remains the same as from the box counting method of black & white pictures. The main disadvantage of the new method may be seen in smaller set of data used for regression analysis as in box counting method. There can be obtained only 10 independent values per image with 1024 × 1024 pixels matrix that can be used to determine the fractal dimension D and fractal measure K. Such disadvantage can be turned into an advantage as removing redundant use of pixels minimizes the error introduced by reusing the values of wrongly determined values of pixels. 2 Box counting method Analysis of black & white images is usually performed by box counting method [2]. Figure 1 shows how is the method used. A net of different size (1 × 1, 2 × 2, 3 × 3, 4 × 4, etc. pixels) is placed on the analysed image and the total count of black boxes NB, white boxes NW and black&white NBW boxes is determined. Figure 1. Determination of the fractal measure and fractal dimension using the box counting method. The function of the count of pixels on the net size ε = 1/n, where ε is the size of pixels and n is the scale, is used to determine three different types of fractal dimension: black area, white area and their edge (DBBW, DWBW, DBW). To calculate the dimension DBBW and DWBW, the count of black and black & white (BW) N BBW = NB + NBW or white and black & white (BW) N WBW = NW + NBW boxes is utilised. The variation of fractal dimension with the count of boxes can be written
N −D D ( ε ) = K ⋅ε N( n) = K ⋅n , 3 , (1) respectively, where K is the fractal measure and D is the fractal dimension (without their indexes). From these terms the fractal dimension is given by the slope of function (1) d ln N D = − d lnε ( ε ) d ln N( n) = d ln n The area filled by the examined set (points) can be expressed as S ( n) and the length of lines in figure as L ( n) N = n ( n) D−2 2 N = n = K ⋅ n ( n) D−1 = K ⋅ n . (2) respectively. The Figure 2 shows the dependency obtained from the model image of a tree for black & white squares. The image shows that the function is almost linear (structure of the image is fractal). There is clearly a visible error for small sizes of the boxes (introduced by the interpolation of values) and also error for the very big sizes of the boxes (introduced by saturation of the number of boxes). log(N BW) 8 7 6 5 4 3 2 1 0 Square numbers WM BCM -6 -5 -4 -3 -2 -1 0 log(n ) Figure 2. Determination of the fractal measure and fractal dimension using the box counting (see Figure 1) and for wavelet method (see Figure 3 - Figure 5). Acquired fractal measures KBBW, KWBW, KBW can be used for determination of the black – SB (white – SW respectively) areas of threshold figure S B = K BBW KBBW − KBW + K − K WBW BW , W = K and the length of divided lines L of threshold figure S K WBW − KBW + K − K BBW WBW BW (3) (4) (5)
Page 1: FRACTAL ANALYSIS OF THE IMAGES USIN
Page 5 and 6: pixels. Figure 3 shows top left cor
Page 7 and 8: the number of the remaining of pixe

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