Objective

2

Fractals have been widely used in image processing

signal processing

physics

biology

system science

medicine

geography

material science

and architecture. Fractals have also become an important tool to describe and study complex and irregular geometric features quantitatively. Fractal dimensions are formally defined as the Hausdorff-Besicovitch dimension. However

estimating fractal dimensions can be conducted in several ways

each of which uses a slightly different definition of the dimension. A few of these methods include the box-counting method (BCM)

wavelet transform

power spectrum (using the Fourier transform)

Hurst coefficient

Bouligand-Minkowski

variation

and capacity dimension methods. Given its high efficiency and easy implementation

BCM has become a widely used method to calculate the fractal dimension of binary images. However

BCM is influenced by many factors

such as range of box sizes

selection of fitting points for calculation

method of box covering

and rotation angle of the image

which lead to the instability of the box-counting dimension. Among the factors that affect the box dimension

the box-counting dimension of binary images change greatly due to image rotation and then leads to the deviation of the box-counting dimension. When the image has rotation

the box dimension calculated by the BCM method is usually smaller than the theoretical fractal dimension. The traditional method (BCM) only has a good estimation of nonrotating images. The estimation of rotating images deviates greatly when the traditional method is used and leads to the large difference in the box-counting dimension of the binary image with different rotating angles of the same object. The average deviation of the box-counting dimension caused by rotation is 3%~5%

and the maximum deviation can reach approximately 8%. To reduce the influence of the image's rotation on the box-counting dimension

the rotation angle of the image must become 0. If the binary image is directly rotated

then the new binary image generated after rotation is inevitably accompanied by interpolation. Although the rotation of the image is corrected

it also causes image deformation and deviation of the box-counting dimension. To avoid rotating the binary images directly

this study proposes a new method to calculate the fractal dimension of binary images.

Method

2

BCM is mostly used to calculate the box-counting dimension of bitmap because the bitmap is relatively easy to obtain

and the box-counting method of vector graphs is rarely studied. Vector graphics have more advantages compared with bitmaps because binary images cannot avoid the thickness problem caused by the pixel size and the interpolation problem caused by the bitmap image rotation. On the contrary

vector images are characterized by no thickness and easy rotation without interpolation. Based on these characteristics

this study proposes a new method called skeleton rotating method to calculate the box-counting dimension. The innovation of this method is the conversion of the binary image into a vector graph and the calculation of the box-counting dimension on the basis of the vector graph. This method mainly includes the following steps: First

the central point of the pixel point of the binary image is regarded as a series of points on the plane. Lines are used to connect the adjacent points on the plane to form a skeleton (vector graph) of the binary image. Then

the minimum containing the rectangle and rotation angle

θ

of the skeleton are calculated using the genetic algorithm. To ensure the accuracy of the calculation results

the genetic algorithm parameters are set as follows: initial number of population

100; mutation probability of offspring

0.1; total generations of reproduction

10 000. Next

the skeleton is rotated by the angle

θ

to obtain a nonrotating vector graph

and the skeleton is covered with boxes of different sizes in a suitable range of box sizes. Simultaneously

we need to record the number of boxes covered with different sizes and the size of the boxes when covering the skeleton to obtain multiple fitting points (-ln

r

ln

N

r

). Lastly

the fitting points are fitted in accordance with the least squares method to obtain the box-counting dimension.

Result

2

In order to compare skeleton rotating method more comprehensively

this paper analyzes and verifies the self-similar image

character scanning image and plant image. In the self-similar image

three different types of self-similar fractal image are calculated (each type contains 45 images with different rotation angles). Compared with the BCM method

the average fitting error (least square fitting error) of the skeleton rotating method was reduced by 0.725 2%

3.060 5% and 2.298 5% respectively

while the variation range (the difference between the maximum and minimum calculated value) decreased by 0.057 3

0.088 3 and 0.085 9

respectively. In the character scanning image

compared with the BCM

the change range of the box dimension of the character scanning image is reduced by 0.012 75 and the average fitting error is reduced by 0.001 28. In the plant image

compared with the BCM

the change of box dimension is reduced by 0.017 04 and the average fitting error is reduced by 0.000 5.

Conclusion

2

The skeleton rotating method converts the binary image into a vector graph on which the estimation of the box-counting dimension is based. The vector graph is used as a basis because it is convenient to rotate and has no thickness. When the vector graph is used for estimating the box-counting dimension

the interpolation problem caused by the rotation of the binary image is avoided

and the influence of the rotation of the binary image on the box-counting dimension is reduced. The results of the skeleton rotating method are better than that of the BCM.