三维空间数字图像的分形维数计算方法

Computation of Fractal Dimension for Digital Image in a 3-D Space

数字图像可以表示为像素位置及对应颜色强度构成的三维空间曲面,该曲面的空间分布反映了图像纹理的特点,在大多数情况下具有分形特征.针对图像的不同区域,用分形维数定量描述了这一图像子区的纹理特点.研究结果表明,计盒维数适于分析较大的图像区域以保证有足够的盒子数目,利用迭代算法提高计盒维数的计算效率,在统计盒子数时可采用差分覆盖、全覆盖等不同的方法;布朗运动维数适于分析较小的图像区域以保证落在无标度区域内,计算布朗运动维数时可通过遍历围绕图像各像素点的正方形边界的右边及下边进行.图像中的纹理越复杂,分形维数也就越大,纹理越简单,分形维数也就越小,相似的纹理具有大致相同的分形维数.

A digital image can be represented in a 3-D curved surface which is displayed by the position of pixels and corresponding color intensity of these pixels. The spatial distribution of the curved surface reflects the feature of image texture and usually appears to be a fractal. The fractal dimension was used to quantitatively describe the feature of image texture in a certain region. The results show that the counting-box dimension is suitable for analyzing larger region of images so that the adequacy of box number is assured. The iterative programming was introduced to improve the efficiency of computation. Difference covering, whole covering and some other covering methods can be adopted during the box-counting process. Brownian-motion dimension is suitable for analyzing smaller region of images so as to be limited in a non-scale range. The pixels in the concerning region are considered one by one and only the right and bottom sides of a certain boundary surrounding each pixel were used during the calculation process of Brownian-motion dimension. The more complex the image texture is, the larger the fractal dimension will be. The simpler the image texture is, the smaller the fractal dimension becomes. The similar image textures have approximately same fractal dimensions.