Fractal Dimensions in practice

[ali]

Dear All,
when we want to use box-count theorem we have to limit box size to zero
but in practice we face with quantized signal(Images,...) and it is impossible(FD is going to zero) , do you know any practical algorithm to find the best aprozimation of Fractal Dimensions(FD).
Ali

 

[me2please]

Box-counting dimension is one of the most practical methods to estimate the dimension because it's easy to perform.

The dimension is defined as a limit of scale going to 0 but in practice you estimate the dimension from the counting at some various scales. From the definition of the dimension (log count)/(log scale) as the scale is getting smaller, the plot of (log count) v.s. (log scale) is converging to a straight line. Therefore, you could estimate this straight line relationship from the counting at various scales. (The scale doesn't have to be too small though.)

 

[ali]

Dear me2please,
thank you for your answer , would you please you tell me a little more about how to find appropriate sizes in box counting?

 

[me2please]

Appropriate size would depend on the characteristic of your signal. Ideally, you can start at any moderate scale (box size), do the count and reduce the size of the box and repeat.

The part of the plot (log count) v.s. (log scale) that exhibits linear relationship is the appropriate part to use for estimation.

However, in your case the signal is quantized, and that means your signal only contains details up to quantization level. That should be the smallest box size. If the box is smaller than the qunatization level, it should observe the detail of the signal at that scale which is already quantized out. Therefore, if you assume your signal to be fractal, using the box size smaller than the level of observable details could introduce error into the estimation.