Johnson-Mehl process anyone?

[Gabriel Landini]

I am interested in constructing a graph that partitions the plane (image)
based on a number of short seed lines. These lines grow all at the same speed
until they intersect with another line (and they stop) or reach the image
border.

I think that this is similar to a Johnson-Mehl process, but I have not seen
this described in the literature as growing from lines.
Has anybody seen anything like this? If so, does it have a name?

Is there a better approach other than growing all the lines at the same time
(or sequentially 1 small increment at a time) and checking for intersections?

Many thanks

Gabriel

[Michael Schmid]

Hi Gabriel,

if the lines keep their width and don't stop, I guess that you would  
simply have the 1D case of Johnson-Mehl-Avrami-Kolmogorov growth (aka  
Avrami equation, Kolmogorov-Johnson-Mehl-Avrami, KJMA, JMAK, ...).

So, if you have enough nuclei, all starting at a random time, and  
never stopping, I guess that you would simply get an Avrami exponent  
of 2 (1 for linear growth of lines + 1 for the number of lines; the  
latter including phantom lines, i.e., lines that are within other  
lines).

The problem that I see: how to treat lines stopping at intersections  
- no idea. But I am not really an expert on Avrami etc.

Best wishes,

Michael
________________________________________________________________

On 4 Mar 2009, at 17:57, Gabriel Landini wrote:

> I am interested in constructing a graph that partitions the plane  
> (image)
> based on a number of short seed lines. These lines grow all at the  
> same speed
> until they intersect with another line (and they stop) or reach the  
> image
> border.
>
> I think that this is similar to a Johnson-Mehl process, but I have  
> not seen
> this described in the literature as growing from lines.
> Has anybody seen anything like this? If so, does it have a name?
>
> Is there a better approach other than growing all the lines at the  
> same time
> (or sequentially 1 small increment at a time) and checking for  
> intersections?
>
> Many thanks
>
> Gabriel

[Gabriel Landini]

On Wednesday 04 March 2009 18:07:43 Cihat wrote:
> After you have all of your seed lines, you can intersect them with each
> other and the boundaries of the plane as well mathematically and find
> the points of intersection. For a given line, you could list the t
> values for each intersection point and choose the smallest one. If your
> seed line is growing in one direction, the point (x0,y0) should be the
> fixed point which is on the non-growing side [so that your t values will
> always be positive]. If it is growing in both directions, then the
> stopping t value in the list will be the one with the smallest absolute
> value.

@ Cihat:
Thank you, Cihat for the suggestion.
The problem is (i think) that for a particular seed, the nearest intersection
point to any of all other possible lines is not necessarily the first
encountered. The seed of the intersected line could be very far away from the
intersecting line (which could be close).

@ Michael:
> if the lines keep their width and don't stop, I guess that you would  
> simply have the 1D case of Johnson-Mehl-Avrami-Kolmogorov growth (aka  
> Avrami equation, Kolmogorov-Johnson-Mehl-Avrami, KJMA, JMAK, ...).

Thanks! so this has a name (or rather several :-) ). Interesting that I still
cannot find through google an image similar to what I am imagining this would
look like (something like a broken glass).

Thank you again. I will think a bit more about this.

Cheers

Gabriel