The heat equation in relation to smoothing algorithm
Recently I learned about a technique in image processing, which has its roots in something called the 'heat equation' from physics. The original creators of this technique were inspired by the physics of how heat diffuses through an object.
The objective of course is to 'smooth out' the image, for general noise removal. This was done by taking a Gaussian kernel, and convolving it with the image. However, the professor says that the heat-equation is actually a generalization of this process, and in fact, we can get much better techniques using the heat-equation framework.
Essentially, if the original image we have is $I$, then the heat equation framework says that:
$$ I(t) = \nabla \cdot (\ D(x,y) \ \nabla I) $$
where the $\nabla$ means spatial derivative, (I think). The $I(t)$ indicates the image new image at some point in time as it evolves - as heat flows - as this algorithm is run. 749
Finally, the $D(x,y)$ is the "diffusion co-efficient", and if $D=1$, (or any constant), then the above simply collapses to a Gaussian kernel convolving the image $I$.
Now, what I am hoping for is the following: I am hoping someone here can add some intuitive insight into how/why this is working, vis-a-vis a physical analogy to 'heat flow' in the image.
The way I currently understand it, is that we have an image. The bigger the amplitude of certain pixels, the 'hotter' those pixels are. In fact every image pixel is as hot as its amplitude. Now, we also know from physics and entropy, that the heat will try to dissipate, so that eventually, the "temperature" across the image becomes equal. (This I take it, is what is happening when a gaussian convolves the image - this is the 'smearing' we are seeking for removal of noise...).
Now, with the diffusion co-efficient being a constant or 1, this 'heat flow' occurs everywhere. However, if the diffusion co-efficient is, say, a binary function of the spatial image, the the co-ordinates of where $D(x,y)$ equal 0, are where no heat can flow through, and so those are pixels that are spared from the heat flow...
Is my understanding of this physically inspired algorithmic technique correct? Are we really doing nothing but 'simulating' heat flow, except in the frame work of an image? Thank you so much. 749