We already talk about it very early on in this class.

So, one penalty is the mean square error. We want to get the piece wise smooth

version of this image, but not very far from it.

Now, you might wonder, okay, if you're going to penalize for the mean

square error, why not to keep the image itself?

That's very clever, but we don't achieve any segmentation.

So, how do I know I don't achieve any segmentation?

Because I'm going to add a term that penalizes for having too many boundaries

in the segmented image. So, these are the regular edges of this

image. Very strong edges.

These are boundaries of the segments that we have here.

I don't want to have too many. I don't want to have all these.

I'm going to penalize for the number of edges for how many, how much do I pay for

having boundaries. So that would be another term.

I'm going to basically, on one hand, I want to penalize for very large

differences, and on the other hand, I'm going to penalize for edges.

So, if you have too much edges, you are going to pay a price.

Bec, so, if I keep the image as it is, I have no error here but I have lot of

edges. Basically, every pixel becomes a segment,

so I'm paying a very large penalty for edges.

If I have a flat image, I don't pay any penalty for edges, but I pay a very high

penalty for error. And then, I have to do a compromise

between these two and that's what Mumford-Shah basic concept is, to write

formulations that compromise between a representation of the image that is too

far from the original image. We want to simplify representation not

too far from the original image, and we also don't want to pay a high price and

get too many segments. Now, there are many ways of doing this.

There's some beautiful mathematical theory behind different formulations that

do this compromise. Some theory relates even to compression.

You have to compress this image and this edges, and then you try to optimize for

that, yours and my, need to compare for the

error. And some very beautiful techniques with, in the framework of what's called

variation and formulations, and, energy formulations really, a lot of very

beautiful mathematical theory, which actually relates to the mathematical

theory that we're going to be discussing next week when we talk about geometric

differential equations and geometric varation of problems.

But, here's the concept.

And very often in image processing, you have a concept and the multiple ways

of implementing that concept. And I want to make sure that you

basically learn during this class, the concept behind Mumford-Shah.

I should also mention to you that this concept of approximation and penalty for

too many edges applies also beyond image segmentation.

And people have extended the framework of Mumford-Shah type of formulations to

image registration and many, many other image and video processing problems.

Thank you very much. See you in the next video.