# Global partition of image sets using neighbourhood measures

I have the following general problem,

I presuppose an image is the addition of $K$ other images

$$f(x) = \sum_{k=1}^{K} f_k(x)$$

but I don't know what they are.

At each point in an image I obtain $K$ values that represent each of these images, but the order is unknown. In my particular case, these are orientation values.

$$\mathbf{g} = \{ g_j \mid j = 1 ... K\}$$

At each point, I wish to uniquely assign each value $g_j$ to each of $K$ images, based on the similarity to their neighbours. For example, for a 3x3 image with $K=2$ and $\mathbf{g}$ given by

[0,1]  [0,1]  [1,0]
[1,0]  [1,0]  [1,0]
[0,1]  [1,0]  [0,1]

the two minimal energy sets would be, for some energy function that compares differences,

0 0 0        1 1 1
0 0 0   and  1 1 1
0 0 0        1 1 1

I imagine this would require some kind of iterative optimisation process, but before reinventing the wheel I wonder if any one has come across this problem before and knows a good solution?

• Why are the two sets $3\times 3$? Why not scalars $0$ and $1$ ? Is there a reason there are $K$ values and $K$ sets? Doesn't that just mean each value goes into its own set? Are the three $3 \times 2$ objects values or sets? (I assume values, but it's not clear). – Peter K. Oct 2 '15 at 2:34
• @PeterK. I've updated the question. This for wavelet reconstruction - at each point I can split the wavelet coefficient vector into $K$ component vectors and thus reconstruct $K$ images that when added together give the original image. However, at each point I don't know which of the $K$ component vectors should be assigned to which image for reconstruction, as the order can change. I figure that if I have some kind of energy functional that measures the difference between nearby vectors I use this to assign them. – geometrikal Oct 2 '15 at 3:12
• Can you train it and use supervised knowledge? – Tolga Birdal Oct 2 '15 at 7:21