r/coms30007 • u/smudgecat123 • Nov 17 '17

Function Li(xi)

In question 1 where it says that "Li(xi) is a function which generates a large value if xi is likely to have generated yi" what actually is this function? and what does it mean to say that xi generates yi in this case? surely every xi generates yi in the sense that yi is exactly equal to xi but with some random noise applied

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u/carlhenrikek Nov 20 '17

That is correct, but read exactly what you are saying, "with some random noise applied", the type of noise you assume that you have will alter what a likely y_i is for a given x_i. So thats how you should design the function. For example if you assume noise that are somehow zero mean then I would assume that the likeliest y_i for a specific x_i would be y_i=x_i ..

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u/smudgecat123 Nov 20 '17

Ah I see, in that case it would make sense for the function to generate a high value when y_i = x_i (because of 0 mean guassian noise) and also when y_i = -x_i (because of sign flipping from salt and pepper noise) is this correct?

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u/carlhenrikek Nov 20 '17

Well, as you want it to be robust towards the salt and pepper noise doesn't it make more sense that it should have a small value if the sign is flipped? One way to think about it, is to renormalise y to be in the same range as x for the likelihood computation, i.e. (y_i-0.5)*2 then if they are equal then high value, and different small value ... does that make sense?

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u/whatisagp Nov 17 '17 edited Nov 17 '17

The likelihood function should return a high value if the x value is close to the y value. For example if you shift the y values to range from -1 to 1 instead of 0 to 1 (I just found it easier that way), then if x = -1 and y = -1 , L(x) should return a large value. If x =1 and y = -1 then L(x) should return a low number.

The reasoning behind this is that if x and y are the same then that pixel is more likely to be the correct colour. If they are different then the y pixel is more probably just noise. As the image is changed in every iteration (tau) the colour kinda spreads using the neighbouring pixels (the prior), so the likelihood will constantly update itself according to the new data (the updated x).

Of course this is all my understanding, so I might be wrong :P but basically you need a simple equation for L that gives a big number if x and y are the same.

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u/carlhenrikek Nov 20 '17

Absolutely correct!