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Optical Flow

The estimation of optocal flow is used to find the movement between to images taken at two different time. For an image X, we can define each pixel as:

I ( x , y , t )

Assuming that each point of the scene is unchanged over short time, we obtain:

I ( x , y , t ) = I ( x + δ x , y + δ y , t + δ t )

Assuming that the mouvement is little, the developement in Taylor series gives:

I ( x + δ x , y + δ y , t + δ t ) = I ( x , y , t ) + δ I δ x δ x + δ I δ y δ y + δ I δ t δ t + H O T

With HOT the higher order terms, they will be ignored. then we have:

δ I δ x δ x + δ I δ y δ y + δ I δ t δ t = 0

We divide all terms by δ t

δ I δ x δ x δ t + δ I δ y δ y δ t + δ I δ t δ t δ t = 0

So, we find:

δ I δ x V x + δ I δ y V y + δ I δ t = 0

or V x and V y are the component of velocity of optical ow . δ I δ x , δ I δ y e t δ I δ t are the derivatives of the image at x, y et t. They are renamed as I x , I y and I t . That give:

I x ⋅ V x + I y ⋅ V y = - I t

or:

▽ x , y I ⋅ V ⟶ = - I t

The problem is its an equation with two unknown. We have to find a way to estimate the both unknown. We will see different method.

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