Vision

Linear Image Filter

A linear image filter is a linear transformation mapping continous functions to continuous functions.

Discrete Convolution

Suppose that

X \subseteq ℕ_{1}

and

f, g : X \to ℝ

are functions then we define the function

f * g : X \times X \to ℝ

(f * g) (i, j) = \sum_{k, l \in X} f (k, l) h (i - k, j - l)

Continuous Convolution

Suppose that

X \subseteq ℝ^{n}

and that

f, g : X \to ℝ

are functions then we define the function

f * g

(f * g) (x) = \int_{- \infty}^{\infty} f (λ) h (x - λ) d λ

Linear Shift Invariant Filter

Suppose that a

l

is a linear filter, and suppose that we have

l (f) = g

then we say that

l

is shift invariant when for any

p \in dom (f)

we have

l (f (x - p)) = g (x - p)

Box Function

We define

{box}_{ϵ} = χ_{[- \frac{ϵ}{2}, \frac{ϵ}{2}]}

where we've used the characteristic function

Notice that with the box function we can create unit boxes by considering the functions $\frac{{box}_{α}}{α}$ for $α \in ℝ^{> 0}$

Every Convolution Is a Linear Shift Invariant Filter

As per title.

TODO: Add the proof here.

From the above proposition we can start to reason about what types of filters can or can't be written as a convolution, for example the converse of the above implise that if a filter is not linear and shift invariant then it cannot be a convolution. For example suppose you have an input image and then smooth it by doing the input convolved with a box, then you use a homography matrix to warp the image, if you compare the result with the image which is a result of doing the homography mapping without the smoothing first then it doesn't work, explain more

Dirac Comb

III (t) = \sum_{n = - \infty}^{\infty} δ (t - n T)

Dirac Comb Convolved With a Function

f (t) * III (t) = \sum_{n = - \infty}^{\infty} f (t - n T)

TODO: Add the proof here.