and 
is written as
where ``*'' denotes the convolution operation.
This is best understood by interpreting x-t as a neighbor of x (as
separated by t) and 
is the weight given to that neighbor.
All of these neighbors are multiplied by their respective weights and the
sum is the output of the convolution at the point x.
For n-variable functions, convolution is written as
Many operations can be modelled by convolution. For example, consider
the probabilities of each possible output for a roll of a single die.
This is a uniform distribution and can be described as a step function
Now consider the probabilities of each output for rolling two dice:
Extending this to n dice
At the limit as n goes to infinity, 
becomes a normal (Gaussian) distribution.
Indeed, a theorem called the Central Limit Theorem states that any
function convolved with itself an infinite number of times becomes a Gaussian.
(No wonder the statisticians like normal distributions so much-if you
get enough samples together, it doesn't matter what the distribution of a
single sample looks like, the overall result is a normal distribution.)
|
In-class exercise:
|
Substituting s = x - t and ds = dx,
and since we may freely change the variable of integration,
Eqs. 7.9 and 7.10 are the Convolution Theorem. Convolution in either domain is equivalent to multiplication in the other.
Question: If the convolution of a unit square pulse with itself is a triangle, what is the Fourier Transform of this triangle? Why?
Answer: 
. It is the square of the Fourier Transform of the unit square.
What this means for imaging systems is that whenever we convolve an image with a kernel, we pointwise multiply the Fourier Transform of the image by the Fourier Transform of the kernel. This attenuates or amplifies each frequency indepently and is called a filter.
Terminology: a convolution function in the spatial domain is a kernel; a pointwise multiplication function in the frequency domain is a filter.
Linear systems are relatively easy to work with, non-linear ones aren't. Most image processing assumes that imaging systems behave linearly, but in reality most imaging systems don't.
Shift invariance is a desirable property for an imaging system-we don't want apples turning into oranges as move them around within the scene. Most imaging systems aren't perfectly shift invariant, but we often treat them as such.
Let us denote this point spread function as the kernel 
with Fourier Transform 
.
Then the image 
of the scene 
is
According to the Convolution Theorem,
the Fourier Transform 
of the image as related to
the Fourier Transform 
of the original scene is given by
In other words, the imaging system simply amplifies or attenuates each
sinusoidal basis function independently.
The function 
is called the Transfer Function (TF) of the system.
If a system involves several stages, each with their own transfer
functions 
, they simply multiply together:
Since multiplication is commutative and associative, so is the
application of a transfer functions. You can think of a pipeline of
separate transfer functions 
or as one big system with an
overall transfer function 
.
This also means that convolution is also commutative and associative.
Questions:
The magnitude part of the Transfer Function is called the Modulation Transfer Function (MPT) and the phase part is called the Phase Transfer Function (PTP).
We'll talk about specific types of filters as throughout the course, but it is important now to understand this classification.
Question: Most imaging systems act like low-pass filters. Can you explain why?
such that
and that we want to eliminate the effects of the system.
We can do this by dividing out the transfer function:
This process is called deconvolution--the undoing of the effects of a convolution. However, as you will see in your second programming assignment, this process is very sensitive to error and noise.
© Bryan S. Morse, 1995
by 
.
by
,
remembering to flip the second list.
