Basic Image Processing
January 26, 30 and February 1
Last week, we mentioned the important fact:
Mathematically, every linear, shift-invariant system S is given by
a convolution.
The idea is to define a function called the point-spread function
(of the system) h(x, y), which are the system’s outputs of the delta
functions
Formally, given a signal (2D signal, image) f(x, y), we can write
The linearity implies that the system’s output commutes with the
integral to give
Its turns out every 2D linear shift-invariant system has the
following functions as its eigenfunctions
S
Recall that
The imaginary and real parts can be interpreted as waves
S
H(u, v) is called the modulation-transfer function.
Fourier Transform
Analogous to expanding a function in the “Dirac” basis:
We can expand the function in the “Fourier” basis
The “coefficient function” is given by the Fourier transform of
f(x, y)
From
We have
H(u , v) is the modulation-transfer function of the system.
What does it do? What is f(0, 0) ?
Spatial domain
Frequency domain
Fourier transform
Domain on
which the
function is
defined
Coefficients of
the wave
components
Inverse Fourier transform
Important Properties of Fourier Transform
1. Fourier Transform of a convolution is a product.
Convolution in the spatial domain becomes multiplication in
the frequency domain.
Using inverse transform we have
Product in the spatial domain becomes convolution
in the frequency domain.
Important Properties of Fourier Transform
2. Raleigh’s theorem (Parseval’s theorem)
Important Properties of Fourier Transform
3. Derivatives and Fourier Transform
Differentiation accentuates (make stroner) high-frequency
components and suppresses the low-frequency components.
Laplacian
What is the Fourier transform of the Laplacian?
Image Noise
Modeling: Noise is usually assumed to be additive and random
The observed intensity is the sum of the true intensity and a
spurious and random signal.
Signal-to-noise ratio, or SNR
Ratio between std of signal and noise
Noise Modeling
In the absence of information, the noise n(i, j) is usually modeled
By a white, Gaussian, zero-mean stochastic proces.
is treated as a random variable, distributed
according to a zero mean Gaussian distribution
function of fixed standard deviation.
Gaussian distribution
Noise Modeling
Impulsive noise (or peak noise): occur usually in addition to
the one normally introduced by acquisition.
x and y are two (uniformly distributed) random
variables with range [0, 1].
Original image
Salt and pepper noise added
Filtering and Smoothing
Problem: Given an image I corrupted by noise n, attenuate n
as much as possible (ideally, eliminate it altogether) without
change I significantly.
Linear Filter: Convolving the image with a constant matrix,
called mask or kernel.
Mean Filter
If the entries of A are non-negative, the filters performs average
smoothing.
Mean filter: take A to be the following matrix (with m=3)
Effect: replaces a pixel value with the mean of its
neighborhood.
Intuitively, averaging takes out small variations: averaging
m^2 noise values divides the standard deviation of the noise
by m.
The Fourier transform of a 1-D mean filter kernel of with 2W is
the sinc function
Signals frequencies falling inside the main lobe are weighted
more than the frequencies falling in the secondary lobes, the
mean filter can be regarded as an approximate “lowpass”filter.
Gaussian smoothing: Kernel is a 2-D Gaussian.
Fourier transform of a Gaussian is still a Gaussian and no
secondary lobes. This makes the Gaussian kernel a better
low-pass filter than the mean filter.
Another important fact about Gaussian kernel: It is
separable.
Another important fact about Gaussian kernel: It is
separable.
In practice, this means that the 2D convoluation can be
computed first by convolvingg all row and then all columns
with a 1-D Gaussian have the same standard deviation.
To be continued …