Chapter 2 Signals and Systems
Marcus Borengasser
8/24/10
Chapter 2 Introduction
The object being imaged is an input signal
– Typically a 3D signal
• The imaging system is a transformation of the input signal to an
output signal
• The image produced is an output signal
– Typically a 2D signal (an image, e.g. an X-ray) or a series of
2D signals (e.g. images from a CT scan)
Chapter 2 Introduction
Input signal: μ(x; y) is the linear attenuation
coefficient for x-rays of a body component
along a line
• Imaging Process: integration over x
variable:
• Output signal: g(y)
Signals – Point Impulse
The concept of a point source in one dimension is known as the
1-D point impulse, which is defined by the following two properties:
The 2-D point impulse is analogously characterized by
Signals – Point Impulse
Signals – Line Impulse
When calibrating medical imaging equipments, it is sometimes
easier to use a line-like rather than a point-like object. For example,
it may be easier to position a wire than a small bead to assess the
resolution of a projection radiography system. For this reason, we
would like a mathematical model for a line source.
The set of points defined by
Is a line whose unit normal is oriented at an angle θ relative to the
x-axis and is at distance l from the origin.
The line impulse associated with line L is given by
Signals – Comb and Sampling Functions
As a first step toward characterizing sampling mathematically, the comb
function is introduced. It is called the comb function because the set of
shifted point impulses comprising it resembles the teeth of a comb.
The 2-D comb is given by
It is useful in signal sampling to space the point impulses in the comb
function by amounts Δx in the x-direction, and Δy in the y-direction.
This yields the samplings function, defined by
Signals – Rect and Sinc Functions
Two signals that are frequently used in the study of medical imaging
systems are the rect and sinc functions. The rect function is given by
The sinc function is given by
Signals – Rect and Sinc Functions
Signals – Sinusoidal Signals
Six instances of the sinusoidal signal s(x,y) = sin[2π(u0x + v0y], 0 ≤ x, y ≤ 1
for various values of the fundamental frequencies u0, v0. Notice that small
values result in slow oscillations in the corresponding direction, whereas
large values result in fast oscillations.
Signals – Separable Signals
Separable signals form another class of continuous signals. A signal f(x,y)
Is a separable signal if there exist two 1-D signals f1(x) and f2(y) such that
A 2-D separable signal that is a function of two independent variables x and
y can be separated into a product of two 1-D signals, one of which is only a
function of x and the other only of y.
Separable signals are limited, in the sense that they can only model signal
variations independently in the x- and y-directions.
Of major importance is the fact that operating on separable signals is much
simpler than operating on purely 2-D signals, since for separable signals,
2-D operations reduce to simpler consecutive 1-D operations.
Systems – Linear Systems
A simplifying assumption is that of linearity. A system S is a linear system if,
when the input consists of a weighted summation of several signals, the
output will also be a weighted summation of the responses of the system
to each individual input signal.
Systems – Impulse Response
Systems – Shift Invariance
An additional simplifying assumption is shift invariance. A system S is shiftinvariant if an arbitrary translation of the input results in an identical translation
in the output
if
Systems – Connections of LSI Systems
LSI systems can be stand-alone or connected with other LSI systems. Two
types of connections are usually considered: (1) cascade or serial connections;
and (2) parallel connections.
Systems – Separable Systems
Separable systems form an important class of LSI systems. As with
separable signals, a 2-D LSI system with a point spread function (PSF, or
impulse response function) h(x,y) is a separable system if there exist two
1-D systems with PSFs h1(x) and h2(y), such that
h(x,y) = h1(x)h2(y),
Calculation of the output
of a 2-D separable
system by using two 1-D
steps in cascade.
Systems – Stable Systems
A medical imaging system is stable if small inputs lead to outputs that do not
diverge. Although there are many ways to characterize system stability, only
consider BIBO stability. A system is a bounded-input bounded-output (BIBO)
stable system if, when the input is a bounded signal – i.e. when
for some finite B - there exists a finite B’ such that
In which case the output will also be a bounded signal. It can be shown
that an LSI system is a BIBO stable system if and only if its PSF is
absolutely integrable, in which case
The Fourier Transform (FT)
Besides decomposing a signal into point impulses, an alternative way to
decompose a signal is in terms of complex exponential signals. It can be
shown that, if
then
The signal F(u,v) is known as the (2-D) Fourier transform of f(x,y), whereas
the signal decomposition below is known as the (2-D) inverse Fourier
transform.
The Fourier Transform (FT)
A Fourier transform is a linear transformation that allows calculation of the
coefficients necessary for the sine and cosine terms to adequately represent
the image.
Because of the computational load in calculating the values for all the sine and
cosine terms along with the coefficient multiplications, a highly efficient version
of the discrete Fourier transform was developed - the Fast Fourier Transform.
The Fourier Transform (FT)
Fourier transformations are typically used for the removal of noise such as
striping, spots, or vibration in imagery by identifying periodicities (areas of
high spatial frequency). Fourier editing can be used to remove regular errors
in data such as those caused by sensor anomalies (e.g., striping). This
analysis technique can also be used across bands as another form of
pattern/feature recognition.
The Fourier Transform (FT)
The raster image generated by the Fourier transform calculation is not an
optimum image for viewing or editing.
•Each pixel of a Fourier image is a complex number (i.e., it has two components
real and imaginary). For display as a single image, these components are
combined in a root-sum of squares operation.
•Since the dynamic range of Fourier spectra vastly exceeds the range of a
typical display device, the Fourier magnitude calculation involves a logarithmic
function.
•A Fourier image is symmetric about the origin (u,v = 0,0). If the origin is
plotted at the upper left corner, the symmetry is more difficult to see than if
the origin is at the center of the image. Therefore, in the Fourier magnitude image,
the origin is shifted to the center of the raster array.
The Fourier Transform (FT)
Three images of
decreasing spatial
variation (from left
to right) and the
associated magnitude
spectra.
The Fourier Transform (FT) - Filtering
Homomorphic filtering is based upon the principle that an image may be
modeled as the product of illumination and reflectance components:
[NO REFLECTANCE ON MEDICAL IMAGES]
I(x,y) = i(x,y) x r(x,y)
where
I(x,y) = image intensity at pixel x,y
i(x,y) = illumination of pixel x,y (dominant at low frequencies)
r(x,y) = reflectance at pixel x,y (dominant at higher frequencies)
The illumination image is a function of lighting conditions. The reflectance image
is a function of the object being imaged. A log function can be used to
separate the two components (i and r) of the image:
ln I(x,y) = ln i(x,y) + ln r(x,y)
The Fourier Transform (FT) - Filtering
Properties of the FT - Linearity
The Fourier transform satisfies a number of useful properties. Most of them
are used in both theory and applications to simplify calculations.
If the Fourier transforms of two signals f(x,y) and g(x,y) are F(u,v) and G(u,v),
respectively, then
where a1 and a2 are two constants. This property can be extended to a
linear combination of an arbitrary number of signals.
Properties of the FT - Translation
If F(u,v) is the Fourier transform of a signal f(x,y), and if
then
In this case,
and
Therefore, translating a signal f(x,y) does not affect its magnitude spectrum but
subtracts a constant phase of 2π (ux0 + vy0) at each frequency (u,v).
Sampling – Sampling Signal Model
Continuous signals must be transformed into collections of numbers. This
process, called discretization or sampling, means that only the representative
signal values are retained. One way to do this is with a rectangular sampling
scheme. According to this scheme, a 2-D continuous signal is replaced by a
discrete signal whose values are the values of the continuous signal at the
vertices of a 2-D rectangular grid.
Δx and Δy are the sampling periods in the x and y directions, respectively.
A coarse and a fine rectangular
sampling scheme. Although
coarse sampling results in fewer
samples, it may not allow
reconstruction of the original
continuous signal.
End-of-Chapter Problems
Problem 2.1 (a) Determine whether the following signal is separable.
Problem 2.2 (b) Determine whether the following signal is periodic and, if it is, what
Is the smallest period.
So the smallest period is 1 in both x and y directions.
End-of-Chapter Problems
Problem 2.5 show that an LSI system is BIBO stable if and only if its PSF is
absolutely integrable.
End-of-Chapter Problems
Problem 2.5 Continued
References
Prince, J.L. and Links, J.M., 2006. Medical Imaging: Signals and Systems,
Prentice Hall, 480 p.