Chapter 2 Signals and Systems

Marcus Borengasser8/24/10

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 sometimeseasier 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 combfunction 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 andy 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 signalvariations independently in the x- and y-directions.

Of major importance is the fact that operating on separable signals is muchsimpler 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 systemto each individual input signal.

Systems – Impulse Response

Systems – Shift Invariance

An additional simplifying assumption is shift invariance. A system S is shift-invariant 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. Twotypes 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 withseparable signals, a 2-D LSI system with a point spread function (PSF, orimpulse 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 outputof a 2-D separablesystem by using two 1-Dsteps in cascade.

Systems – Stable Systems

A medical imaging system is stable if small inputs lead to outputs that do notdiverge. Although there are many ways to characterize system stability, onlyconsider 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 shownthat an LSI system is a BIBO stable system if and only if its PSF isabsolutely integrable, in which case

The Fourier Transform (FT)

Besides decomposing a signal into point impulses, an alternative way todecompose a signal is in terms of complex exponential signals. It can beshown that, if

then

The signal F(u,v) is known as the (2-D) Fourier transform of f(x,y), whereasthe signal decomposition below is known as the (2-D) inverse Fouriertransform.

The Fourier Transform (FT)A Fourier transform is a linear transformation that allows calculation of thecoefficients necessary for the sine and cosine terms to adequately representthe 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 asstriping, spots, or vibration in imagery by identifying periodicities (areas ofhigh spatial frequency). Fourier editing can be used to remove regular errors in data such as those caused by sensor anomalies (e.g., striping). Thisanalysis 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 anoptimum image for viewing or editing.

•Each pixel of a Fourier image is a complex number (i.e., it has two componentsreal 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 logarithmicfunction.

•A Fourier image is symmetric about the origin (u,v = 0,0). If the origin isplotted 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 ofdecreasing 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 bemodeled 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,yi(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 imageis 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 themare 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 butsubtracts 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. Thisprocess, called discretization or sampling, means that only the representativesignal 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 rectangularsampling scheme. Although coarse sampling results in fewersamples, it may not allow reconstruction of the originalcontinuous 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, whatIs the smallest period.

So the smallest period is 1 in both x and y directions.

End-of-Chapter ProblemsProblem 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.