BMME 560 & BME 590IMedical Imaging: X-ray, CT, and

Nuclear Methods

Introductory Topics Part 1

Today

• Introductions

• About the course

• Introductory Topics– Linear shift-invariant systems in two dimensions– Fourier analysis in two dimensions

• Syllabus

• Assignment 0

Medical Imaging Systems

Hardware

SoftwareApplications

Medical Imaging Systems

Hardware

SoftwareApplications

X-ray sourcesScreen-film detectorsDigital DetectorsGamma Cameras

FilteringCT reconstructionImage quantification

MammographyFluoroscopyFunctional imagingCancer staging

What we will cover

• Imaging Concepts• Physics of radiation• X-ray

– Sources and detectors

– Imaging techniques

– Applications

• Tomographic Reconstruction

• Nuclear Medicine– Radionuclides

– SPECT systems

– PET systems

– Applications

• X-ray CT– Instrumentation

– Applications

There is no MRI, ultrasound, or optical in this course!

What are the fundamentals?

• Understanding the complete imaging system is the synthesis of several disciplines:– Physics– Mathematics– Biology– Chemistry

Prerequisites

• Physics– Atomic structure and basic nuclear physics

• Probability and Statistics– Probability distributions

• Linear Systems– System characterization– Fourier analysis

What is a Signal?

• Most general:– A representation of the change of some number of

dependent variables with respect to changes in some number of independent variables

What is a Signal?

• Usually, we think of– A single independent

variable (time)

– A single dependent variable (voltage, current, pressure, temperature, displacement, …)

0 200 400 600 800 1000 1200-6000

-4000

-2000

0

2000

4000

6000

8000

10000

12000Electroencephalogram Signal

time (msec)

volta

ge (

mv)

What is an image?

• A signal with at least two independent variables– Independent variables are spatial– Dependent variable is intensity

• Intensity of what?

The Real World

• The real world is an image– There are three continuous spatial variables.– There may be other independent variables (time,

wavelength, energy, …)– The dependent variable depends on what you are

measuring.

An Imaging System

• Consider an imaging system that maps the real world “image” onto another, (usually) more convenient image space– To display things that humans cannot see unaided– To record the state of the real world for storage

and retrieval– To permit manipulation of data

SystemReal worldSimpler image space

Systems Concepts

• A system has at least one input and at least one output

• The system is characterized by its mapping of the input to the output, a transformation T[]– Let us consider a system that has 2D images as

input and output

Systemf(x,y) g(x,y) = T[f(x,y)]T[.]

Linear and Nonlinear Systems

• This must be true for a linear system:– Given: A system characterized by T[.]

– If an input f1(x,y) gives output g1(x,y) and an input f2(x,y) gives output g2(x,y),

– Then:

– For all inputs f1(x,y) and f2(x,y)

Systemf(x,y) g(x,y)

Shift-Invariance

• This must be true for a shift-invariant system:– Given: A system characterized by T[.]

– If an input f(x,y) gives output g(x,y)

– Then:

– For all inputs f(x,y)

Systemf(x,y) g(x,y)

Linear, Shift-Invariant Systems

• LSI systems are characterized by their impulse response– In imaging, we call it a point spread function

(PSF).

System

Linear, Shift-Invariant Systems

• Mathematically, this is represented by a 2D convolution

( , ) [ ( , )] ( , ) ( , )

( , )* ( , )

g x y T f x y f m n h x m t n dmdn

f x y h x y

Systemf(x,y) g(x,y)h(x,y)

The Delta Function

• The delta function is also called an impulse function.

• In 2D, this is

• In imaging, we also call it a point source.• Note that

( , ) if = 0 and = 0

= 0 otherwise

x y x y

( , ) 1x y x y

LSI Systems

• Any image can be considered a weighted sum of point sources at different locations.

• Linearity:– The output is the sum of responses to each point

source.

• Shift-invariance– The output of each point source is the point spread

function, shifted to the given location.

Compare two PSFs

• What can you say about these two systems from their PSFs?

System 1 System 2

Which system produced which?

Fourier Transform

2 ( )

2 ( )2

( , ) ( , )

1( , ) ( , )

(2 )

j ux vy

j ux vy

F u v f x y e x y

f x y F u v e u v

Remember: F is a complex quantity!!

What are the units of frequency?

• When the independent variable is time, the units are cycles per second

• When the independent variable is spatial, the units are?

Fourier Transform

• The convolution property of the Fourier transform allows us to model an LSI system in another way:

Systemf(x,y) g(x,y)h(x,y)

f(x,y) g(x,y)[.] 1[.]H(u,v)

System Frequency Response

• The Fourier transform of the PSF is the system frequency response, or transfer function, or Optical Transfer Function (OTF):

• Modulation Transfer Function (MTF)

• Phase Transfer Function (PTF)

( , ) ( , )h x y H u v

( , ) ( , ) / (0,0)MTF u v H u v H

( , ) ( , )PTF u v H u v

Compare Two MTFs

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency, cycles per pixel

MT

F

System A

System B

Zero frequencyIs here

Key Point

• Small features in an image require high spatial frequencies.

• What do we observe from a system with low MTF at high frequencies?

Cascaded Systems

• The frequency response of a series of systems is the product of their individual responses

Systemf(x,y) g(x,y)

f(x,y) g(x,y)

H(u,v,)= H1(u,v) H2(u,v) H3(u,v)

H1(u,v) H2(u,v) H3(u,v)

Cascaded Systems

• Which of the three systems most influences the net response of the cascaded system?

f(x,y) g(x,y)H1(u,v) H2(u,v) H3(u,v)

Today

• Introductions

• About the course

• Introductory Topics– Linear shift-invariant systems in two dimensions– Fourier analysis in two dimensions

• Syllabus

• Assignment 0