BMME 560 & BME 590I Medical 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 Software Applications

Medical Imaging Systems Hardware Software Applications X-ray sources Screen-film detectors Digital Detectors Gamma Cameras Filtering CT reconstruction Image quantification Mammography Fluoroscopy Functional imaging Cancer 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, …)

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 System Real world Simpler 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 System f(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 f 1 (x,y) gives output g 1 (x,y) and an input f 2 (x,y) gives output g 2 (x,y), –Then: –For all inputs f 1 (x,y) and f 2 (x,y) System f(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) System f(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 System f(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

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 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: System f(x,y) g(x,y) h(x,y) f(x,y) g(x,y) 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)

Compare Two MTFs Zero frequency Is 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 System f(x,y) g(x,y) f(x,y) g(x,y) H(u,v, )= H 1 ( u,v ) H 2 ( u,v ) H 3 ( u,v ) H 1 (u,v)H 2 (u,v)H 3 (u,v)

Cascaded Systems Which of the three systems most influences the net response of the cascaded system? f(x,y) g(x,y) H 1 (u,v)H 2 (u,v)H 3 (u,v)

Today Introductions About the course Introductory Topics –Linear shift-invariant systems in two dimensions –Fourier analysis in two dimensions Syllabus Assignment 0