Case Study Computed Tomography (CT)

Non-invasive medical imaging techniques

最传统的方法

Standard X-rayLost spatial

informationVery limited use

X-ray

Ultrasound (B超 )

Use very high frequency sound (MHz)Safe Image in real timeEquipment is not

expensive

LimitationStrong sound

absorption by bonesHard in brain studies,

or anywhere with bones

基本原理（物理的）：波的折射与反射

Computed Tomography A parallel sheet of X-ray Mathematical tools needed for successful

computed tomography (CT) imaging Basic principle is discovered more once 1979 Nobel Prize in Physiology and Medicine

was awarded to G. Hounsfield & A.M. Cormack jointly for realizing and bringing to medical use

Computed Tomography

Not expensive to use Some drawbacks

Possible tissue damage from ionization

Low contrasts between different type of soft tissues

Computed Tomography

There are many applications other than medical tomography:Astronomy---binary stars, coronal studyOceangraphy---acoustic probing of ocean

conditionsGeophysics---mantle flow, atmospheric studiesPorous Media---

Hubble Image

Hubble Image

Hubble Image

Hubble Image

Test Object

Model problem

Model problem Coordinate axes satisfy

Radon Transformation

Assume the density function of the test object is given by

The scan data can be written as

Question: how to recover from

Least Square Method To understand the idea

behind, consider3*3 small objectHomogenous blocksThe block densities

either 0 or 1 How to achieve the

reconstruction?

Least Square Method Density 1:

Density 0:

Scan Only 2 Directions

More Directions

How Matrix Forms

Reconstruction with least square

Efficiency

Computed Tomography

冠状动脉