Transcript Computed Tomography

Case Study
Computed Tomography (CT)
Non-invasive medical imaging
techniques
最传统的方法
 Standard
X-ray
 Lost spatial
information
 Very limited use
X-ray
Ultrasound (B超)

Use very high frequency
sound (MHz)
 Safe
 Image in real time
 Equipment is not
expensive

Limitation
 Strong sound
absorption by bones
 Hard 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 study
 Oceangraphy---acoustic probing of ocean
conditions
 Geophysics---mantle flow, atmospheric studies
 Porous 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, consider
 3*3 small object
 Homogenous blocks
 The 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