Matrices & Terms Systems of Equations Medical Imaging

Matrix Motivation

Katrina Palmer

Appalachian State University

October 29th, 2009

Matrices & Terms Systems of Equations Medical Imaging

Outline

1 Matrices & Terms

2 Systems of Equations

3 Medical Imaging

Matrices & Terms Systems of Equations Medical Imaging

Terminology

Matrices (or one matrix) 8 23 −17 −9

[red greenblue purple

]Element (or entry)Size of a matrixRows & ColumnsOperations with matrices

Matrices & Terms Systems of Equations Medical Imaging

How do matrices relate to systems of equations?

8x + 2y = 53x − y = 4

↔[

8 23 −1

] [xy

]=

[54

]

Matrices & Terms Systems of Equations Medical Imaging

Set up the matrix equation for each of these

5x + 3y = 43x + 2y = 0

3x + 4y = 107x + 9y = 20

5x + 7y + 4z = 13x − y + 3z = 16x + 7y + 5z = 1

x + 2y + 3w = 0y + z + w = 1y + w = 2x + 2y + 2w = 3

Matrices & Terms Systems of Equations Medical Imaging

5x + 3y = 43x + 2y = 0

↔[

5 33 2

] [xy

]=

[??

]

3x + 4y = 107x + 9y = 20

↔[

? 47 ?

] [xy

]=

[1020

]

Matrices & Terms Systems of Equations Medical Imaging

5x + 7y + 4z = 13x − y + 3z = 16x + 7y + 5z = 1

↔

? ? ?3 −1 36 7 5

xyz

=

111

x + 2y + 3w = 0y + z + w = 1y + w = 2x + 2y + 2w = 3

↔

1 2 0 3? ? ? ?0 1 0 11 2 0 2

xyzw

=

0123

Matrices & Terms Systems of Equations Medical Imaging

Solving Methods

What is the solution tox + y = 2−3x + y = 6

?

Matrices & Terms Systems of Equations Medical Imaging

Other Methods?

1 Elimination (Addition)2 Substitution3 Matrices!!

Consider the algebraic solution to the equation 5x = 12.

15 · 5x = 1

5 · 12 or 5−1 · 5x = 5−1 · 12

x = 125

So, the solution to Ax = B is x = A−1 · B

Matrices & Terms Systems of Equations Medical Imaging

example

Use Matrices to solve:

x + 2y + 3w = 0y + z + w = 1y + w = 2x + 2y + 2w = 3

↔

1 2 0 30 1 1 10 1 0 11 2 0 2

xyzw

=

0123

A =

1 2 0 30 1 1 10 1 0 11 2 0 2

, and B =

0123

, then type A−1B

Matrices & Terms Systems of Equations Medical Imaging

What is medical imaging? and what makes it so special?

CT, MRI, PET, fMRI, etc..yarn demo

Matrices & Terms Systems of Equations Medical Imaging

Activity

Material Needed:Small tupperware containersCloth - cut into squaresBlocksNeedles

Matrices & Terms Systems of Equations Medical Imaging

CT details

What is send through the body?What kind of data is collected?

Matrices & Terms Systems of Equations Medical Imaging

Beer’s Law

N Nµ

d

in out

density

(Amt of Energy out) = (Amt of Energy in) · (decay factor)

Nout = Nine−µd (known quantities?)

Matrices & Terms Systems of Equations Medical Imaging

Does it make sense that the object in all one density?

Nout = Nine−(µ1+µ2+µ3)d

N Nµ

d

in out

d d

1 2 3µ µ

How many xrays would I need to take in order to find threedensities?

Matrices & Terms Systems of Equations Medical Imaging

Reconstruction Idea

µ1 + µ2 = 7µ3 + µ4 = 3µ1 + µ3 = 6µ2 + µ4 = 4

Matrices & Terms Systems of Equations Medical Imaging

One possible solution is u1 = 4, µ2 = 3, µ3 = 2, and µ4 = 1

Matrices & Terms Systems of Equations Medical Imaging

Actual Densities

Matrices & Terms Systems of Equations Medical Imaging

CT review

What would you get a CT scan for?Does a CT scan harm you in any way?What information does a CT scan collect?What mathematical tool does CT scan images use?

Matrices & Terms Systems of Equations Medical Imaging

Dog Image

Matrices & Terms Systems of Equations Medical Imaging

Dog Matrix

Matrices & Terms Systems of Equations Medical Imaging

Dog Image

Matrices & Terms Systems of Equations Medical Imaging

Matrices & Terms Systems of Equations Medical Imaging

Matrices & Terms Systems of Equations Medical Imaging