matlab and medical image

Matlab and Medical Image Analysis basics / Human Image Perception Kostas Marias... What is MATLAB?• MATLAB is a tool for doing numerical computations with matrices and vectors.. Matl

Matlab and Medical Image

Analysis basics / Human Image

Perception

Kostas Marias

Introduction

What is MATLAB?

• MATLAB is a tool for doing

numerical computations with

matrices and vectors It is

very powerful and easy to use

In fact, it integrates

computation, visualization and programming all together in an easy-to-use environment and

can be used on almost all the

platforms: windows, Unix, and

Apple Macintosh, etc.

• MATLAB stands for Matrix “

Laboratory ”

What is Matlab?

• A software environment for interactive numerical computations

• Examples:

– Matrix computations and linear algebra

– Solving nonlinear equations

– Numerical solution of differential equations

– Mathematical optimization

– Statistics and data analysis

– Signal processing

– Modelling of dynamical systems

– Solving partial differential equations

– Simulation of engineering systems

Matlab Background

•Matlab = Matrix Laboratory

•Originally a user interface for numerical linear algebra routines (Lapak/Linpak)

•Commercialized 1984 by The Mathworks

•Since then heavily extended (defacto-standard)

•Alternatives Complements

Octave (free; GNU) Mathematica (symbolic)Lyme (free; Palm)

Environment

Display Windows

Matlab Workspace

• Variables are stored here in

the form of numerical matrices which can be of different

– All numerical operations need to be

performed on double precision, takes 8 bytes per value

– Other types are efficient for storage (<= 2 bytes)

– Who, whos – current variables in workspace

– Save, load – save or load variables to *.mat

file

– Clear all – clear all variables

– using one number to index: A(8) – result = 6 – matrix is written out as a single column.

– A(:,2) – obtain second column of the matrix

Programming in Matlab

• Script Files

– List of commands to be executed sequentially

Useful when same sequence is executed many times They should be saved with extension script.m and should be placed on work path to be called.

• >>script runs the sequence

• Functions

– Similar to script, but takes arguments

function [output]=funcname(input)

– Any function can be listed with: >>type funcname

• See Matlab editor

(File>New>m-file)

What kind of graphics is possible in Matlab ?

What kind of graphics is possible in Matlab ?

Vectors and Matrices

• Vectors (arrays) are defined as

• >> v = [1, 2, 4, 5]

• >> w = [1; 2; 4; 5]

• Matrices (2D arrays) defined similarly

• >> A = [1,2,3;4,-5,6;5,-6,7]

A and a are two different variables

• Transponate conjugates complex entries; avoided by

• >> B=A.’

Indexing Matrices

•Indexing using parentheses

•>> A(2,3)

•Index submatrices using vectors

of row and column indices

•>> A([2 3],[1 2])

•Ordering of indices is important!

•>> B=A([3 2],[2 1])

•>> B=[A(3,2),A(3,1);A(2,2);A(2,1)]

Indexing Matrices

•Index complete row or column using

the colon operator

Numerical Linear Algebra

• Basic numerical linear algebra

• The number of input/output

arguments can often be varied

• >> [V,D]=eig(A)

• Can change plot properties in

• >> h=plot(x,y); set(h, ’LineWidth’, 4);

• Many other plot functions available

• >> v=1:4; pie(v)

Basic concepts from

Inner (dot) Product

v

w

α

2 2

1 1 2

1 2

( w x x y y x y x y

The inner product is a

The inner product is a SCALAR! SCALAR!

).(

, (

w x1 x2 y1 y2 v w

w v

w

m m m

m

n

a a

a

a a

a

a a

a

a a

a A

3 32

31

2 22

21

1 12

11

m n m

n m

n A B

C × = × + ×Sum:

ij ij

ij a b

c = +

A and B must have the same dimensions

p m m n p

n n

n B B A

A × × ≠ × ×

Identity Matrix:

A AI

0

0 1

0

0 0

m n

T n

T A B B

A + ) = +

(

If AT = A A is symmetric

Determinant: A must be square

32 31

22 21

13 33

31

23 21

12 33

32

23 22

11 33

32 31

23 22

21

13 12

11

det

a a

a

a a a

a

a

a a a

a

a

a a a

a a

a a

a

a a

11 22

21

12 11

22 21

12 11

a a

a

a a

I A

A A

12 22

12 21 22

11

1

22 21

12

a a

a

a a

a a

a a

a

a a

MATLAB works with

essentially only one

An Example

•A = [1, 2, 3; 7, 8,

9]

  Use ; to indicate ‘ ’ the end of each row

to separate elements of

a row

• To make the * , ^ , \ ‘ ’ ‘ ’ ‘ ’

A = [1, 2, 3; 4, 5, 6]

B = [0.5; 1;3]

 C = A * B

v.s.

 C = A * B

Array Operations

•Scalar-Array Mathematics

For addition, subtraction,

multiplication, and division of an array

by a scalar simply apply the operations

to all elements of the array.

Array Operations (con’t…)

• Element-by-Element Array-Array Mathematics.

Algebraic operations in Matlab:

subtraction of the individual elements of the matrices Sometimes it is desired to simply multiply or divide each element of an matrix by the corresponding element of another matrix 'array operations”

Array or element-by-element operations are executed when the operator is preceded

by a '.' (period):

a * b multiplies each element of a by the respective element of b

a / b divides each element of a by the respective element of b

a \ b divides each element of b by the respective element of a

a ^ b raise each element of a by the respective b element

For example, if matrices G and H are G = [ 1 3 5; 2 4 6]; H = [-4 0 3; 1 9 8];

D=G * H = [ -4 0 15

2 36 48 ]

Algebraic operations in Matlab:

•Subscripts: the element

in row i and column j

of A is denoted by A(i, j)

•Example: A = magic(3); A(1,1) + A(1,2) +

A(1,3)

The Colon Operator ‘:’

The colon : is one of MATLAB s ‘ ’ ’ most important operators It

has many formats:

• 3 : -2 : -11 is a row vector containing integers from 3 to –

11 with a increment of 2 –

• Subscript expressions involving colons refer to portions of a matrix:

A(1:3 , 2) is the first to the third elements of the second

column of A.

What does A([1:3], [2:4]) mean?

Working with Matrices:

MATLAB provides four functions that generate basic matrices:

• Zeros: all zeros A =

zeros(1,3)

• Ones: all ones A = ones(2,4)

• Rand: uniformly distributed

•Concatenation: join small (compatible) matrices to make

returns the larger of the number of rows or columns in A.

Matrices (con’t…)

Transpose B = A’

Identity Matrix eye(n)  returns an n x n identity matrix

eye(m,n)  returns an m x n matrix with ones on the main diagonal and zeros elsewhere.

Addition and subtraction C = A + B

C = A – B Scalar Multiplication B = α A, where α is a scalar.

Matrix Multiplication C = A*B

Matrix Inverse B = inv(A), A must be a square matrix in this case.

rank (A)  returns the rank of the matrix A.

Matrix Powers B = A.^2  squares each element in the matrix

C = A * A  computes A*A, and A must be a square matrix Determinant det (A), and A must be a square matrix.

more commands

A, B, C are matrices, and m, n, α are scalars.

Solutions to Systems of Linear Equations

• Example: a system of 3 linear equations with

2 3 1

1 2 3

x x

x x

10

b

Let :

Solutions to Systems of Linear Equations

• Solution by Matrix Division:

The solution to the equation

Ax = b can be computed using left division.

• The polynomials are represented by

their coefficients in MATLAB.

• Consider the following polynomial:

• For s is scalar: use scalar operations

– A = s^3 + 3*s^2 + 3*s + 1;

• For s is a vector or a matrix: use

array or element by element operation

Polynomials (con’t…)

>> s = linspace (-5, 5, 100);

Polynomials (con’t…)

Operation MATLAB command Description

same length.

Polynomial

Multiplication

c = conv(a,b) returns the coefficient vector for the polynomial resulting from

the product of polynomial A and B.

Polynomial

Division

[q,r] = deconv(a,b) returns the long division of A and B.

q is the quotient polynomial coefficient, and r is the remainder polynomial coefficient.

Derivatives polyder(a) returns the coefficients of the derivative of the polynomial A

polyder(a, b) returns the coefficients of the derivative of the product of

Programming with MATLAB:

•Files that contain code

in the MATLAB language

are called M-files You create M-files using a

text editor, then use

them as you would any

other MATLAB functions or command There are two types of M-files: Scripts and Functions.

• Scripts: a bunch of code

grouped together; doesn t accept ’ argument or return output.

• Example: create a file called magicrank.m that calculates

the rank of magic squares:

• Add the file into search path

and type the statement:

magicrank

• The M-file is a text file that

consists a group of MATLAB

commands.

• MATLAB can open and execute the commands exactly as if they were entered at the MATLAB command

window.

• To run the M-files, just type

the file name in the command

window (make sure the current

working directory is set

correctly)

All MATLAB commands are M-files.

So far, we have executed the commands in the command window But a more practical way is to create a M-file

M-files  Functions

• M-files are macros of MATLAB commands that are stored as ordinary text files with the extension m, that is

• Get input by prompting on m-file:

T = input('Input the value of T: ')

•Functions are M-files that

can accept input arguments

and return output

arguments The name of the M-file and of the function should be the same

•For example, the M-file

rank.m is available in the

if … elseif … else … end

for … end

for i = 1:m for j = 1:n H(i,j) = 1/ (i+j)

end

end

Suppressing Output:

If you simply type a statement and press Enter, MATLAB

automatically displays the

results on screen If you end the line with a semicolon ; , ‘ ’

MATLAB performs the computation but does not display any

result.

• Example: C = randn(5,1)

v.s

subplot

Plotting (con’t…)

plot the polynomial using linear/linear scale, log/linear scale, linear/log scale, & log/log scale:

subplot (2,2,1), plot (x,y);

title ('Polynomial, linear/linear scale');

ylabel ('y'), grid;

subplot (2,2,2), semilogx (x,y);

title ('Polynomial, log/linear scale');

ylabel ('y'), grid;

subplot (2,2,3), semilogy (x,y);

title ('Polynomial, linear/log scale');

xlabel('x'), ylabel ('y'), grid;

subplot (2,2,4), loglog (x,y);

title ('Polynomial, log/log scale');

xlabel('x'), ylabel ('y'), grid;

Plotting (con’t…)

Plotting (con’t…)

• Adding new curves to the existing graph:

• Use the hold command to add lines/points to an existing plot.

– hold on – retain existing axes, add new curves to current axes Axes are rescaled when necessary.

– hold off – release the current figure window for new plots

n Grids and

Labels:

Command Description

grid on Adds dashed grids lines at the tick marks

grid off removes grid lines (default)

grid toggles grid status (off to on, or on to off)

title (‘text’) labels top of plot with text in quotes

xlabel (‘text’) labels horizontal (x) axis with text is quotes

ylabel (‘text’) labels vertical (y) axis with text is quotes

text (x,y,’text’) Adds text in quotes to location (x,y) on the current axes, where (x,y) is in

units from the current plot.

Additional commands for plotting

color of the point or curve Marker of the data points Plot line styles

Symbol Line Style

– solid line : dotted line – dash-dot line – – dashed line

3D - Plotting example

• x=[0:10]; y=[0:10]; z=x *y; ’

• mesh(x,y,z); title( 3-D ‘

Graph ); ’

– Type ‘hold on’ to add new plot to current graph

– Type ‘hold off’ to resume stomping

•Make your graph beautiful:

– title(‘apples over oranges’)

– xtitle(‘apples’)

– ytitle(‘oranges’)

Useful Constants

•Inf infinity

•NaN Not and number (div

by zero)

•eps machine epsilon

•ans most recent unassigned answer

•pi 3.14159 …

•i and j Matlab supports

imaginary numbers!

Ανθρώπινη όραση, αντίλήψη ιατρικής εικόνας

Retina - the “focal plane array” on the

back surface of the eye the detects and

measures light.

Photoreceptors - the nerves in the

retina that detect light.

Fovea - a small region in the retina

with high spatial resolution.

Blind spot - a small region in the retina

where the optic nerve is located that

has no photoreceptors.

Long, medium, and short cones - the three specific types of codes used to produce color vision These cones are sensitive to long (L or red) wavelengths, medium (M or green) wavelengths, and short (S or blue) wavelengths.

Αντίληψη εικόνας: Το ανθρώπινο σύστημα

Goals in the field of perception:

– Understand contrast and how humans detect changes in images – Understand photometric properties of the physical world.

– Understand the percept of “color”.

– Learn how to use this understanding to design imaging systems.

Αντίληψη εικόνας: Βασικές Κατευθύνσεις

A Simple Visual Stimulus

A single uniform dot of luminance L in a large uniform background of

luminance LB.

How much difference is necessary for a “standard observer” to notice the

difference between L and LB?

– The just noticeable difference (JND) is the difference that allows an observer

to detect the stimulus 50% of the time.

– ΔJND is the difference in L and LB required to achieve a just noticable

difference.

L

LB

Αντίληψη εικόνας: Βασικά μεγέθη

“Image perception is intimately connected to the psychophysical properties of the visual system of the observer, i.e the ability of the observer to respond to low-contrast and fine-detail stimuli.”

In medical imaging, information is transferred to the observer in two steps:

1 Data acquisition and image formation in the imaging system

2 Processing and display of image data

Visual performance in medical imaging can be divided into three categories:

1 Detection of an abnormality (presence)

2 Recognition of an abnormality (shape and size)

3 Identification (relation to likely disease) of the abnormality

Αντίληψη εικόνας στη Ιατρική

πληροφορίας αλλά και της εμπειρίας του.

Medical Image Analysis

με βάση της οπτικής πληροφορίας αλλά και της εμπειρίας του

Ο παρατηρητής αποφασίζει και παρέχει πληροφορίες σχετικά με

το απεικονιζόμενο αντικείμενο

•The human brain instinctively searches for geometrical

patterns of objects in an image, such as border lines —

preferring well-known figures such as circles, rectangles and triangles — and tries to form those objects

•Τhe viewing distance must be long enough, so that the

pixels cannot be distinguished individually and thus be seen as squares Otherwise, the faculty of vision gives priority to

identification of the square-shaped objects rather than to combining them into a larger object within the image

•The resolution properties of the eye are dependent on

both the contrast level and the light intensity impinging

on the eye The resolution drops from ~7line pairs per mm for film-

lightbox combination (x-rays) to less than 3 for a common CRT monitor!!!

Αντίληψη εικόνας στην Ιατρική

"There are times when an experienced

physician experienced physician sees a

visible lesion clearly and times when he

does not This is the baffling problem,

apparently partly visual and partly

psychologic They constitute the still

unexplained human equation in

diagnostic procedures.” Henry Garland,

M.D., 1959

“In the last ten years, our basic

knowledge of physics of

radiological images has

increased to such an extent that

it cries out to be linked with

observer performance studies.”

Kurt Rossmann, 1974

Αντίληψη εικόνας στην Ιατρική : Γνώμες…