MATLAB Fundamentals-An Overview UGC ASC Burdwan Univ Sept2013

MATLAB® Fundamentals: MATLAB® Fundamentals: An OverviewAn Overview

byDr Amitava ChatterjeeDr. Amitava Chatterjee

Electrical Measurement and Instrumentation Laboratory,Electrical Engineering Department,

J d U i it K lk t I diJadavpur University, Kolkata, India.URL: http://www.jaduniv.edu.in/profile.php?uid=322

Short Term Course on Computational Tools and Open Source Software,

UGC-Academic Staff College, Burdwan University, Burdwan, India.02/09/2013

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MATLAB®1 – A Bird’s Eye ViewWh t i MATLAB??What is MATLAB??

MATLAB is an interactive software package which was developed to perform numerical calculations on vectors and matrices.

The name MATLAB stands for MATrix LABoratory. MATLAB was written originally to provide easy access to matrix software developed by the LINPACK (linear system package)and EISPACK (Eigen system package) projects matrices.

This software package has been commercially available since 1984and is now widely accepted as a standard tool at most universities and industries worldwide.

In recent times, MATLAB has evolved as an excellent tool for

1MATLAB® is a registered trademark of MathWorks, Inc.

teaching and research. AMITAVA C

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MATLAB – A Bird’s Eye ViewWh MATLAB??Why MATLAB??

MATLAB facilitates easy handling of many of the computations involved. In fact, the general feeling is that MATLAB should not be viewed as another complicated programming language, but more as a powerful calculator. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning.y q g It has powerful built-in routines that enable a very wide variety of computations. Specific applications are collected in packages referred to as p pp p gtoolbox. It contains a wide variety of toolboxes which allow it to perform a wide range of applications from science and engineering. Thea wide range of applications from science and engineering. The toolboxes range from statistics to simulation, to control systems, to signal processing, to optimization, and so on.

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MATLAB Fundamentals

The graphical interface to the MATLAB® workspace.

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MATLAB FundamentalsSt ti MATLABStarting MATLAB

THE COMMAND WINDOWTHE COMMAND WINDOW

THE COMMAND HISTORY

THE WORKSPACE

THE CURRENT DIRECTORY

Some Useful Commands to Get Started Some Useful Commands to Get Started

>> demo

>> help <command>

>> doc <command>

>> lookfor <command>AMITAVA C

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MATLAB FundamentalsSimple Arithmetic Operationsp p

Use it as a calculator:

>> 4.23*6.8 + 25/6

You can enter 345 as:

>> 3^45

You can enter a complex number 4 + 6i in MATLAB as:

>> 4 + 6i or >> 4 + 6*i

Write on MATLAB command:

>> x = 4 + 5*7

Change the previous command:

>> x = 4 + 5*7; What is the difference ??

Th i th dThen issue the command:

>> disp(x)AMITAVA C

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MATLAB FundamentalsVector and Matrix Operationsp

Create a Row Vector (one dimensional array):

>> ur = [5 7 9 5 18 5];[ ]

Verify the size of the Row Vector:

>> size(ur)

Create a Column Vector (one dimensional array):

>> uc = [12; 8; 21; 6; 35; 67];

Verify the size of the Column Vector:

>> size(uc)

Perform Matrix Multiplication:

>> um1 = ur*uc or >> um2 = uc*ur

V if th i f th M t iVerify the size of the Matrix:

>> size(um1) or >> size(um2) AMITAVA C

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MATLAB FundamentalsVector and Matrix Operations

Create a 5-by-3 Matrix:

>> m1 = [5 7 9; 5 18 5; 6 23 56; 54 6 70; 21 4 65]

Extract only the first row of Matrix m1:

>> row1 = m1(1,:)

Extract only the second column of Matrix m1:

>> col2 = m1(:,2)

Extract a submatrix of m1 comprising elements from 2nd andExtract a submatrix of m1 comprising elements from 2nd and 3rd rows and 1st and 2nd columns only:

>> subm1 = m1(2:3,1:2)

Transpose m1:

>> m1t = m1

Create a 5-by-5 Matrix comprising all zeros or all ones:

>> xz = zeros(5,5) or xo = ones(5,5) AMITAVA C

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S El t C d f M t i d ASome Elementary Commands for Matrices and Arrays

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Matrices and Arrays: Operations and ManipulationAMITAVA C

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Matrices and Arrays: Matrix Analysis and Linear EquationsAMITAVA C

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Variables and NumbersMATLAB Fundamentals

MATLAB does not require any type declarations or dimension statements. When MATLAB encounters a new variable name, it automatically creates the variable and allocates the appropriate amount of storage.

In MATLAB, X and x are not the same variable.

MATLAB by default displays only 4 decimals in the result of the

Issue the command: >> xx = 125.6578

calculations, but it actually performs numerical calculations in double precision, i.e. in 15 digits.

Issue the command: >> xx 125.6578

Study how the format command changes the display:

>> format longg

Then issue the next command: >> xx

: Again revert back the display format : >> format short

Then verify the display of the same variable: >> xx

Study the commands, whos and who. Is there any difference ??

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The Colon OperatorMATLAB Fundamentalsp

The colon (:) is one of the most important MATLAB operators. It occurs inseveral different forms.

If you want to create a row vector of integers with unit spacing, issue the command: >> var1 = 1:10

If you want a similar array with nonunit spacing specify theIf you want a similar array with nonunit spacing, specify the increment (either positive or negative). Study the commands:

>> var2 = 3:6:33 or >> var3 = 33:-6:3

The spacing can be floating point numbers too:

>> var4 = 5:0.25:15 or >> var5 = 15:-0.25:5 or >> var6 = 0:pi/6:pi

You can delete a row or a column from a matrix using “:” and “[ ]”. You can delete the second column of a matrix as:

>> Xd = m1>> Xd = m1

>> Xd(:,2) = [ ]AMITAVA C

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Some Useful CommandsMATLAB Fundamentals

f CTo clear the Command Window, type clc

To abort a MATLAB computation, type ctrl-cp , yp

To continue a line, type . . .

To delete a variable X, type clear X

To delete all variables from the work space, type clear all

To erase any statement from the command line, type ctrl-u

To turn WARNINGS issued by MATLAB on/off type warning on/offTo turn WARNINGS issued by MATLAB on/off, type warning on/off

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Save and Load CommandsMATLAB Fundamentals

Workspace variables do not persist upon exiting MATLAB. So one needs to store data for later use using save command. You can save the entire workspace as a binary MAT file:

>> save myworkspace.mat

Next time when you re-open MATLAB, you can load the entire k i l t dworkspace previously stored:

>> load myworkspace.mat

Save a variable as a MAT file:Save a variable as a MAT file:

>> save var1

Load this variable from the MAT file:Load this variable from the MAT file:

>> load var1

Similarly save a variable in a file in ASCII format and load a file, already stored in ASCII format, in MATLAB workspace:

>> save var1.dat var1 -ascii and >> load var1.datAMITAVA C

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Plot Commands (two-dimensional)MATLAB Fundamentals

Plot the graph of e-0.2xsin(x) over the domain 0 to 6.>> x = 0:pi/20:6*pi;>> plot(x,exp(-0.2*x).*sin(x),'r'),gridp ( , p( ) ( ), ),g>> axis([0 20 -0.4 0.8])>> xlabel('time') >> ylabel('amplitude')>> title('Plot of the Exponentially Damped Sine Function') title( Plot of the Exponentially Damped Sine Function )

0.6

0.8Plot of the Exponentially Damped Sine Function

0 2

0.4

itude

0

0.2

ampl

0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

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Plotting multiple data sets (or functions) in one plot. >> x = 0:pi/20:6*pi;>> x = 0:pi/20:6*pi;

>> y1 = 5*cos(x);

>> y2 = 0.5*sin(x); y ( );

>> y3 = 2.5*cos(x);

>> plot(x,y1,'r--',x,y2,'b-',x,y3,'k:')

>> axis([0 18 -5 5])

>> xlabel('0 \leq x \leq 6\pi')

>> ylabel('Cosine functions')>> ylabel( Cosine functions )

>> legend('5*cos(x)','0.5*sin(x)','2.5*cos(x)')

>> title('Typical example of multiple plots')

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Plotting multiple data sets (or functions) in one plot. >> x = 0:pi/20:6*pi; >> y1 = 5*cos(x); >> y2 = 0.5*sin(x); >> y3 = 2.5*cos(x); …..

4

5Typical example of multiple plots

5*cos(x)0 5*sin(x)

>> title('Typical example of multiple plots')

2

3

s

0.5 sin(x)2.5*cos(x)

-1

0

1

osin

e fu

nctio

ns

-3

-2

Co

0 2 4 6 8 10 12 14 16 18-5

-4

0 x 6

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Plotting multiple data sets (or functions) using subplot. >> x = 0:pi/20:6*pi; >> y1 = 5*cos(x);>> y1 = 5*cos(x); >> y2 = 0.5*sin(x); >> y3 = 2.5*sqrt(x); >> y4 = 12*exp(-0.5*x);>> subplot(2,2,1), plot(x,y1,'r--') >> subplot(2 2 2) plot(x y2 'b-')>> subplot(2,2,2), plot(x,y2, b )>> subplot(2,2,3), plot(x,y3,'k:') >> subplot(2,2,4), plot(x,y4,'m-.')>> subplot(2,2,1), axis([0 18 -5 5]) >> subplot(2,2,1), xlabel('0 \leq x \leq 6\pi') >> subplot(2,2,1), ylabel('Cosine function') p ( , , ), y ( )>> subplot(2,2,1), legend('5*cos(x)')>> subplot(2,2,1), title('Typical example of a subplot-1') …

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Plotting multiple data sets (or functions) using subplot. >> x = 0:pi/20:6*pi; >> y1 = 5*cos(x); >> y2 = 0.5*sin(x); >> y3 = 2.5*sqrt(x); >> y4 = 12*exp(-0.5*x); >> subplot(2,2,1), plot(x,y1,'r--') >> subplot(2,2,2), plot(x,y2,'b-') …

5

n

Typical example of a subplot-1 0.5

Typical example of a subplot-2

5*cos(x) 0.5*sin(x)

0

Cos

ine

func

tion

0

Sin

e fu

nctio

n0 5 10 15

-5

0 x 6

C

0 5 10 15

-0.5

0 x 6

Typical example of a subplot-3 Typical example of a subplot-4

5

10

e ro

ot fu

nctio

n

5

10

entia

l fun

ctio

n2.5*sqrt(x) 12*exp(-0.5*x)

0 5 10 150

Squ

are

0 x 6

0 5 10 15

0

0 x 6

Exp

one

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Plot Commands (three-dimensional)MATLAB Fundamentals

Plot the Mexican Hat.>> [x y] = meshgrid(-8:0.5:8);>> r = sqrt(x.^2 + y.^2)+eps;q ( y ) p ;>> z = sin(r)./r;>> mesh(z)

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Plot Commands (three-dimensional)MATLAB Fundamentals

Plot the Mexican Hat.>> [x y] = meshgrid(-8:0.5:8);>> r = sqrt(x.^2 + y.^2)+eps;q ( y ) p ;>> z = sin(r)./r;>> surf(z), shading flat or >> figure, surf(z), shading flat

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Scripts and FunctionsProgramming in MATLAB

p A script file is an external file that contains a sequence of MATLABstatements. Script files are called M-files and they have .m in their filename extensionfilename extension.

Scripts do not accept input arguments or return output arguments. They operate on data in MATLAB workspaceThey operate on data in MATLAB workspace.

M-files can also be written in form of functions that can accept input arguments and return output arguments Internal variables remainarguments and return output arguments. Internal variables remain local to the function. If you want more than one function to share a single copy of a variable, simply declare the variable as global in all the functionsthe functions.

It is easy to edit a script file if we wish to remove any error(s) and then the sequence of statements can be executed whenever desiredthen the sequence of statements can be executed whenever desired by typing the <file name> in MATLAB command line in the command window.

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How to Write and Execute Script M-files ??Programming in MATLAB

p fLet us consider that we wish to write the Mexican Hat

program as an M-file.

Use the MATLAB editor to create a new file:

File New Script (or M-file)

E t th i f MATLAB t t tEnter the series of MATLAB statements:[x y] = meshgrid(-8:0.5:8);r = sqrt(x.^2 + y.^2)+eps;z = sin(r)./r;mesh(z)

Save the file, for example, as mexhat.m.Save the file, for example, as mexhat.m.

Run the file, in the command line, by typing:

>> mexhat

When the execution completes, the variables remain in the workspace. Verify this by typing: >> whos

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M-file FunctionsProgramming in MATLAB

fLet us consider that we wish to write an M-file function that

will calculate squares and cubes of the sum of two numbers.

Use the MATLAB editor to create a new function:

File New Function

E t th i f MATLAB t t tEnter the series of MATLAB statements:function [outp1 outp2] = sqre_cube(inp1,inp2)outp1 = inp1^2 + inp2^2+ 2*inp1*inp2;outp2 = inp1^3 + inp2^3 + 3*inp1*inp2*(inp1 + inp2);outp2 inp1 3 inp2 3 3 inp1 inp2 (inp1 inp2);end

Save the file as sqre_cube.m.

Run the file, in the command line, by typing:

>> [x y] = sqre_cube(5,4)

R th fil i i th d li b t iRun the file again, in the command line, by typing:

>> [x y] = sqre_cube(6,7)AMITAVA C

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Control Flow and OperatorsProgramming in MATLAB

pMATLAB has several flow control constructs:

The “if … end” structure

The “if … else … end” structure

The “if … elseif … else … end” structure

The “for … end” loop

The “while … end” loop

Th “b k” t t t ( b d t t i t hil l The “break” statement (can be used to terminate a while loop or to exit a for loop).

The “continue” statement (can be used to exit a for loop to pass immediately to the next iteration of the loop, skipping the remaining statements in the loop).

The “return” statement, the “switch” statement etc.The return statement, the switch statement etc.

One can study the syntax by typing the help command (e.g. help for)

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pIF Conditionally execute statements.

The general form of the if statement is:The general form of the if statement is:IF expression

statementsELSEIF expression

statementsELSEELSE

statementsEND

FOR Repeat statements a specific number of times.

The general form of a for statement is:The general form of a for statement is:

FOR variable = expression, statement, ..., statement ENDAMITAVA C

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Arithmetic Operators and Special CharactersAMITAVA C

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p

Array Operators

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Relational and Logical OperatorsAMITAVA C

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Operator PrecedenceAMITAVA C

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fprintf Statement for Displaying Output on the ScreenProgramming in MATLAB

fp f f p y g pIn MATLAB, the command fprintf can be utilized both for displaying the output on the screen as well as for writing the

t t t filoutput to a file.

A script M file that executes a sequence of statements andA script M-file that executes a sequence of statements and displays the result on the screen:

x array = [5 5 8 7 12 4 34 7 67 9 -3 2 6 8];x_array = [5.5 8.7 12.4 34.7 67.9 -3.2 6.8];fprintf('\nmaximum value = %8.4f', max(x_array));fprintf('\nminimum value = %8.4f', min(x_array));fprintf('\naverage value = %8.4f', mean(x_array));fprintf('\n');

O il h th f t f di l if h / h i h One can easily change the format of display if he/she wishes to.

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fprintf Statement for Saving Output to a FileProgramming in MATLAB

fp f f g pOne can save the results of some computation to a file in text format, using the following steps:

Open a file using fopen

Write the output using fprintf

Cl th fil i f lClose the file using fclose

Change the previous script M-file as shown below and g p pexecute it in the command prompt:

x_array = [5.5 8.7 12.4 34.7 67.9 -3.2 6.8];outp = fopen('array.txt','wt');fprintf(outp,'\nmaximum value = %8.4f', max(x_array));fprintf(outp,'\nminimum value = %8.4f', min(x_array));fprintf(outp,'\naverage value = %8.4f', mean(x array));fprintf(outp, \naverage value %8.4f , mean(x_array));fprintf(outp,'\n');fclose(outp);

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Programming in MATLAB

Some Useful Workspace and File CommandsAMITAVA C

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Programming in MATLAB

Some Predefined Variables and Math Constants

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Introduction to SimulinkMATLAB/Simulink

kSimulink is a user-friendly environment created in MATLABplatform, essentially for the purpose of graphical, model-based simulation of dynamic systems.based simulation of dynamic systems.

The key features of Simulink are: Graphical editor for building and managing hierarchical blockdiagramsdiagrams Libraries of predefined blocks for modeling continuous-timeand discrete-time systems Simulation engine with fixed-step and variable-step ODEg p psolvers Scopes and data displays for viewing simulation results Project and data management tools for managing model files

d d tand data Model analysis tools for refining model architecture andincreasing simulation speed MATLAB Function block for importing MATLAB algorithms intoMATLAB Function block for importing MATLAB algorithms intomodels ….

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How to Start SimulinkMATLAB/Simulink

kType simulink in command prompt. or

click here to start

choose current to start

simulinkcurrent

directory

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Simulink Library BrowserMATLAB/Simulink

k y

Library Browserygives access tovarious standard oradditional blocksthat can be utilizedthat can be utilizedto build complicatedmodels. AMITAVA C

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Creating a New Simulink ModelMATLAB/Simulink

g kFrom the Simulink Library Browser Menu, select: File New Model. An empty model opens in the Simulink Editor.Simulink Editor.

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Edit the blank model by dragging and dropping blocks from the library browser, adding appropriate sources and sinks, and making suitable connections to build your own model.

drag and drop from the Sources in the Library

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double click the sine block to change its parameter values

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add another block in your model AMITAVA C

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source 1

source 2

sink (oscilloscope)

source 2

MUXconnector

The Complete ModelAMITAVA C

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set the configuration parameters before running the simulation

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click here to start simulation total simulation time

autoscale

scope double click on thebl k t scope

parametersscope block todisplay outputwaveform

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Saving the Simulink ModelMATLAB/Simulink

the simulink modelcreated is saved, forfuture use, as sim1.mdlin your chosendirectory

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Reusing an Existing Simulink ModelMATLAB/Simulink

double clicksim1.mdl toopen thispsimulink model

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Creating GUIsGraphical User Interfaces (GUIs)

g GIn MATLAB, one can use GUIDE, the MATLAB Graphical User Interface Development Environment to create GUIs.

A GUI is essentially a graphical display which allows a user to pictorially interact with a program. It comprises the following basic elements:

Components – Push buttons, sliders, radio buttons, check boxes etc., which enable the user to interact with a program.program.

Figure – A graphical window which houses all these components.

Callbacks – Code modules that are used to enable user interaction with the program.

To write a GUI, type in the command prompt:>> guide

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Creating GUIsGraphical User Interfaces (GUIs)

g G

Select this and click OK to create a blank, new GUI.AMITAVA C

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The Layout EditorGraphical User Interfaces (GUIs)

The blank GUI is displayed in the Layout Editor, which acts as thecontrol panel for all of the GUI Tools.

Layout Area

ComponentPalette

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Lay out your GUI, by dragging components like Slider, Edit Text, CheckBox, Push Button etc. from the Component Palette into the Layout Area.

Layout Area

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Layout Area

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Layout Area

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An ExampleGraphical User Interfaces (GUIs)

Design a GUI in MATLAB to determine square root of a number, usingNewton’s method. In the GUI, enter the number to be square-rooted, press theStart button, and the GUI should show the result in an iterative manner.

Right click anywhere inRight click anywhere inthe Layout Area to reachthe figure’s PropertyInspector. AMITAVA C

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Click the figure’sClick the figure sPropertyInspector and setthe figure’s Nameproperty toproperty toNewton.

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Make the GUIResizable. ClickTools GUIOptions. AMITAVA C

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UnderProportionalwhen the windowwhen the windowis resized, theGUI componentsalso get resized. AMITAVA C

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Place aStatic Textcontrol.Right clickto reach itsPropertyInspector.p AMITAVA C

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Change itsString propertyto Enter thenumber tosquare rootand PressStart. The Tagof thiscomponent istext1.

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Add anEdit Textcontrol. Setits Stringproperty toblank. TheTag of thiscomponentis edit1.

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Next, activate the figure using Editor in View option and saving it asNewton.fig.

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This automatically opens the application M-file Newton.m.

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Add anotherStatic TextStatic Textcontrol. Set itsString propertyto Output anditsitsHorizontalAlignment to left.The Tag of thiscomponent iscomponent istext2.

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Add a PushButton. Set itsString propertyto Start. TheTag of thiscomponent ispushbutton1. AMITAVA C

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Now, activate the figure again using Editor in View option, to updateNewton.m. Then the pushbutton1_Callback sub function is programmed as:

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Save your work and activate the figure again to test it. Then Run the figure.

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Save your work and activate the figure again to test it. Then Run the figure.

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The GUI Newton.fig shows the following result.

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Useful Images Processing CommandsMATLAB in Image Processing

Read an image using imread command in command line:

>> img1 = imread('pout.tif');

Check the row and column dimensions of the image:Check the row and column dimensions of the image:

>> [img_rows, img_cols] = size(img1)

Display the image using imshow command in command line:

>> imshow(img1)

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Images are written to disc using the function imwrite:

>> imwrite(img1, 'pout_test.tif');

Obtain an image file details using the command imfinfo:Obtain an image file details using the command imfinfo:

>> imfinfo pout_test.tif

Create a histogram to view the intensity distribution:

>> figure, imhist(img1)

Histogramof pout.tif

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Improve the contrast of this image using the function histeq which performs histogram equalization:

>> img2 = histeq(img1);g ( g )

Display the histogram equalized image in a new figure window:

>> figure, imshow(img2)

Histogram equalizedpout.tif

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View the intensity distribution of the histogram equalized version of pout.tif:

>> figure, imhist(img2)

Histogram ofhistogramequalizedequalizedversion ofpout.tif

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Read a new grayscale image moon.tif and display this image:

>> img_moon = imread('moon.tif');

>> imshow(img moon)>> imshow(img_moon)

Use the command imadjust to perform intensity adjustment for moon.tif:

>> img_moon1 = imadjust(img_moon, [0.5 0.75], [0 1], 2);

fi i h (i 1)>> figure, imshow(img_moon1)

Intensityadjustedadjustedversion ofmoon.tif

moon.tif

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To perform image segmentation of moon.tif, use function graythresh that automatically computes an appropriate threshold:

>> thr_intensity = graythresh(img_moon);

Use the command im2bw to convert grayscale moon.tif into a binary image using thresholding and display the thresholded image:

>> img_moon_bw = im2bw(img_moon,thr_intensity);

>> figure, imshow(img_moon_bw)

Automaticthresholdingthresholdingbasedsegmentedmoon.tif

moon.tif

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An example of linear spatial filtering:>> img_pepper = imread('peppers.png');>> figure, imshow(img_pepper)

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i b2 (i )>> img_pepper_gr = rgb2gray(img_pepper);>> figure, imshow(img_pepper_gr)

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i b2 (i )>> img_pepper_gr = rgb2gray(img_pepper);>> figure, imshow(img_pepper_gr)>> img_pepper_grn = imnoise(img_pepper_gr,'gaussian',0,0.01);>> figure, imshow(img pepper grn) figure, imshow(img_pepper_grn)

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An example of linear spatial filtering:…>> img_pepper_grn = imnoise(img_pepper_gr,'gaussian',0,0.01);

fi i h (i )>> figure, imshow(img_pepper_grn)>> H_mask1 = fspecial('average');>> H_mask2 = fspecial('gaussian');>> img pepper grf1 = imfilter(img pepper grn,H mask1,'replicate'); img_pepper_grf1 imfilter(img_pepper_grn,H_mask1, replicate );>> img_pepper_grf2 = imfilter(img_pepper_grn,H_mask2,'replicate');>> figure, imshow(img_pepper_grf1)

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An example of linear spatial filtering:…>> img_pepper_grn = imnoise(img_pepper_gr,'gaussian',0,0.01);

fi i h (i )>> figure, imshow(img_pepper_grn)>> H_mask1 = fspecial('average');>> H_mask2 = fspecial('gaussian');>> img pepper grf1 = imfilter(img pepper grn,H mask1,'replicate'); img_pepper_grf1 imfilter(img_pepper_grn,H_mask1, replicate );>> img_pepper_grf2 = imfilter(img_pepper_grn,H_mask2,'replicate');>> figure, imshow(img_pepper_grf1)>> figure, imshow(img_pepper_grf2)

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ReferencesL i MATLAB 7 R l 14 MATLAB® & Si li k®Learning MATLAB 7, Release14, MATLAB® & Simulink®

Student Version, The MathWorks, Inc., 2005.

Ed Overman A MATLAB Tutorial Department of Ed Overman, A MATLAB Tutorial, Department of Mathematics, The Ohio State University, 2012.

David Houcque, Introduction to MATLAB for Engineering David Houcque, Introduction to MATLAB for Engineering Students, Northwestern University, 2005.

Simulink® Getting Started Guide, R2013a, MATLAB® & g , ,Simulink®, The MathWorks, Inc., 2013.

MATLAB/Simulink Tutorial, ECEN 2060, Colorado University, 2008.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image P i i MATLAB P Ed ti 2005Processing using MATLAB, Pearson Education, 2005.

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Documents

MATLAB Fundamentals-An Overview UGC ASC Burdwan Univ Sept2013