%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Introduction to Matlab % (adapted from http://www.stanford.edu/class/cs223b/matlabIntro.html) % % Stefan Roth <roth (AT) cs DOT brown DOT edu>, 09/08/2003 % Last modified: 09/10/2003 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (1) Basics % The symbol "%" is used to indicate a comment (for the remainder of % the line). % When writing a long Matlab statement that becomes to long for a % single line use "..." at the end of the line to continue on the next % line. E.g. A = [1, 2; ... 3, 4]; % A semicolon at the end of a statement means that Matlab will not % display the result of the evaluated statement. If the ";" is omitted % then Matlab will display the result. This is also useful for % printing the value of variables, e.g. A % Matlab's command line is a little like a standard shell: % - Use the up arrow to recall commands without retyping them (and % down arrow to go forward in the command history). % - C-a moves to beginning of line (C-e for end), C-f moves forward a % character and C-b moves back (equivalent to the left and right % arrow keys), C-d deletes a character, C-k deletes the rest of the % line to the right of the cursor, C-p goes back through the % command history and C-n goes forward (equivalent to up and down % arrows), Tab tries to complete a command. % Simple debugging: % If the command "dbstop if error" is issued before running a script % or a function that causes a run-time error, the execution will stop % at the point where the error occurred. Very useful for tracking down % errors. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (2) Basic types in Matlab %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (A) The basic types in Matlab are scalars (usually double-precision % floating point), vectors, and matrices: A = [1 2; 3 4]; % Creates a 2x2 matrix B = [1,2; 3,4]; % The simplest way to create a matrix is % to list its entries in square brackets. % The ";" symbol separates rows; % the (optional) "," separates columns. N = 5 % A scalar v = [1 0 0] % A row vector v = [1; 2; 3] % A column vector v = v' % Transpose a vector (row to column or % column to row) v = 1:.5:3 % A vector filled in a specified range: v = pi*[-4:4]/4 % [start:stepsize:end] % (brackets are optional) v = [] % Empty vector %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (B) Creating special matrices: 1ST parameter is ROWS, % 2ND parameter is COLS m = zeros(2, 3) % Creates a 2x3 matrix of zeros v = ones(1, 3) % Creates a 1x3 matrix (row vector) of ones m = eye(3) % Identity matrix (3x3) v = rand(3, 1) % Randomly filled 3x1 matrix (column % vector); see also randn % But watch out: m = zeros(3) % Creates a 3x3 matrix (!) of zeros %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (C) Indexing vectors and matrices. % Warning: Indices always start at 1 and *NOT* at 0! v = [1 2 3]; v(3) % Access a vector element m = [1 2 3 4; 5 7 8 8; 9 10 11 12; 13 14 15 16] m(1, 3) % Access a matrix element % matrix(ROW #, COLUMN #) m(2, :) % Access a whole matrix row (2nd row) m(:, 1) % Access a whole matrix column (1st column) m(1, 1:3) % Access elements 1 through 3 of the 1st row m(2:3, 2) % Access elements 2 through 3 of the % 2nd column m(2:end, 3) % Keyword "end" accesses the remainder of a % column or row m = [1 2 3; 4 5 6] size(m) % Returns the size of a matrix size(m, 1) % Number of rows size(m, 2) % Number of columns m1 = zeros(size(m)) % Create a new matrix with the size of m who % List variables in workspace whos % List variables w/ info about size, type, etc. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (3) Simple operations on vectors and matrices %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (A) Element-wise operations: % These operations are done "element by element". If two % vectors/matrices are to be added, subtracted, or element-wise % multiplied or divided, they must have the same size. a = [1 2 3 4]'; % A column vector 2 * a % Scalar multiplication a / 4 % Scalar division b = [5 6 7 8]'; % Another column vector a + b % Vector addition a - b % Vector subtraction a .^ 2 % Element-wise squaring (note the ".") a .* b % Element-wise multiplication (note the ".") a ./ b % Element-wise division (note the ".") log([1 2 3 4]) % Element-wise logarithm round([1.5 2; 2.2 3.1]) % Element-wise rounding to nearest integer % Other element-wise arithmetic operations include e.g. : % floor, ceil, ... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (B) Vector Operations % Built-in Matlab functions that operate on vectors a = [1 4 6 3] % A row vector sum(a) % Sum of vector elements mean(a) % Mean of vector elements var(a) % Variance of elements std(a) % Standard deviation max(a) % Maximum min(a) % Minimum % If a matrix is given, then these functions will operate on each column % of the matrix and return a row vector as result a = [1 2 3; 4 5 6] % A matrix mean(a) % Mean of each column max(a) % Max of each column max(max(a)) % Obtaining the max of a matrix mean(a, 2) % Mean of each row (second argument specifies % dimension along which operation is taken) [1 2 3] * [4 5 6]' % 1x3 row vector times a 3x1 column vector % results in a scalar. Known as dot product % or inner product. Note the absence of "." [1 2 3]' * [4 5 6] % 3x1 column vector times a 1x3 row vector % results in a 3x3 matrix. Known as outer % product. Note the absence of "." %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (C) Matrix Operations: a = rand(3,2) % A 3x2 matrix b = rand(2,4) % A 2x4 matrix c = a * b % Matrix product results in a 3x4 matrix a = [1 2; 3 4; 5 6]; % A 3x2 matrix b = [5 6 7]; % A 1x3 row vector b * a % Vector-matrix product results in % a 1x2 row vector c = [8; 9]; % A 2x1 column vector a * c % Matrix-vector product results in % a 3x1 column vector a = [1 3 2; 6 5 4; 7 8 9]; % A 3x3 matrix inv(a) % Matrix inverse of a eig(a) % Vector of eigenvalues of a [V, D] = eig(a) % D matrix with eigenvalues on diagonal; % V matrix of eigenvectors % Example for multiple return values! [U, S, V] = svd(a) % Singular value decomposition of a. % a = U * S * V', singular values are % stored in S % Other matrix operations: det, norm, rank, ... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (D) Reshaping and assembling matrices: a = [1 2; 3 4; 5 6]; % A 3x2 matrix b = a(:) % Make 6x1 column vector by stacking % up columns of a sum(a(:)) % Useful: sum of all elements a = reshape(b, 2, 3) % Make 2x3 matrix out of vector % elements (column-wise) a = [1 2]; b = [3 4]; % Two row vectors c = [a b] % Horizontal concatenation (see horzcat) a = [1; 2; 3]; % Column vector c = [a; 4] % Vertical concatenation (see vertcat) a = [eye(3) rand(3)] % Concatenation for matrices b = [eye(3); ones(1, 3)] b = repmat(5, 3, 2) % Create a 3x2 matrix of fives b = repmat([1 2; 3 4], 1, 2) % Replicate the 2x2 matrix twice in % column direction; makes 2x4 matrix b = diag([1 2 3]) % Create 3x3 diagonal matrix with given % diagonal elements %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % (4) Control statements & vectorization % Syntax of control flow statements: % % for VARIABLE = EXPR % STATEMENT % ... % STATEMENT % end % % EXPR is a vector here, e.g. 1:10 or -1:0.5:1 or [1 4 7] % % % while EXPRESSION % STATEMENTS % end % % if EXPRESSION % STATEMENTS % elseif EXPRESSION % STATEMENTS % else % STATEMENTS % end % % (elseif and else clauses are optional, the "end" is required) % % EXPRESSIONs are usually made of relational clauses, e.g. a < b % The operators are <, >, <=, >=, ==, ~= (almost like in C(++)) % Warning: % Loops run very slowly in Matlab, because of interpretation overhead. % This has gotten somewhat better in version 6.5, but you should % nevertheless try to avoid them by "vectorizing" the computation, % i.e. by rewriting the code in form of matrix operations. This is % illustrated in some examples below. % Examples: for i=1:2:7 % Loop from 1 to 7 in steps of 2 i % Print i end for i=[5 13 -1] % Loop over given vector if (i > 10) % Sample if statement disp('Larger than 10') % Print given string elseif i < 0 % Parentheses are optional disp('Negative value') else disp('Something else') end end % Here is another example: given an mxn matrix A and a 1xn % vector v, we want to subtract v from every row of A. m = 50; n = 10; A = ones(m, n); v = 2 * rand(1, n); % % Implementation using loops: for i=1:m A(i,:) = A(i,:) - v; end % We can compute the same thing using only matrix operations A = ones(m, n) - repmat(v, m, 1); % This version of the code runs % much faster!!! % We can vectorize the computation even when loops contain % conditional statements. % % Example: given an mxn matrix A, create a matrix B of the same size % containing all zeros, and then copy into B the elements of A that % are greater than zero. % Implementation using loops: B = zeros(m,n); for i=1:m for j=1:n if A(i,j)>0 B(i,j) = A(i,j); end end end % All this can be computed w/o any loop! B = zeros(m,n); ind = find(A > 0); % Find indices of positive elements of A % (see "help find" for more info) B(ind) = A(ind); % Copies into B only the elements of A % that are > 0 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(5) Saving your work save myfile % Saves all workspace variables into % file myfile.mat save myfile a b % Saves only variables a and b clear a b % Removes variables a and b from the % workspace clear % Clears the entire workspace load myfile % Loads variable(s) from myfile.mat %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(6) Creating scripts or functions using m-files: % % Matlab scripts are files with ".m" extension containing Matlab % commands. Variables in a script file are global and will change the % value of variables of the same name in the environment of the current % Matlab session. A script with name "script1.m" can be invoked by % typing "script1" in the command window. % Functions are also m-files. The first line in a function file must be % of this form: % function [outarg_1, ..., outarg_m] = myfunction(inarg_1, ..., inarg_n) % % The function name should be the same as that of the file % (i.e. function "myfunction" should be saved in file "myfunction.m"). % Have a look at myfunction.m and myotherfunction.m for examples. % % Functions are executed using local workspaces: there is no risk of % conflicts with the variables in the main workspace. At the end of a % function execution only the output arguments will be visible in the % main workspace. a = [1 2 3 4]; % Global variable a b = myfunction(2 * a) % Call myfunction which has local % variable a a % Global variable a is unchanged [c, d] = ... myotherfunction(a, b) % Call myotherfunction with two return % values %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(7) Plotting x = [0 1 2 3 4]; % Basic plotting plot(x); % Plot x versus its index values pause % Wait for key press plot(x, 2*x); % Plot 2*x versus x axis([0 8 0 8]); % Adjust visible rectangle figure; % Open new figure x = pi*[-24:24]/24; plot(x, sin(x)); xlabel('radians'); % Assign label for x-axis ylabel('sin value'); % Assign label for y-axis title('dummy'); % Assign plot title figure; subplot(1, 2, 1); % Multiple functions in separate graphs plot(x, sin(x)); % (see "help subplot") axis square; % Make visible area square subplot(1, 2, 2); plot(x, 2*cos(x)); axis square; figure; plot(x, sin(x)); hold on; % Multiple functions in single graph plot(x, 2*cos(x), '--'); % '--' chooses different line pattern legend('sin', 'cos'); % Assigns names to each plot hold off; % Stop putting multiple figures in current % graph figure; % Matrices vs. images m = rand(64,64); imagesc(m) % Plot matrix as image colormap gray; % Choose gray level colormap axis image; % Show pixel coordinates as axes axis off; % Remove axes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %(8) Working with (gray level) images I = imread('cit.png'); % Read a PNG image figure imagesc(I) % Display it as gray level image colormap gray; colorbar % Turn on color bar on the side pixval % Display pixel values interactively truesize % Display at resolution of one screen % pixel per image pixel truesize(2*size(I)) % Display at resolution of two screen % pixels per image pixel I2 = imresize(I, 0.5, 'bil'); % Resize to 50% using bilinear % interpolation I3 = imrotate(I2, 45, ... % Rotate 45 degrees and crop to 'bil', 'crop'); % original size I3 = double(I2); % Convert from uint8 to double, to allow % math operations imagesc(I3.^2) % Display squared image (pixel-wise) imagesc(log(I3)) % Display log of image (pixel-wise) I3 = uint8(I3); % Convert back to uint8 for writing imwrite(I3, 'test.png') % Save image as PNG figure; g = [1 2 1]' * [1 2 1] / 16; % 3x3 Gaussian filter mask I2 = double(I); % Convert image to floating point I3 = conv2(I2, g); % Convolve image with filter mask I3 = conv2(I2, g, 'same'); % Convolve image, but keep original size subplot(1, 2, 1) % Display original and filtered image imagesc(I); % side-by-side axis square; colormap gray; subplot(1, 2, 2) imagesc(I3); axis square; colormap gray; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function y = myfunction(x) % Function of one argument with one return value a = [-2 -1 0 1]; % Have a global variable of the same name y = a + x;

function [y, z] = myotherfunction(a, b) % Function of two arguments with two return values y = a + b; z = a - b;