LAB 5, 02/26/14
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1. 	Two-dimensional arrays (or matrices) are handled much in the same way
	as one-dimensional arrays (or vectors).  If

	A	is an arbitrary matrix and

	I, J 	are (one-dimensional) index vectors, 

	then
		
	A(I,J)	is a matrix of size 

		length(I) x length(J), 
	whose 		
		(i,j)th entry = (I(i),J(j))th entry of A

	Replacing either I or J by a colon (:) yields all applicable indices, 
	i.e., (all rows or columns) in increasing order without repetition.


	Examples:

		A = 1:20;             % row vector for time being

		A = reshape(A,4,5)    % application of RESHAPE function
                              % new matrix is filled column-by-column

		A([1 2], [3 4 5])

		A(1:3, 4:-1:2)

		A(:, 3)

		A(:, [3 3])           % repetitions allowed

		p = [2 3 5 4 1];      % permutation of 1:5

		A(:, p)               % column permutation

		A(p, :)               % error

		q = [2 3 4 1];        % permutation of 1:4

		A(q, :)               % row permutation


2.	As we saw earlier, 

		A*B = matrix product between two compatible matrices (i.e.,
		      A is m-by-p and B is p-by-n)

		A.*B = array product betwen two arrays of the same size

	If B is a square matrix, then B*B and B.*B are in general different.

	Example:

		B = reshape(1:9,3,3)

		B*B

		B.*B

		B*B == B.*B		% relational operator ("is equal")


	If c is a scalar (1*1), e.g.,
        
        c = 5; %then

		c*B  % is the same as c.*B

		c+B  % adds c to each element of B


3.	Handy matrices:

		ONES(M,N)        matrix of 1s of size MxN

		ZEROS(M,N)       matrix of 0s (of size MxN)

		RAND(M,N)        random entries, values ranging from 0 to 1

		EYE(N)           NxN identity matrix

		DIAG(VECTOR)     returns a diagonal (and square) matrix with
                         VECTOR on its main diagonal

		DIAG(SQ_MATRIX)  returns the diagonal (in form of a vector)
                         of a square matrix

	Examples:

		U = ones(3,3)

		Z = zeros(3,3)

		5.^Z - U		% explain the result

		R = rand(size(U))

		2*R-1			% random entries from -1 to 1

		R == R.'		% why is this a very likely result?

		ans - eye(3)

		v = diag(R)

		diag(v)


4.	We have seen examples of MATLAB graphics in the context of 2-D plots.
	MATLAB has extensive capabilities that extend to 3-D plots, images,
	video, etc.  The structure and representation of all graphics in MATLAB
	are based on matrices.  Here, we explore one basic connection between 
	graphics and matrices.

	For simplicity, we will consider matrices with entries between 0 and 1.
	The function 

		IMSHOW(MATRIX)

	where MATRIX has dimensions MxN, displays the entries of the matrix as
	an MxN grayscale image, where the the intensity of the (i,j)th pixel is
	given by the (i,j)th entry of MATRIX - with 0 corresponding to black
	and 1 corresponding to white.

		m = 100;

		U = ones(m,m);

		imshow(U)

		H = 0.5*U; imshow(H)

		Z = 0*U; imshow(Z)

		E = eye(m); imshow(E)

		imshow(0.5*E + 0.5*U)		% all values in [0.5,1]

		R = rand(m,m); imshow(R)

		imshow(0.5*E + 0.5*R)		% all values still in [0,1]

	(MATLAB also uses the color map feature to represent different shades 
	of gray by different colors. A menu of preset color maps can be accessed 
	on the figure window through EDIT > FIGURE PROPERTIES.)


5.	Explain the form of the images generated below:

		m = 200;

		a = 0 : 1/m : 1;

		u = ones(size(a));

		X = u.'*a;

		Y = a.'*u;

			imshow(X)

			imshow(Y)

			imshow(X>Y)

			imshow(X.^2<Y)

		C = (cos((X+Y-1)*pi/2)).^6 ;

			imshow(C)

		F = a.'*a ;		% what is an equivalent formula for F?

			imshow((1-F).^6)
		

6.	Similarly, a grayscale digital image can be represented by a single
	matrix. (Color images require three matrices, one each for the
	intensities of the constituent colors Red, Green and Blue).  MATLAB 
	can read most image file types.

	The array cameraman contains the intensity values of a 512x512
	grayscale image:

		A = imread('cameraman.tif')

		subplot(2,1,1), imshow(A)

	Some standard image transformations, performed on A:

		A1 = A(61:188, 171:298);      % zooming in
		
		subplot(2,1,2), imshow(A1)			

		A2 = rot90(A,1);	      % counterclockwise rotation
                                  % 1: 90 deg, 2: 180 deg, etc.
			imshow(A2)	

			imshow(fliplr(A))     % left-to-right flip

			imshow(flipud(A))     % upside-down flip

			imshow(A.')           % transposition

		A3 = circshift(A,256) ;	  % circular shift; 256=512/2,
                                  % so top, bottom halves swapped
			imshow(A3)

		A4 = (circshift(A3.',256)).'; % explain the result
	
			imshow(A4)

		A5 = A>120;               % conversion to B&W (no gray)

			imshow(A5)

		A6 = double(A).^0.8;      % brightens image

		imshow(A6)                % does not work, since A6 is of type
                                    double, pixel values will be capped at 
                                    1.0, and thus its range of display will 
                                    be 0--1.

		imshow(uint8(A6))         % works for integers, range of dispay
                                    is 0--255.

		A7 = double(A).^1.05;     % darkens image

		imshow(uint8(A7))