Lecture 18 Chapter 12 Matrices continued. Outline from Chapter 12 -1 12.2 Matrix Operations 12.2.1 Matrix Multiplication 12.2.2 Matrix Division 12.2.3.

Lecture 18

Chapter 12 Matricescontinued

Outline from Chapter 12 -1

12.2 Matrix Operations12.2.1 Matrix Multiplication12.2.2 Matrix Division12.2.3 Matrix Exponentiation

12.3 MATLAB Implementation12.3.1 Matrix Multiplication12.3.2 Matrix Division

12.4 Rotating Coordinates12.4.1 2-D Rotation12.4.2 3-D Rotation

Outline from Chapter 12 -2

• 12.5 Solving Simultaneous Linear Equations– 12.5.1 Intersecting Lines– 12.5.2 Curve Fitting

Matrix Multiplication

• A x = b• Number of columns in A = number of rows in x• Number of rows in b is number of rows in A• Number of columns in b is number of columns

in x.• b(i, j) = (å k=1 to nColsA) A(i,k)*x(k,j)

Matrix Division

• With Ax = b, want to know x.• A -1 A x = A -1 b uses the inverse of A,

which does not always exist• Requirements for inverse to exist:– Number of rows of A = number of columns of A– i.e., A is square

Matrix Exponentiation

• Raising matrix A to non-zero integer power• For positive power, multiply A by itself that

many times.• For negative power, first obtain positive power

then invert.• Non-integer scalar powers are also

meaningful, but outside the scope of the book.

MATLAB Matrix Multiplication

• Use the operator *:b = A * x

>> A=[3 4; 4 3];>> b=[7 7]b = 7 7>> inv(A) * b??? Error using ==> mtimesInner matrix dimensions must agree.>> b = reshape(b, 2,1)b = 7 7>> inv(A) * bans = 1 1>>

MATLAB Matrix Division

• If A is square and nonsingular, then, without roundoff error, X = inv(A)*B is theoretically the same as X = A\B and Y = B*inv(A) is theoretically the same as Y = B/A. But the computations involving the backslash and slash operators are preferable because they require less computer time, less memory, and have better error-detection properties.

MATLAB Matrix Division -2• >> inv(A) * b

• ans =

• 1• 1

• >> A\b

• ans =

• 1.0000• 1.0000

• >>

Besides the multiplication (by the inverse) on the left that is illustrated on the left, there is also multiplication on the right, shown below:

c = 1 2 2 3 3 4>> c * inv(A)ans = 0.7143 -0.2857 0.8571 -0.1429 1.0000 -0.0000>> c / Aans = 0.7143 -0.2857 0.8571 -0.1429 1.0000 0

An Application of Matrix Multiplication

• Rotate Coordinates in 2-D, about the origin:– xNew, yNew = rotationMatrix * xOld,yOld

• Rotate coordinates in 2-D, about some other point:– Translate point of rotation to origin– Rotate– Translate back

3D Rotationfunction newCoords = rotate3D(theta, phi, psi, oldCoords)%perform a rotation, centered on origin, about axes%rotations are performed in order about z, about y, about x%to use other orders, call function multiple times, employ 0-angle rotationszRotation = [cos(theta) -sin(theta) 0;... sin(theta) cos(theta) 0;... 0 0 1];yRotation = [cos(phi) 0 sin(phi);... 0 1 0 ;... sin(phi) 0 cos(phi)];xRotation = [1 0 0;... 0 cos(psi) -sin(psi);... 0 sin(psi) cos(psi)];newCoords = zRotation * yRotation * zRotation * oldCoords;end

Simultaneous Linear Equations

• Soccer ball: – nP = number of pentagons– nH = number of hexagons– nP + nH = number of faces– nE = (5* nP + 6*nH)/2– nV = 2*nE/(how many edges impinge on a vertex?)

• Euler’s formula: If you take ANY solid without holes that's made of flat faces, and F = number of faces, E = number of edges, and V = number of vertices: F - E + V = 2

N Independent Linear Equations in M Unknowns

• N < M: we can obtain a subspace, i.e., a restriction of the values that correspond to solutions

• N=M: we can obtain single values for each of the unknowns

• N>M: might be over-specified, i.e., might be no solution

N Independent Linear Equations in N Unknowns

• Looks like:– a1x1 + b1x2 + …+p1 xN = y1

– a2x1 + b2x2 + …+p2 xN = y2

– …

– aNx1 + bNx2 + …+pN xN = yN

• Converts to, for coefficients a, b, etc.:[a1 b1 …+p1 ;

a2 b2 …+p2 ;

… a2 b2 …+p2 ]

• For x: [x1 x2 …xN ] as a column vector, same for y• Ax = y

Curve Fitting:Polynomial of Order N to N+1 points

• Given N+1 points, (xi, yi), find the polynomial f(x), of order N, that has the smallest difference squared, (yi-f(xi))2, summed over the N+1 points.

• http://mathworld.wolfram.com/LeastSquaresFittingPolynomial.html

• C, the coefficients, are what we want to know• C = A \ B• A, B are given at the link above.

Chapter 13 Outline -1

• 13.1 Nature of an Image• 13.2 Image Types– 13.2.1 True Color Images– 13.2.2 Gray Scale Images– 13.2.3 Color Mapped Images– 13.2.4 Preferred Image Format

• 13.3 Reading, Displaying and Writing Images

Chapter 13 Outline -2

• 13.4 Operating on Images– 13.4.1 Stretching or Shrinking Images– 13.4.2 Color Masking– 13.4.3 Making a Collage– 13.4.4 Creating a Kaleidoscope– 13.4.5 Images on a Surface

Nature of an Image

• Interpreted information -> drawn graph of n-dimensions (representation based on a concept)

• Vs• Uninterpreted data displayed in n-dimensions

(may lack explanation)

2-D Slice of an Image

• Many images are of more than 2 dimensions, such as magnetic resonance image (MRI), CAT scan, etc.

• Textbook is talking about 2 dimensions, which could be a slice from a higher dimensional object, or could be the whole image.

• 2D array has a number of cells, given by the number of rows x number of columns.

• In the context of what can be displayed, if a cell corresponds to a dot on the screen, it is called a pixel.

• Color pixels contain a degree of red, blue and green (a positive integer for each, uint8).

Reading in Image File

• One function, imread, reads many types of files.• There are three types of target variables– True color (3 uint8’s per pixel)– Grey scale (1 uint8 per pixel)– Color mapped (1 uint8 per pixel, chooses a 3 uint8

color from the color map)• Pic = imread(myPicture.jpg’)• image(Pic)• imwrite(Pic, ‘newPicture.jpg’, ‘jpg’)

Operating on Images

• Stretching (sampling and resampling the array)• Color Masking