Mathematics of Imaging: Lecture 3

Mathematics of Imaging: Lecture 3

Linear Operators in Infinite Dimensions Consider the linear operator on the space of continuous functions defined on IR. J (f)(x) = x 0 f(s) ds Every function in the range of J has a continuous derivative, but not every continuously differentiable function is in the range, since J f(0) = 0. The nullspace of J is trivial. The first derivative operator is a left inverse to J It fails to be a right inverse, since ( ) d dx J (f)(x) = f(x) ( J d ) (f)(x) = f(x) f(0). dx

Moving Average or Window Operator Consider the transformation on the space of locally integrable functions: M δ (f)(x) = x+δ x δ f(s) ds Clearly M δ is a linear operator; the value of M δ (f) at x is obtained by taking the window [x δ, x + δ] and computing the average value of f there, then sliding the window along. What is the null space of M δ? Can we say anything about the natural domain or the range? Can this operator be represented as convolutino with a certain function g?

Complex Numbers Defined Abstractly The statement that x is a complex number means that either x is a number or x is an ordered number-pair with second term different from zero. If each of {a, b} and {c, d} is an ordered number-pair and bd 0 then (1) a + {c, d} = {c, d} + a = {a + c, d} (2) if b + d = 0, then {a, b} + {c, d} = a + c (3) if b + d 0, then {a, b} + {c, d} = {a + c, b + d} (4) 0{c, d} = {c, d}0 = 0 (5) b{c, d} = {, d}b = {bc, bd} (6) if ad + bc = 0, then {a, b}{c, d} = ac bd (7) if ad + bc 0, then {a, b}{c, d} = {ac bd, ad + bc} The symbol i is used to denote the complex number {0, 1}. Simply put, complex addition holds no surprises. Since i 2 = 1, if standard foil is to hold, (a+bi)(c+di) = ac+adi+bci+bdi 2 = (ac bd)+i(ad+bc).

Real and Imaginary Parts, Conjugates If each of x and y is a complex number, then each of x + y and xy is a complex number. If {a, b} is an ordered number-pair then a + ib = a + bi and (1) if b = 0 then a + ib = a, but (2) if b 0 then a + ib = {a, b}. If {a, b} is an ordered number-pair then (1) Re(a + ib) = a and is called the real part of a + ib (2) Im(a + ib) = b and is called the imaginary part of a + ib (3) (a + ib) = a + i( b) and is called the conjugate of a + ib (4) a + ib = (a 2 + b 3 ) 1 2 and is called the modulus of a + ib. If x is a complex number then x + x = 2Re(x), x = x + 2iIm(x), and the modulus of x (a nonnegative number) has the property: x 2 = xx = (Re x) 2 + (Im x) 2.

Complex Plane and Distance Function As an aid to the intuition we introduce certain geometric terminology: a point is a complex number, the real line is the set of all numbers, the number-plane is the set of all complex numbers, the right half-plane is the set to which x belongs only in case x is a point and Re(x) > 0, the left half-plane is similarly characterized by the condition that Re(x) < 0, and the upper half-plane and lower half-plane by the conditions that Im(x) > 0 and Im(x) < 0, respectively. Moreover, by the distance from the point x to the point y we mean the number y x ; the following algebraic facts should be verified: if each of x and y is a point then (1) x y = y x and xy = x y. (2) x y 2 = x 2 2Re(x y) + y 2. (3) x y x z + z y for each point z.

Complex Exponential First definition of e z where z is complex is as exp(z) e z = n=0 z n n! Show this converges because the modulus of exp(x) is majorized by exp( z ). We also have the identity, exp(z) exp(w) = exp(z + w). By analogy with the real hyperbolic sine and cosine, we define sin(z) = eiz e iz 2i cos(z) = eiz + e iz 2 If θ is real, then e iθ = cos(θ) + i sin(θ) and any complex number can be written in polar form z = z e iθ where z is the modulus and θ is the

argument (angle in ( π, π] that z makes with real axis, arctan Im(z)/Re(z) ). Multiplication of complex numbers has a simple interpretation now: to multiply two complex numbers multiply there moduli and add their arguments: zw = z e iθ w e iφ = z w e i(θ+φ) There are several ways to complexify a vector space over the reals (see Halmos, Finite Dimensional Vector Spaces). Can start with n-tuples of complex numbers and form C n. Means some changes to abstract definition of inner product: Hermitian symmetry f, g = g, f. Cauchy-Schwartz still true, f, g f g.

Assignment 1 Your first assignment has been posted on webct and my webpage - it is due next Tuesday. It is reproduced here so we can discuss the questions at the end of this period. 1. Initial Value Problems Review techniques for solving second order linear ODE s. Consider the problem y + y = 1, y(1) = 3, y (1) = 2. (a) First consider the homogeneous problem y + y = 0. Guess a solution of the form y(x) = e rx. Find the characteristic equation, and write the general solution as the sum of two linearly independent solutions: y(x) = c 1 y 1 (x) + c 2 y 2 (x). (Answer will be in terms of complex exponentials.) (b) Using the definition of complex exponential e iθ = cos(θ) + i sin(θ), θ real, rewrite your solution in terms of real-valued functions.

(c) Find a particular solution y par (x) by guessing one of the same general form as the right-hand side. In this case, guess a polynomial. (d) Show the general solution to the nonhomogeneous equation is the sum of the general solution to the homogeneous equation (called the complementary solution) and a particular solution. (e) If Ly = y + y, what is the natural domain of L? Can you say anything about the range? What is the nullspace of L? (f) Find the exact solution of the IVP, i.e., the one satisfying the initial conditions given above. 2. Boundary Value Problems Consider solving the problem y + λy = 0 on [0, 1] with boundary conditions y(0) = 0 = y(1). (a) Examine the quantity y, y = 1 0 y (x)y(x) dx and using integration by parts, show that for a nontrivial, real solution to exist, λ must be positive.

(b) By analogy with the problems above, rewrite it as y + ( λ) 2 y = 0 and guess a solution of the form y(x) = c 1 sin( λ x) + c 2 cos( λ x). Argue that only when λ = k 2 π 2, k a positive integer, is there a solution. (c) Let H be the set of all functions on [0, 1] with a continuous second derivative. Is this a linear (vector) space? What about the subset H 0 of H consisting of those members of H which are zero on the boundary. (d) What are the eigenvalues, eigenvectors of the operator A, where Ay = y with zero boundary conditions. (e) Show that the eigenvectors are pairwise orthogonal: y k, y j 1 0 y k (x)y j (x) dx = 0, k j