Find an expression for the signature of each of the following boundaries, and plot the signatures.

i) An equilateral triangle

ii) A rectangle

iii) An ellipse

Explain how the MPP algorithm behaves under the following conditions: b

i) 1-pixel wide, 1-pixel deep indentations.

ii) 1-pixel wide, 2-or- more pixel deep indentations.

iii) 1-pixel wide, 1-pixel longprotrusions.

iv) 1-pixel wide, n-pixel long protrusions.

Suppose that an image f(x, y) is convolved with a mask of size n x n (with cofficients 1/n²) to produce a smoothed image .

i) Derive an expression for edge strength (edge magnitude) of the smoothed image as a function of mask size. Assume for simplicity that n is odd and that edges are obtained using the partial derivatives

ii) Show that the ratio of the maximum edge strength of the smoothed image to the maximum edge strength of the orginal is 1/n. In other words, edge strength is inversely proportional to the size of the smoothing mask.

A binary image contains straight lines oriented horizontally, vertically, at 45°, and at - 45°. Give a set of 3 x 3 masks that can be used to detect 1-pixel breaks in these lines. Assume that the intensities of the lines and background are 1 and 0, respectively.

The arithmetic decoding process is the reverse of the encoding procedure. Decode the message 0.23355 given the coding model.

Prove the following result:

Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set {r₁,r₂,..., r_n} of a non-zero real numbers with  and an orthonormal set {w₁, w₂,..., w_n} in H such that

Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?

Give an example of a positive operator on .

Give an example of a compact linear map on l².

Let X be a Banach space and Y be a closed subspace of X. Let π: X → X/Y be canonical quotient map. Show that is open.

Define the spectral radius of a bounded linear operator A ∈ BL(X). Find the spectral radius of A in BL, where A is given by the matrix

with respect to the standard basis of .

Let {A_n} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.

Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.

Let X,Y be normed spaces and suppose BL(X,Y) and CL(X,Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X,Y) is linear subspace of BL(X,Y). Also, Show that if Y is a Banach space, then CL(X,Y) is a closed subspace of BL(X,Y).

Let X be a normed space and Y be proper subspace of X. Show that the interior Yº of Y is empty.

Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on .

ii) A normal operator on a Hilbert space which is not unitary.

Let X = C₀₀ with  Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.

Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σ_a(A) where σ_a (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.

Given an example of an Hilbert space H and an operator A on Η such that σ_e(A)is empty. Justify your choice of example.

Let H=R³ and F be the set of all x = (x₁, x₂, x₃) in H such that x₁ = 0. Find F¹. Verify that every x ∈ H can be expressed as x = y + z where y ∈ Fand z ∈ F¹.

Let X be an inner product space and x, y ∈ X. Prove that x | y if and only if