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Homework 2
I. FOURIE R DES CR IP TORS
Let a 2D shape be represented by Npoints on its outer contour. The co-ordinates of these points will be considered as
complex numbers
U=
x0+jy0
x1+jy1
.
.
xN−1+jyN−1
(1)
where j=√−1. Considering Uto be a periodic signal, its discrete fourier transform (DFT) can be written as
F(k) =
N−1
X
n=0
U(n)exp(−j2π
Nnk)(2)
where U(n) = xn+jyn, for k= 0 . . . N −1.
Consider the situation when the points in the contour undergo
1) Translation: ˜
U(n) = U(n) + t∀n= 0 . . . N −1.
2) Scaling: ˜
U(n) = sU(n)∀n.
3) Rotation: ˜
U(n) = exp(jθ)U(n)∀n.
4) Change in starting point ˜
U(n) = U((n+m)mod(N)) ∀n(cyclical shift by munits).
Question 1: Derive expressions for the DFT coefficients ˜
F(k)of ˜
U(n)in terms of F(k)and the transformation parameters
{t, s, θ, m}. (5 marks)
Question 2: If you had to match shapes using their Fourier descriptors, suggest methods to achieve invariance to the above
transformations i.e. 2 shapes which are related by the above transformations should still be declared to be the same shape even
though their raw DFT coefficients may be different. (5 marks).
II. NORMA LI ZE D CUT S
Question 1: Starting from the definition of Normalized cuts in Shi and Malik’s paper, work your way towards the final
criterion function given in equation (5) of the paper
minxNcut(x) = miny
yT(D−W)y
yTDy (3)
(Derivation for the part before the definitions of α(x), β(x), γ will be made available on the course site). (5 marks)
Question 2: Solve the above minimization problem using the Rayleigh quotient result. (5 marks)