With the fast progress of research and applications in image understanding and computer vision, the need for accurate line detection and segmentation is increasing. The work presented in this thesis has been an attempt to contribute to one of the primitive and yet important steps in many object recognition and feature detection tasks, i.e. line detection and segmentation. On one hand, the proposed algorithm illustrated promising outcomes with good accuracy. On the other hand, it highlighted the gaps and limitation that are worthy of future investigation. It also provided grounds for improvement and expansion of this work.
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Appendix A
Fast Fourier transform
A.1
Forward transform
The fast Fourier transform (FFT) is a computationally optimised algorithm for calculating the discrete Fourier transform (DFT). Let us consider the continuous Fourier transform as a form of integral given by
F (u) = Z ∞
−∞
f (x)e−j2πuxdx (A.1)
where f (x) is a continuous function in the time domain. As a result of an imaginary component j the transform yields to a complex domain, refereed as the Fourier or frequency domain. Recalling that the imaginary exponent could be written as:
ejθ = cos θ + j sin θ (A.2)
For a digital signal that has been sampled from its continuous form f (x) into a discrete form f (n), a DFT is defied as:
F (k) =
NX−1 n=0
Here{f0, f1, ..., fN−1} are discrete samples of input signal f(n) and
{F0, F1, ..., FN−1} are the corresponding result of the DFT. N is the total
number of available discrete components in f (n) and is usually a power of 2. Therefore, to implement the algorithm using a computer program it is sufficient to write a double loop code and calculate the sums of the products of input samples and imaginary exponents. The complexity of such operation is of O(N2) order. However, the complexity can be reduced to the order of O(N log2N ) using the algorithm explained in the following.
Let us consider the DFT of a signal with N = 8 samples given as:
F (k) = f (0) + f (1) e−j2πk/8+ f (2) e−j2π2k/8+ f (3) e−j2π3k/8
+ f (4) e−j2π4k/8+ f (5) e−j2π5k/8+ f (6) e−j2π6k/8+ f (7) e−j2π7k/8 (A.4)
Equation. A.4 can be split into two similar sums by separating the odd and even elements and factoring out the e−j2πk/8 from f (1) component.
F (k) = " f (0) + f (2) e−j2π2k/8+ f (4) e−j2π4k/8+ f (6) e−j2π6k/8 # + e−j2πk/8 " f (1) + f (3) e−j2π2k/8+ f (5) e−j2π4k/8+ f (7) e−j2π6k/8 # (A.5)
F (k) = " f (0) + f (4) e−j2π4k/8 ! + e−j2π2k/8 f (2) + f (6) e−j2π4k/8 !# + e−j2πk/8 " f (1) + f (5) e−j2π4k/8 ! + e−j2π2k/8 f (3) + f (7) e−j2π4k/8 !# (A.6) Equation. A.6 shows there are log28 = 3 levels of summation, i.e. the deepest level in parenthesis, the middle level in brackets and the outer or the last level. Also, for every level the exponential component is the same. Equation. A.6 can be further simplified as:
F (k) = " f (0) + f (4) e−jπk ! + e−jπk/2 f (2) + f (6) e−jπk !# + e−jπk/4 " f (1) + f (5) e−jπk ! + e−jπk/2 f (3) + f (7) e−jπk !# (A.7)
Note that one of the interesting properties of a complex number ejθ is the periodicity at every 2π radian and can be used to speed-up the computation. Given that
ejθ = ej(θ+2π) (A.8) Period for each of the exponential terms inside the parenthesis in Equa-