SUBSAMPLING

Subsampling is the process by which a discrete image is obtained from a continuous image. This is different from the analogue-digital conversion which relates to the image values,ie, to the discretization process.

Subsampling Theorem (Shannon)

Given a signal S it is possible to reconstruct it exactly from the sequence of sample S(n T) if the signal is B band-limited (ie, the Fourier transform of the signal has compact support contained in [-B,+B]), and the sampling frequency 1/T = B/pi.

F( x ) = Sumn F(n T) sin( B (x - n T) ) / ( B ( x - n T) )

The space of the B band-limited functions is called Paley-Wiener space PB, and is a subspace of L2(R).

Proff
Since F^ is B band-limited, the inverse Fourier transform is

F(x) = Int[-B,B] e(i v x) F^(v) dv

This formula written for F(n T) (remember that T=pi/B) is like the inverse Fourier transform for a discrete function. Therefore, taking into account that F^ is supported in [-B,B], we have
F^(v) = 1/(2 B) Sumn F(n T) e(-i v n T) X[-B,B](v)

where X is the function that has value 1 inside the interval [-B,B], and zero outside.
By taking the inverse Fourier transform we have that F is the sum weighted by the coefficients F(n T) of the inverse Fourier tarnsforms of 1/(2B) e(-i v n T)X[-B,B](v) . By the identities of the Fourier transform, these are the n T translates of the inverse Fourier transform of 1/(2B)X[-B,B](v).
The inverse Fuorier transform of 1/(2B) X is readily computed, and is the sinc function:
sinc( Bx ) = sin( B x) / (B x )

 

Practically a signal is never band-limited from a strictly mathematical point of view. However it is usually the case that the high frequency components are negligable so that the signal is "physically" band-limited.

The choice of the sampling step T (called also spatial resolution) is a key feature of any vision system. The optical resolution is determined by the physics of the imaging system. The scanner nominal resolution is usually not a true resolution because they often interpolate the sensed values.

Marco Corvi - Page hosted by geocities.com.