Physics and Engineering of Radiation Detection

430 Chapter 7. Position Sensitive Detection and Imaging

Image

Multiplication

Spike Function Sampled Data

Frequency Spectrum Spike Function

Convolution

Convoluted Spectrum

Convoluted Spectrum Box Function Reconstructed Spectrum

Multiplication

Convolution

Frequency Spectrum Spike Function Convoluted Spectrum

Convoluted Spectrum Box Function

Multiplication

Reconstructed Spectrum

(d)

(c)

(b)

(a)

(e)

Aliasing

Overlap

Overlap

Transformation

Fourier

Figure 7.1.5: (a) Sampling in spatial domain is equivalent to multiplication of the
image by a spike function. (b) Sampling in frequency domain is equivalent to con-
volution of the frequency spectrum with a spike function. (c) Multiplication of the
convoluted frequency spectrum by a sinc function leads to reconstruction of the
original frequency spectrum provided the sampling condition has been satisfied. (d)
Convolution of the frequency spectrum with spike function leading to sampling at a
frequency lower than the Nyquist frequency. The resulting copies of the frequency
spectrum overlap at the two ends. (e) Multiplication of the frequency spectrum
obtained in the last step with a box function to determine the original frequency
spectrum leads to aliasing.

The above examples clearly demonstrate the importance of Nyquist condition
(fs≥NNyq) in image sampling and reconstruction. However, even though this is a
necessary condition but is in no way the sufficient condition for producing an image