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2
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3
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4
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5
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- Using these two formulas, the real domain image is first transformed
into an intermediate image using N one-dimensional Fourier Transforms.
- This intermediate image is then transformed into the final image, again
using N one-dimensional Fourier Transforms.
- Expressing the two-dimensional Fourier Transform in terms of a series of
2N one-dimensional transforms decreases the number of required
computations.
- Even with these computational savings, the ordinary one-dimensional DFT
has N2 complexity.
- This can be reduced to N log
N if we employ the Fast Fourier
Transform (FFT) to compute the one dimension DFTs.
- This is a significant improvement, in particular for large images.
- There are various forms of the FFT and most of them restrict the size of
the input image that may be transformed, often to N=n2
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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- A number of practical difficulties are encountered in reconstructing a
signal from its samples.
- The sampling theorem assumes that a signal is band limited. In practice,
however, signals are time-limited, not band limited.
- As a result, determining an adequate sampling frequency which does not
lose desired information can be difficult.
- When a signal is under sampled, its spectrum has overlapping tails; that
is F(s) no longer has complete information about the spectrum and it is
no longer possible to recover f(t) from the sampled signal.
- In this case, the tailing spectrum does not go to zero, but is folded
back onto the apparent spectrum.
- This inversion of the tail is called spectral folding or aliasing
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16
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17
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18
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19
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22
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Notes
