Aliasing
Distortion
Sampling theorem guarantees exact recovery of
a continuous image, which is lowpassand band limited to frequencies FxS/2 and FyS/2, from its samples taken atFxS and FyS
samples per unit distance. If the sampling rates are below the Nyquistrates,
namely, FxS and FyS, then the continuous image cannot be recovered exactlyfrom
its samples by filtering and so the reconstructed image suffers from a
distortionknown as aliasing distortion. This can be inferred from Figure 1.
Figure 1.ashows the Fourier transform of a continuous lowpass image with cutoff
frequenciesof fxC and fyC in the two spatial directions. Figure 1 b shows the
Fourier transformof the sampled image, where the sampling frequencies equal
twice the respectivecutoff frequencies. It is seen from Figure 1 b that there
is no overlap of the replicasand that the Fourier transform of the sampled
image is identical to that of the continuous image in the region [(−
fxC , fxC ) ×(− fyC , fyC)], and therefore, it can be recovered by filtering
the sampled image by an ideal lowpass filter with cutofffrequencies equal to
half the sampling frequencies.
Figure
3.Fourier domain illustration of aliasing distortion due to
sampling a continuous image: (a) Fourier transform of a continuous lowpass
image, (b) Fourier transform of the sampled image with sampling frequencies
exactly equal to the respective Nyquist frequencies, (c) Fourier transform of
the sampled image with undersampling, and (d) Fourier transform of the sampled image
for oversampling.
When the sampling rates are less than
the Nyquist rates, the replicas overlap and the portion of the Fourier
transformin the region specified by [(- fxC , fxC ) ×(- fyC , fyC)] no longer
corresponds to that of the continuous image, and therefore, exact recovery is
not possible, see Figure 1.c. In this case, the frequencies above fxS/2 and
fyS/2 alias themselves as lowfrequencies by folding over and hence the name
aliasing distortion. The frequencies
fxS/2 and fyS/2 are called the fold over
frequencies. When the sampling rates are
greater than the corresponding Nyquist
rates, the replicas do not overlap and there is
no aliasing distortion as shown in
Figure 1.d.
Read a grayscale image,
downsample it by a factor of M, and reconstructthe full size image from the
downsampled image. Now, prefilter the original image with a lowpass filter and
repeat downsampling and reconstruction as before.Discuss the effects. Choose a
value of 4 for M.
Let us read Barbara image, which
is of size 661 × 628 pixels. Figure 2 ais the original image downsampled by 4
without any prefiltering. The reconstructed image is shown in Figure 2 b. The
image has been cropped so as to have a manageable size. We can clearly see the
aliasing distortions in both figures—the patterns in the cloth. Now, we filter
the original image with a Gaussian lowpass filter of size 7 × 7 and then
downsample it as before. Figures 2 c,d correspond to the downsampledand
reconstructed images with prefiltering. We see no aliasing distortions in the
downsampled and reconstructed images when a lowpassprefiltering is applied before
downsampling. However, the reconstructed image is blurry, which is due to lowpass filtering.
Figure 2.Illustration of aliasing distortion in a real image: (a)
original image downsampled
by a factor of 4 in both dimensions with no prefiltering, (b)
reconstructed image and cropped
to size 215 × 275, (c) original image prefiltered by a 7 × 7
Gaussian lowpass filter and then
downsampled
by 4 in both directions, (d) image reconstructed from that in (c). In (b) and
(d), the reconstruction is carried out using bicubic splinMore Matlab Codes:
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