Image Processing Reference
In-Depth Information
Fig. 9.1. The graphs in the
top row
illustrate the reconstruction of a 1D function and down-
sampling by the factor 2. The
bottom row
illustrates the data used that achieve this: the original
discrete signal and the sampled filter
1. Figure 9.1 top, left illustrates a sampled discrete signal (green stems) along with
the original (green solid). The dashed magenta curves show the interpolation
functions that can be generated by amplifying the interpolation function with
the corresponding function values. With the magenta solid curve we represent
thereconstructedsignal,i.e.,thesignalthatisobtainedfromthesamplesbysum-
ming up the dashed magenta curves. The reconstructed signal (magenta curve)
is a version of the original which lacks rapid variations. We wish to sample the
original atalarger step sizethan thedistance between theshown samples. How-
ever, a larger step size means smaller repetition period in the frequency domain,
meaning that high frequencies must be deleted before the repetition takes place
(i.e., sampling in the time domain). If we do not suppress the high frequencies,
we will introduce undesired artifacts to the resulting discrete signal. To achieve
this implicit smoothing effect, the interpolation function must be chosen twice
as wide as it needs to be to reconstruct the original signal.
2. On the top, right of the same figure, we show with magenta samples the desired
values, which represent the samples of the lowpass filtered reconstructed signal
using twice as large a discretization step as the original discretization.
3. Onthebottomweshowhowthisdown-sampling isachieved byprocessingonly
discrete signal samples. On the left we show the discrete signal to be down-
sampled. On the right we show the sampled interpolation function. The down-
sampled signal (shown as the magenta samples in top, right) is obtained by con-
