The process of sampling

In computer graphics in general and especially in volume visualization we often deal with originally (at least piecewise) continuous functions. Usually, these functions cannot be represented adequately, instead often a few samples at certain positions of the function are given. For example, we can consider a (gray-scale) image as a continuous intensity distribution, but since every pixel can have only one value it has to be discretized before it can be depicted. Similarly, in volume visualization the problem often is to visualize an originally continuous density function of some object (e.g., the patient in medical applications) which is given as a discretized three-dimensional data set (for example, CT or MRI scans).

The preparations from the previous chapters allow to investigate the effects of sampling. What happens when a function is sampled is essentially that it is multiplied with a comb function.

$$\mathrm{f_s}(x)=\mathrm{f}(x)\cdot \mathrm{comb_T}(x)$$ (3.33)

Figure 3.5: Some function was sampled with a certain sampling rate. Its frequency response is depicted on top left. The sampling causes the frequency response to get convolved with an impulse train of reciprocal spacing of the sampling rate. The result is depicted on the right. The original function got replicated and the replicas were moved to the positions of the impulses in the impulse train

The subscription s indicates that the function is sampled (actually it is even discretized). The convolution theorem from Sec. 3.5 tells us that multiplication in spatial domain is equal to convolution in frequency domain. This means that the frequency response of the function that is sampled is convolved with the frequency response of the comb function (which is again a comb function but with reciprocal spacing as already stated in Sec. 3.3), mathematically

$$\mathrm{F_s}(x) = \mathrm{F}(x) \ast \mathrm{comb_{1/T}}(x)$$ (3.34)

But what does this mean? As we saw in Sec. 3.4 convolving a function with a shifted impulse signal shifts the function to the position of the impulse signal. An impulse train now consists of one central impulse and a lot of shifted impulse signals. So when a function is convolved with a comb function it gets replicated and the replicas are shifted to the positions of the impulses in the impulse train. Fig. 3.5 depicts this process graphically. On top left the frequency response of some (band-limited) function is depicted. Below the impulse train is depicted. On the right the result of this convolution (many replicas shifted to the positions of the impulses in the impulse train) is given.

This tells us two things: first, if the functions contains too high frequencies (with respect to the sampling distance), the replicas will mix with the original frequency response and we will never be able to tell them apart again. Second, if the function is sampled with a too low frequency (this means, the replicas in the frequency domain are too close), we get the same displeasing effect that frequencies superimpose, so that they cannot be distinguished any more. This problem is depicted in Fig 3.6 where the sampling rate was lowered in contrast to Fig 3.5 so that the distance between the impulses in the impulse train decreased and the resulting replicas on the right side overlap. However, to reconstruct the original function from a set of samples, just the central copy of the frequency response is needed. If the replicas overlapped it is not possible to get this original central copy without distortions introduced by the other replicas.

Figure 3.6: The same function as in Fig. 3.5 was sampled again but this time with a lower sampling rate. This means that the distance between the impulses of the impulse train decreased and the replicas of the original frequency response overlap.

This problem (and all its implied problems) stays essentially the same regardless in what dimensions the functions we are dealing with are given. In volume visualization usually a three-dimensional functions as a description of some three-dimensional object is given. However, when a three-dimensional function is sampled its three-dimensional frequency response is replicated. This function can be differently band-limited in the each dimension but it can also be sampled with a different sampling rate in each dimension. So the problems in more than one dimensions are the same which allows us to often restrict our discussion to one dimension where it is usually more easily understandable.