Discreteness of the image. Transition from continuous signals and conversions to discrete

Replacing a continuous image with a discrete one can be performed different ways... It is possible, for example, to choose any system of orthogonal functions and, having calculated the coefficients of the image representation according to this system (according to this basis), replace the image with them. The variety of bases makes it possible to form various discrete representations of a continuous image. However, the most common is periodic sampling, in particular, as mentioned above, sampling with a rectangular raster. This sampling method can be considered as one of the options for using the orthogonal basis, which uses shifted β-functions as its elements. Further, following, in general, we will consider in detail the main features of rectangular sampling.

Let be a continuous image, and a discrete corresponding to it, obtained from a continuous one by rectangular sampling. This means that the relationship between them is determined by the expression:

where are the vertical and horizontal steps or sampling intervals, respectively. Figure 1.1 illustrates the location of samples on a plane for rectangular sampling.

The main question that arises when replacing a continuous image with a discrete one is to determine the conditions under which such a replacement is complete, i.e. not accompanied by the loss of information contained in the continuous signal. There are no losses if, with a discrete signal, it is possible to restore a continuous one. From a mathematical point of view, the question, therefore, is to restore a continuous signal in two-dimensional intervals between nodes in which its values ​​are known, or, in other words, in the implementation of two-dimensional interpolation. This question can be answered by analyzing the spectral properties of continuous and discrete images.

The two-dimensional continuous frequency spectrum of the continuous signal is determined by the two-dimensional forward Fourier transform:

which corresponds to the two-dimensional inverse continuous Fourier transform:

The last relation is true for any values, including at the nodes of a rectangular lattice ... Therefore, for the values ​​of the signal at the nodes, taking into account (1.1), relation (1.3) can be written in the form:

For brevity, we denote by a rectangular section in the two-dimensional frequency domain. The calculation of the integral in (1.4) over the entire frequency domain can be replaced by integrating over individual sections and summing the results:

Changing variables according to the rule, we achieve independence of the integration region from numbers and:

It is taken into account here that for any integer values ​​and. This expression is very close in form to the inverse Fourier transform. The only difference is the wrong form of the exponential factor. To give it the required form, we introduce the normalized frequencies and change the variables accordingly. As a result, we get:

Now expression (1.5) has the form of the inverse Fourier transform; therefore, the function under the integral sign

(1.6)

is a two-dimensional spectrum of a discrete image. In the plane of unnormalized frequencies, expression (1.6) has the form:

(1.7)

From (1.7) it follows that the two-dimensional spectrum of a discrete image is rectangular periodic with periods and along the frequency axes and, respectively. The spectrum of a discrete image is formed as a result of the summation of an infinite number of spectra of a continuous image, differing from each other in frequency shifts and. Figure 1.2 qualitatively shows the relationship between the two-dimensional spectra of continuous (Figure 1.2.a) and discrete (Figure 1.2.b) images.

Rice. 1.2. Frequency spectra of continuous and discrete images

The very result of the summation essentially depends on the values ​​of these frequency shifts, or, in other words, on the choice of sampling intervals. Let us assume that the spectrum of a continuous image is nonzero in some two-dimensional region in the vicinity of zero frequency, that is, it is described by a two-dimensional finite function. If, in this case, the sampling intervals are chosen so that for,, then the superposition of individual branches during the formation of the sum (1.7) will not occur. Therefore, within each rectangular section, only one term will differ from zero. In particular, for we have:

at , . (1.8)

Thus, within the frequency domain, the spectra of continuous and discrete images coincide up to a constant factor. In this case, the spectrum of the discrete image in this frequency domain contains full information about the spectrum of a continuous image. We emphasize that this coincidence takes place only under the stipulated conditions determined by a successful choice of sampling intervals. Note that the fulfillment of these conditions, according to (1.8), is achieved at sufficiently small values ​​of the sampling intervals, which must satisfy the requirements:

where are the cutoff frequencies of the two-dimensional spectrum.

Relation (1.8) defines a method for obtaining a continuous image from a discrete one. To do this, it is sufficient to perform two-dimensional filtering of a discrete image with a low-pass filter with frequency response

The spectrum of the image at its output contains nonzero components only in the frequency domain and is equal, according to (1.8), to the spectrum of a continuous image. This means that the image at the output of an ideal low pass filter is the same as.

Thus, an ideal interpolation reconstruction of a continuous image is performed using a two-dimensional filter with a rectangular frequency response (1.10). It is not difficult to write down explicitly the algorithm for restoring a continuous image. The two-dimensional impulse response of the reconstruction filter, which can be easily obtained using the inverse Fourier transform of (1.10), has the form:

.

The filter product can be determined using a 2D convolution of the input image and a given impulse response. By representing the input image as a two-dimensional sequence of -functions

after doing the convolution, we find:

The resulting relationship indicates a method for accurate interpolation reconstruction of a continuous image from a known sequence of its two-dimensional samples. According to this expression, for accurate reconstruction in the role of interpolating functions, two-dimensional functions of the form should be used. Relation (1.11) is a two-dimensional version of the Kotelnikov-Nyquist theorem.

Let us emphasize again that these results are valid if the two-dimensional spectrum of the signal is finite and the sampling intervals are small enough. The validity of the conclusions drawn is violated if at least one of these conditions is not met. Real images rarely have spectra with pronounced cutoff frequencies. One of the reasons leading to the unboundedness of the spectrum is the limited size of the image. Because of this, when summing in (1.7), the action of terms from neighboring spectral bands manifests itself in each of the bands. In this case, accurate reconstruction of a continuous image becomes generally impossible. In particular, the use of a rectangular filter does not lead to accurate reconstruction.

A feature of the optimal image reconstruction in the intervals between samples is the use of all samples of a discrete image, as prescribed by procedure (1.11). This is not always convenient; it is often required to reconstruct the signal in the local area, relying on a small number of available discrete values. In these cases, it is advisable to apply quasi-optimal recovery using various interpolating functions. This kind of problem arises, for example, when solving the problem of linking two images, when, due to the geometric mismatch of these images, the available readings of one of them may correspond to some points located in the intervals between the nodes of the other. The solution to this problem is discussed in more detail in the subsequent sections of this manual.

Rice. 1.3. Effect of sampling interval on image reconstruction "Fingerprint"

Rice. 1.3 illustrates the effect of sampling intervals on image reconstruction. The original image, which is a fingerprint, is shown in Fig. 1.3, a, and one of the sections of its normalized spectrum is shown in Fig. 1.3, b. This image is discrete, and the value is used as the cutoff frequency. As follows from Fig. 1.3, b, the value of the spectrum at this frequency is negligible, which guarantees a high-quality restoration. As a matter of fact, observed in Fig. 1.3. A picture is the result of restoring a continuous image, and the role of a restoring filter is played by a visualization device - a monitor or a printer. In this sense, the image of Fig. 1.3.a can be considered as continuous.

Rice. 1.3, c, d show the consequences of the wrong choice of sampling intervals. When they were obtained, the “continuous” image discretization was carried out in Fig. 1.3.а by thinning out its samples. Rice. 1.3, c corresponds to an increase in the sampling step for each coordinate by three, and Fig. 1.3, d - four times. This would be acceptable if the values ​​of the cutoff frequencies were lower by the same number of times. In fact, as can be seen from Fig. 1.3, b, there is a violation of requirements (1.9), especially gross with a fourfold decimation of samples. Therefore, the images reconstructed using the algorithm (1.11) are not only defocused, but also strongly distort the texture of the print.

Rice. 1.4. Impact of sampling interval on the restoration of the "Portrait" image

In fig. 1.4 shows a similar series of results obtained for an image of the "portrait" type. The consequences of stronger decimation (four times in Fig. 1.4.c and six times in Fig. 1.4.d) are manifested mainly in the loss of definition. Subjectively, the quality losses seem to be less significant than in Fig. 1.3. This is explained by the significantly smaller spectrum width than the fingerprint image. Sampling of the original image corresponds to the cutoff frequency. As seen from Fig. 1.4.b, this value is much higher than the true value. Therefore, an increase in the sampling interval, illustrated in Fig. 1.3, c, d, although it worsens the picture, still does not lead to such destructive consequences as in the previous example.

In the previous chapter, we studied linear spatially invariant systems in a continuous two-dimensional domain. In practice, we are dealing with images that have limited sizes and at the same time are counted in a discrete set of points. Therefore, the methods developed so far need to be adapted, extended and modified so that they can be applied in this area as well. Several new points also arise that require careful consideration.

The sampling theorem says under what conditions a continuous image can be accurately reconstructed from a discrete set of values. We will also learn what happens when the conditions for its applicability are not met. All of this has a lot to do with the development of visual systems.

Methods requiring transition to the frequency domain have become popular in part due to fast computation algorithms. discrete conversion Fourier. Care must be taken, however, as these methods assume a periodic signal is present. We will discuss how this requirement can be met and what the violation leads to.

7.1. Limiting image sizes

In practice, images always have finite dimensions. Consider a rectangular image with width and height I. Now there is no need to take integrals in the Fourier transform in infinite limits:

It is curious that in order to restore the function, we do not need to know at all frequencies. Knowing that when is a hard constraint. In other words, a function that is nonzero only in a limited area of ​​the image plane contains much less information than a function that does not have this property.

To verify this, imagine that the screen plane is covered with copies of a given image. In other words, we expand our image to a function that is periodic in both directions

Here is the largest integer not exceeding x. The Fourier transform of such a multiplied image has the form

Using appropriately selected convergence factors in exercise. 7.1 it is proved that

Hence,

whence we see that it is equal to zero everywhere, except for a discrete set of frequencies Thus, to find it is enough for us to know at these points. However, the function is obtained from a simple clipping of the area for which. Therefore, in order to restore it is enough for us to know only for all. This is a countable set of numbers.

Note that the transformation of the periodic function turns out to be discrete. The inverse transformation can be represented as a series, since

Another way to verify this is to consider a function as a function obtained by cutting off some function for which inside the window. In other words, where the window selection function is defined as follows.

Analog and discrete presentation of images and sound

A person is able to perceive and store information in the form of images (visual, sound, tactile, gustatory and olfactory). Visual images can be saved in the form of images (drawings, photographs, etc.), and sound images can be recorded on records, magnetic tapes, laser disks, and so on.

Information, including graphics and sound, can be presented in analog or discrete form. With an analog representation, a physical quantity takes on an infinite set of values, and its values ​​change continuously. In a discrete representation, a physical quantity takes on a finite set of values, and its value changes abruptly.

Let's give an example of analog and discrete representation information. The position of the body on an inclined plane and on a staircase is set by the values ​​of the X and Y coordinates.When a body moves along an inclined plane, its coordinates can take on an infinite set of continuously changing values ​​from a certain range, and when moving along a staircase, only a certain set of values, and changing abruptly (Fig. . 1.6).

An example of an analogue representation of graphic information can be, for example, a painting canvas, the color of which changes continuously, and discrete - an image printed using inkjet printer and consisting of separate dots of different colors. An example of analog storage audio information is an vinyl record(the sound track changes its shape continuously), and the discrete one is an audio CD (the sound track of which contains sections with different reflectivity).

Conversion of graphic and sound information from analog to discrete form is performed by sampling, that is, splitting a continuous graphic image and a continuous (analog) sound signal on individual elements... In the process of sampling, coding is performed, that is, the assignment of each element to a specific value in the form of a code.

Sampling is the transformation of continuous images and sound into a set of discrete values ​​in the form of codes.

Questions to Think About

1. Give examples of analog and discrete ways of presenting graphic and sound information.

2. What is the essence of the sampling process?

Images consisting of discrete elements, each of which can take only a finite number of distinguishable values ​​that change over a finite time, are called discrete. It should be emphasized that elements of a discrete image, generally speaking, can have an unequal area and each of them can have an unequal number of distinguishable gradations.

As shown in the first chapter, the retina transmits discrete images to the higher parts of the visual analyzer.

Their apparent continuity is only one of the illusions of sight. This "quantization" of initially continuous images is determined not by the limitations associated with the resolving power of the optical system of the eye and not even by the morphological structural elements of the visual system, but by the functional organization of the neural networks.

The image is broken down into discrete elements by receptive fields that combine one or another number of photoreceptors. The receptive fields produce the primary isolation of the useful light signal by spatial and temporal summation.

The central part of the retina (fovea) is occupied only by cones; on the periphery, outside the fovea, there are both cones and rods. Under conditions of night vision, cone fields in the central part of the retina have approximately the same size (about 5 "in angular measure). The number of such fields in a fovea, whose angular dimensions are about 90", is about 200. The main role in conditions of night vision is played by rod fields occupying the rest of the retina. They have an angular size of about 1 ° over the entire surface of the retina. The number of such fields in the retina is about 3 thousand. Not only detection, but also examination of poorly illuminated objects under these conditions is carried out by the peripheral areas of the retina.

With increasing illumination, another system of storage cells, cone receptive fields, begins to play the main role. In the fovea, an increase in illumination causes a gradual decrease in the effective field value until, at a brightness of about 100 asb, it decreases to one cone. At the periphery, with an increase in illumination, the rod fields are gradually turned off (inhibited) and the cone fields come into action. Cone fields at the periphery, like foveal fields, have the ability to decrease depending on the light energy incident on them. The largest number of cones, which can have cone receptive fields with increasing illumination, grows from the center to the edges of the retina and at an angular distance of 50-60 ° from the center reaches approximately 90.

It can be calculated that in conditions of good daylight the number of receptive fields reaches about 800 thousand. This value roughly corresponds to the number of fibers in the human optic nerve. The discrimination (resolution) of objects in daytime vision is carried out mainly by the fovea, where the receptive field can be reduced to one cone, and the cones themselves are most densely located.

If the number of storage cells of the retina can be determined in a satisfactory approximation, then there is still insufficient data to determine the number of possible states of the receptive fields. Only some estimates can be made based on the study of the differential thresholds of receptive fields. The threshold contrast in foveal receptive fields in a certain operating range of illumination is of the order of 1. The number of distinguishable gradations is small. In the entire range of restructuring of the cone foveal receptive field, 8-9 gradations differ.

The period of accumulation in the receptive field - the so-called critical duration - is determined on average by about 0.1 s, but at high illumination levels it can apparently decrease significantly.

In fact, the model describing the discrete structure of the transmitted images must be even more complex. The relationship between the size of the receptive field, thresholds and critical duration, as well as the statistical nature of the visual thresholds, should be taken into account. But for now, this is not necessary. It is enough to represent as an image model a set of elements of the same area, the angular dimensions of which are less than the angular dimensions of the smallest detail resolved by the eye, the number of distinguishable states of which is greater than the maximum number of distinguishable brightness gradations, and the discrete change time of which is less than the period of flickering at critical flicker fusion frequency.

If you replace images of real continuous objects outside world such discrete images, the eye will not notice the substitution. * Consequently, discrete images of this kind contain at least no less information than the visual system perceives. **

* Color and volumetric images can also be replaced with a discrete model.
** The problem of replacing continuous images with discrete ones is of great importance for film and television technology. Time quantization is at the heart of this technique. In pulse-code television systems, the image is also split into discrete elements and quantized in terms of brightness.

Analog and discrete image. Graphic information can be presented in analog or discrete form. An example of an analog image is a painting canvas, the color of which changes continuously, and an example of a discrete image, a drawing printed using an inkjet printer, consisting of separate dots of different colors. Analog (oil painting). Discrete.

Slide 11 from presentation "Coding and processing of information"... The size of the archive with the presentation is 445 KB.

Informatics grade 9

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