# Criteria for the Estimation of the Size of Image Isoplanarity Zones

Consider the isoplanarity of images formed in observations through scattering media. Remember that, for ideal optical systems, the pulsed reaction h(r) is understood as an image of a point, and it is applicable to description of an image over the whole frame or the field of view. For vision systems, it is not usually true, even if the

*Isoplanarity Problem in Vision Theory ?* 309

Figure 6.2 Example of simulation of the function *h _{1}(x,y;^-_{[}).*

optical source is assumed to be ideal. The point spread function of vision systems is characterized by the far wider half-width, which can exceed the frame size many times [11].

If we know the influence function determined for the central point of the frame, then this is by no means always possible to reconstruct (with the required accuracy) the image of an elementary volume—a radiant point against the background of an absorbing surface. This theoretically and experimentally established fact was used in Reference 27 to find criteria of the size of the central isoplanarity zone in vision systems. At the same time, this fact indicates the principal difference in the physical meaning of the point spread function used in the theory of analysis of optical systems and the characteristics used in the vision theory, in which the point spread function does not always correspond to the image of a point source.

*Criterion 1* [11,27]. Let the observation be carried out through the scattering medium with an optical (or opto-electronic) system having the large field-of-view angle. In this case, it is worth asking how many isoplanarity zones can be separated in a given image or how many linear-system characteristics (pulsed reactions or optical transfer functions) should be used for the correct construction (with the required accuracy) of object images or for removal of a “trace” of scattering medium from an image.

We can answer this question using criteria [27] for estimates of the size of the *central* isoplanarity zone of images. It should be noted that even if we know the entire set of the Green’s functions necessary for particular formulation of the problem, we cannot answer this question until the corresponding pulsed reactions of the imaging channels are determined with the use of these functions, that is, until real reflective (emissive) properties of the object plane are taken into account.

The image of some or other object can be obtained in different ways. The following procedures are the well known [28]:

- 1. Radiation brightness in the same fixed direction is measured at different points of the plane, where the detector is placed (spatial scanning)
- 2. Angular distribution of brightness is measured at one fixed spatial point (projective images)
- 3. The image is formed with a raster-type system when the angular scanning is carried out along one coordinate, while the detector moves along another axis

In the general case, in the presence of a scattering medium between the object and the receiving device, all the three images are different, even if the optics is assumed ideal.

Thus, for the isoplanar vision system (or invariant to the shift [21]), only one function

can be used for image construction in place of the infinite set of the functions *h(x,y;*

*x',y').*

This simplifies significantly the image reconstruction process. However, the property of isoplanarity is inherent, strictly speaking, to only the imaging system employing the procedure (a) and under condition of horizontal homogeneity of the medium. Nevertheless, in the most applied problems of the vision theory for the procedures (b) and (c), it is possible to find the ranges of *(x,y),* within which Equation 6.9 is fulfilled with a certain degree of accuracy.

Two criteria for the estimation of the size of the central isoplanar zone (isozone) for the scheme of object observation through a scattering layer are proposed in Reference 27. Consider the image of the simplest object Q(w;?,n) = *Q(w)**6**(x* — *Q6**(y* — *n**)*—the point source of radiation placed at the point (?,n), formed by the method of spatial scanning. From here on, we assume that the coordinates in the object and image planes are reduced to the same scale. Then we can write

Here, *?* and *n* are coordinates in the image plane or in the object plane.

This means that in the case of imaging by the procedure (a), the image of point is, accurate to a constant, the point spread function and the image is isoplanar all over the frame.

The point spread function (or the optical transfer function) determined for the center of the frame in case of the schemes (b) and (c) allows reconstruction of the image of any optically homogeneous object at only one, namely, central point of the frame. Or, in other words, if condition (6.9) is not fulfilled, then the images formed by the algorithms (b) and (c) are not isoplanar in the strict sense.

Let us analyze the above mentioned geometrical explanation presented in Figure 6.3, which shows the scheme of formation of the image of a point through the scattering medium by the projective optical system *L* (Figure 6.3a) and the scheme of determination of PSF of the vision system (Figure 6.3b) for the central point of the frame.

We assume that the axis of the system *L* is directed along the vector w0 (Figure 6.3b), Q(w) = Q(v,<^) = Q(v) (here, *v* is the zenith angle, and is the azimuth angle of the direction w), and the scattering medium is homogeneous along the planes *z **= **const.* Then *h(**9**;*w0) = *h(x,*y;w0) = h(r;w0), where *r =* (*x*^{2} + *y*^{2}), that is, the pulsed reaction has the axial symmetry and decreases monotonically with an increase of the argument. Here

*вi* is the angle between the direction w0 and the direction *w** to the point from the center of the entrance pupil of the optical system, *H* is the distance between the plane of the entrance pupil of the optical system and the object plane.

(b)

*L*

(a)

*L*

А/

*/в;*

*Н*

Figure 6.3 Scheme of formation of the image of point (a) and PSF (b) for the central isoplanar zone.

If PSF can be defined as dependence of brightness on the angle *9,,* then the measure of PSF *h(9)* mismatch with the image of the point *q(9)* can be introduced as follows:

It should be noted that in this case h(0) = q(0). Then the angular size *0 _{iso}* of the central isoplanarity zone can be found from the condition

Then, using Equation 6.11, we can determine the radius *Rf _{o} = R^{0}o* (u0) of this zone around the point A

_{0}or (using the scaling coefficient) around the center of the image plane. An increase in the level of error e

_{0}predetermines the obvious growth of R0.

It follows from the data presented in References 11,27 that criterion (6.12) is very sensitive to variations of optical-geometric parameters of the observation scheme. The integral criterion is somewhat more sensitive in this respect. Following to this criterion, the size of the isoplanarity zone can be determined from the condition

where

Here, *r* and *в* are related by Equation 6.11.

The results of laboratory (observation through a cell with the scattering medium) and numerical experiments (for the scheme of vertical observation through the cloudless atmosphere and under overcast conditions) reported in References 11,27 can be reduced to the following conclusions. The size of the central isozone is a complex, ambiguous function of optical-geometric conditions of observation. It depends on the optical thickness of the medium, the distribution of scattering and absorption coefficients at the line of sight, and the scattering phase function. The isoplanarity zone *D(x,y)* converges to the center with a decrease of e_{0} or *6 _{0}.* In particular, in the scheme of numerical experiments considered in References 11,27 at ?

_{0}и 20%, the angular dimensions of the radius of the

*D(x,y)*zone took values from few degrees to tens of degrees depending on the initial optical-geometric conditions. We assume that the radius of the central isozone is smaller than the frame size. This means that we cannot construct the image of even a point object with the required accuracy starting from

*r > R*

^{0}o.

Consequently, keeping within the framework of the theory of linear systems, we should construct the following pulsed reaction *h _{1}(x,y* ;u>j), that is, PSF for the direction (Figure 6.4b) oriented, for example, to the boundary of the central

Figure 6.4 Scheme of formation of the image of a point (a) and PSF (b) for the peripheral isoplanar zone.

isoplanar zone. This allows us to construct the image of a point object at *r > R ^{0}*

*, and at*

_{o}*r = R*

^{0}*it coincides exactly with h^OThuij).*

_{o}It should be noted that PSF for this isoplanar zone differs significantly from h(r;u>0) determined for the image center. This difference consists, first of all, in the fact that the function h_{1}(x,jy;wj) does not have axial symmetry, although it is independent of the azimuth angle between the projection of the direction u>j onto the plane *z* = 0 and any point on the circle of the radius *R ^{0}*

*on this plane (even if we assume axial symmetry of the function Q(w) and homogeneity of optical properties of the medium in the directions perpendicular to the vector u>0).*

_{o}How can we determine the size of the following isoplanar zone of the image? Figure 6.4 shows the scheme of formation of the image of a point object for the second isoplanar zone (Figure 6.4a) and determination of the pulsed reaction h_{1}(x,y;wj) for this zone (Figure 6.4b). For determination of the angular or linear size of the second isoplanar zone, we can use criterion (6.12) upon the replacement of *h**(**ff**)* = h((9;u>0) = *h**(**x,y**) **= h**(**r**)* with h_{1}(0;wj). Then the criterion for estimation of the size of this zone has the following form:

where *p* is the azimuth angle.

Then the angular size *в**] _{ю}* of the second isoplanar zone can be found from the condition

The linear size of the radius *Rl** _{o}* can be determined using either Equation 6.11 or

The integral criterion in this case takes the form where

It should be noted that, using the same values of e_{0} (or *6** _{0})* for estimation of the sizes of all isoplanarity zones, we reconstruct the image of a point with the same error.

Following this line of reasoning, we can estimate the size of the following zones of image isoplanarity.

*Criterion 2* [26]. Thus, there are situations when it is necessary to separate the observed surface into zones of isoplanarity. If the spherically homogeneous atmosphere is considered, then the isoplanarity zones are rings around the center of the observed fragment of the surface.

Assume that the optical system is directed at the boundary of i-th isoplanarity, and the use of PSF of previous zone for construction of the image in the *i*-th zone leads to an intolerable error. This error can be determined by the equation

Here, *E'* is the radiation intensity reconstructed with the use of PSF of the previous zone, I/+1 is the exact value of intensity reconstructed with the aid of PSF with the optical system oriented at the zone boundary, *E* is the surface brightness, *E _{0}* is the surface illumination, and

*p*is the distribution of the reflection coefficient over the surface.

If we expand *E(x — X , y — y')* in Equation 6.18 into the Taylor series, then obtain

where

—

The similar expansion of the convolution can be found, for example, in Reference 21, p. 36.

If we neglect all the expansion terms but the first one, we obtain approximately

Therefore, the relative difference of the intensities reconstructed with different PSFs can be estimated approximately by Equation 6.24.

The integral characteristics *m _{00}(9),* in their turn, can be approximated by the angles of orientation of the receiving system

*в.*In solution of particular problems, we recommend the approximations of the integral characteristics

*m*

_{00}to be constructed. In Reference 26, for the following ranges of optical-geometric parameters of numerical experiments: wavelength A = 0.35—0.8 pm, meteorological visibility range

*S*1—50 km, height of position of the optical systems above the Earth’s surface

_{M}=*H >*10 km, and zenith angle (determining its orientation) 0 = 0—60°, it is proposed to use the following approximation equation:

Then, upon transformation of Equation 6.24 and expression of *в++ _{х},* we obtain

where m—0 is the function inverse to *т _{00}(в)*

If we substitute Equation 6.25 into Equation 6.26, then we get

Thus, we obtain the following criterion for determination of boundaries of isoplanarity zones:

Using Equation 6.28, we obtain the values of the orientation angles of the optical system *в,* which determine the boundary of the isoplanarity zones. Upon the calculation of PSF for given *в _{t}* and using Equation 6.29, we can find the image of the observed surface

where *r(x',y')* is the distribution of the reflection coefficient on the surface, E_{0}(x,y) is the distribution of Earth illumination by the sun.