# Isoplanarity Problem in Vision Theory

## Introduction to the Problem

The problem of the isoplanarity or shift-invariant problem in the vision theory begun to be discussed with the process of imaging of stationary objects through scattering and absorbing media taken as an example [1—10]. The considered approach to estimation of the size of isoplanarity zones is also applicable, from our point of view, to the observation of objects through turbulent media [11—28]. The main equation used in the vision theory for the construction of key functions and characteristics allowing analysis of the influence of a scattering medium on the imaging process is the stationary radiation transfer equation

Here, w is the radiation propagation direction, *I* is the radiation flux intensity, *в**ех(* and */3** _{sc}* are, respectively, the extinction and scattering coefficients, r is the vector of point coordinates in space, w' is the direction of radiation propagation after scattering,

*g*is the scattering phase function, and Ф

_{0}is the source function.

The particular formulation of the problem of vision theory is determined, as known, by the boundary conditions. The general solution of Equation 6.1 is the Green’s function G(r_{0},w_{0}; r,w) with the boundary conditions specified in the form of a omnidirectional point source *6**(r* — r_{0})<5(w — w_{0}) placed at the point r_{0} and emitting in the direction w_{0} (see definitions also in Chapter 2). Let they correspond to the night (no external sources of illumination) observation of objects on the surface z = 0 with a projection-type optical system being on the surface z = z_{1} and oriented, for example, perpendicularly to the object plane (nadir observation). Then they can be written in the form

where *n, n** _{1}* are external normals to the surfaces z = 0 and z = z

_{1}.

For the construction of the projective or scanner image of an object observed through the scattering medium under these observation conditions, it is sufficient to know (at least, for the central isoplanarity zone of the image) the function G(r_{0},w_{0}; r*,w*) only at the center of the entrance pupil of the optical system, that is, at the point r* and for one given direction w*. If w_{0} is fixed, then the function *G =* G(r_{0},w_{0}; r*,w*) = G(r_{0}) is identical to the point spread function h(r) from the viewpoint of the theory of linear systems (it is applicable in this case because Equation 6.1 is linear with respect to the intensity I), and the twodimensional Fourier transform of this function *F* ^{2}[G (r)] = *F* ^{2}[h(r)] = *H(w,* у) is the optical transfer (in general case, complex) function, where w,y are the spatial frequencies. The subscript 0 of r is usually omitted from here on in this chapter to emphasize that it is an argument of the Green’s function or the point spread function (PSF) rather than a fixed point at the line of sight of the optical system.

Are the functions G(r) = h(r) and *H* (w, у) individual characteristics of this particular system independent of the properties of objects and (even ideal) optical systems? Is it possible to construct images of some objects with the aid of the function G(r) (or H(w, у))? Obviously, yes, but only for the central point of the image plane of the ideal optical system oriented in the direction w* and *only *for objects, whose every point emits radiation energy strictly in the direction w_{0}. Evidently, it is hard to find objects with such properties in reality. The only exclusion is the trivial case of observation of a collimated laser beam propagating along the optical axis of imaging system through the scattering medium.

As a rule, elementary parts of natural objects are characterized by the angular directional pattern markedly different from *6(w* — w_{0}). For simplicity, we consider homogeneous (in the optical sense) object surfaces, that is, objects, in which radiation from every point is described by the same angular directional pattern Q(w,r) = Q(w).

Thus, to the describe the process of imaging through the scattering medium with the aid of the Green’s functions, it is necessary to solve Equation 6.1 for every of *N 6(w* — w_{p}r) sources. This set of the radiators should allow reconstruction of the real angular directional pattern Q(w) of radiation from object surface elements, that is

This, in its turn, allows the total PSF to be constructed, for example, as follows:

where p is the “weight” of the *i-th* direction of radiation from a monodirectional point source in Equation 6.3, and G/r) is the Green’s function corresponding to this direction, *AQ, _{i}* is the solid angle, within which it is assumed that

*G*const. The accuracy of h(r) reconstruction obviously depends both on the accuracy of the G/r) estimation method and on the choice of

_{i}(r) =*N*and

*AQ*in Equation 6.4.

_{i}In the cases when the imaging requires consideration of the sizes of scattering spots arising because projective systems are used to construct plane images of volume objects [22], it is necessary to find the Green’s functions already for the entire plane of the entrance pupil and all directions within a hemisphere rather than for the single point r* and the selected direction w*. That is, it is necessary to find either *G =* G(r,w; r*,w*), where r* ? Щ (a point in the coordinate space), w* ? {П+} (unit vector in the directional half-space) or *G* = G(r,w;?), where *?* is the current coordinate along the section of the optical axis of the receiver between the center of the entrance pupil and the intersection with the object plane.

Remember that in the linear-system approach, as defined, for example, in Reference 23, the point spread function is understood as a solution of Equation 6.1 with the following boundary condition:

That is, in this case, Equation 6.1 is solved only once, and this solution is the point spread function, allowing us to construct, using the convolution integral, the image of any optically homogeneous objects (at least, in the central isoplanar zone), whose every pony emits by the law Q(w).

Thus, the method of Green’s functions allows the solution of problems of the vision theory in the more general form than within the framework of the linear- system approach. However, it is not always possible to implement this approach in practice, because usually there is no exact solution of Equation 6.1 in the analytical form (for example, when solving particular problems of vision in the atmosphere or through the atmosphere). Therefore, in the cases when the complexity of solution of Equation 6.1 with boundary conditions (6.2) for the only one given direction

is comparable with solution of this equation with boundary conditions (6.5), the linear-system approach becomes more efficient. It can be easily shown, as in Reference 22 that within the framework of this approach, the finite focal depth of the space imaged by an optical system can be taken into account rather readily. To the full extent, the advantages of the linear-system approach show themselves in solution of Equation 6.1 by the Monte Carlo technique, the principle of which is described in Chapter 5.

It is obvious that the above statements are also true for the case of observation under daytime conditions. The similar reasoning can also be given for transfer properties of the channel of additional illumination of the object plane [11] with the only difference that for this channel there is no need to take into account any characteristics of the optical imaging system. This influence function (or the corresponding optical transfer function) allows us to take into account the processes of radiation reflection (rereflection) by the surface of objects with the following scattering in the medium toward this surface.