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Weak solutions to the near-field reflector problem with spatial restrictions approached with generalized reflectors constructed from ellipsoids

We motivate then formulate a novel variant of the near-field reflector problem and call it the near-field reflector problem with spatial restrictions. Let $O$ be an anisotropic point source of light and assume that we are given a bounded open set $U$. Suppose that the light emitted from the source at $O$ in directions defined by the aperture $D\subseteq S^2$, of radiance $g(m)$ for $m\in D$, is reflected off $R\subset \overline{U}$, creating the irradiance $f(x)$ for $x\in T$. The inverse problem consists of constructing the reflector $R\subseteq \overline{U}$ from the given position of the source $O$, the input aperture $D$, radiance $g$, `target' set $T$, and irradiance $f$. We focus entirely on the case where the target set $T$ is finite.