  Designing invisible objects without the usage of extreme materials is a
long-sought goal for photonic applications. Invisibility techniques
demonstrated so far typically require high anisotropy, gain and losses, while
also not being flexible. Here we propose an invisibility approach to suppress
the scattering of waves from/to given directions and for particular frequency
ranges, i.e. invisibility on demand. We derive a Born approximation-based
generalized Hilbert transform for a specific invisibility arrangement relating
the two quadratures of the complex permittivity of an object. The theoretical
proposal is confirmed by numerical calculations, indicating that near-perfect
invisibility can be attained for arbitrary objects with low-index contrast. We
further demonstrate the cases where the idea can be extended to high-index
objects or restricted to within practical limits by avoiding gain areas. The
proposed concept opens a new route for the practical implementation of
complex-shaped objects with arbitrarily suppressed scatterings determined on
demand.
