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 approximationbased 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.Full invisibility, or cloaking, was proposed using transformation optics or, equivalently, conformal mapping 1,2 . The idea is elegant and fascinating; however, it can hardly cross the limits of science fiction, since the complexity of the required metamaterials severely limit practical realizations. Therefore, actual cloaking schemes generally scarify the perfect wavefront reconstruction or operate under a narrow bandwidth, as for instance in carpet cloaking 3-7 , plasmonic cloaking 8,9 , or mantle cloaking with thin patterned metasurfaces 10 , metallic scatterer 11 or dielectric coating 12 based cloaking, among others 13,14 .A completely different approach to the concept of invisibility, referred as "unidirectional invisibility", relays on systems described by non-Hermitian Hamiltonians 15-17 . The concept is based on the property of an object to be invisible when probed by a wave from one side only. Such effect is accomplished by specific complex-modulated potentials (in optical terms: specific refraction index and gain/loss distributions), that allow suppressing the scattering of radiation from an object.Unidirectional invisibility was first proposed for parity-time (PT) symmetric periodical systems (defined by symmetric index modulations accompanied by anti-symmetric gain/loss distributions), close to so-called PT-symmetry breaking point. Initially proposed for narrow frequency bands (due to the resonances of the periodic structure), and for particular incidence directions 18-20 , the idea was extended to broad band radiation (both in frequency and in propagation direction) also by considering non-PT-symmetric potentials 21-25 .More recently, "unidirectional invisibility" has been related to the more general class of non-Hermitian potentials fulfilling the spatial Kramers-Kronig (KK) relations 26 . In the same way as the causality in time imposes KK relations in frequency, analogously, the KK theory may be directly extended to attain unidirectional invisibility...