The problem that is presented and investigated is a natural nonlinear extension of the following linear problem. Let H ⊕ H′ and K ⊕ K′ be two orthogonal Hilbert decompositions of a real Hilbert space X. Let P, P′, Q, Q′ and N′ be the operators of orthogonal projection of X onto H, H′, K, K′ and H′ ∩ K′ respectively. Denoting by Z′ the Hilbert space, Z′ = {( a′, b′) ϵ H′ × K′: N′ a′ = N′ b′}, let F be the linear mapping of X into Z′, F( x) = ( P′ x, Q′ x). Under the condition ∥ PQ∥ < 1, which proves to be equivalent to H ∩ K = {0} and H + K closed, F is bicontinuous. The problem is then to choose a constructive procedure for the calculation of a = ( P ∘ F −1) · ( a′, b′), and to analyse the continuity of P ∘ F −1. One may use an iterative technique depending on a real relaxation parameter ω. Let the “separation angle” between H and K be defined by ( H, K) = Arc cos ∥ PQ∥. The present analysis stresses the fundamental part played by the separation angles α = ( H, K), α′ = ( H, K′), β = ( H, SH) and β′ = ( H′, SH) where S (= 2 Q − I) denotes the operator of orthogonal symmetry with respect to K. In the special case where X and H are complex spaces, and K′ = iK, the analysis of the problem is governed by the separation angles β and β′ only. These angles are involved in what may then be called “the conjugate image effect of H with respect to the orthogonal decomposition of X, K ⊕ iK.” Then, α = α′ = β 2 , and the optimal value of ω is known a priori ( ω 0 = 2). This particular problem, which proves to be related to the central problem of Holography, defines what we have called “Abstract Holography”. (One of the main objects of our analysis is to show what underlies the principle of “Wavefront Reconstruction,” which is referred to in Classical Holography, and how it is possible to circumvent certain related difficulties by using an optimal iterative procedure).