Immersion extension-lift over a Morse function

Let V be a compact connected oriented surface with boundary and f:∂V×[0,1)→R a non-singular function such that f|∂V×{0} is a Morse function. Let ι:∂V×[0,1)→V be a collaring of ∂V and π:R2→R an orthogonal projection. In this paper, we study existence of an orientation preserving immersion F:V→R2 such that π∘F∘ι=f. We also study image homotopy classes of F when we fix f and study relation between two image homotopy classes when f is deformed under a Morse homotopy.