We discuss in details the electron scattering pattern on skyrmion-like magnetic textures in two-dimensional geometry. The special attention is focused on analyzing the scattering asymmetry, which is a precursor of the topological Hall effect. We present analytical results valid for the limiting regimes of strong and weak coupling, we also describe the numerical scheme that gives access to the exact solution of the scattering problem. Based on the numerical computations we investigate the properties of the asymmetric scattering for an arbitrary magnitude of the interaction strength and the topology of a magnetic texture. We analyze in details the conditions when the topological charge of a magnetic texture is indeed essential for the appearance of the Hall response. We also describe how the electron scattering asymmetry is modified due to an additional short-range impurity located inside a magnetic skyrmion.
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