Motivated by the analysis of a Positron Emission Tomography (PET) imaging data considered in Bowen et al. [Radiother. Oncol. 105 (2012) 41-48], we introduce a semiparametric topographical mixture model able to capture the characteristics of dichotomous shifted response-type experiments. We propose a pointwise estimation procedure of the proportion and location functions involved in our model. Our estimation procedure is only based on the symmetry of the local noise and does not require any finite moments on the errors (e.g., Cauchy-type errors). We establish under mild conditions minimax properties and asymptotic normality of our estimators. Moreover, Monte Carlo simulations are conducted to examine their finite sample performance. Finally, a statistical analysis of the PET imaging data in Bowen et al. is illustrated for the proposed method.