This work investigates the capabilities of methods based on morphological data demixing in appli-cation to anomaly detection and triangulation in solid media with significant structural complex-ity. The operating principle of morphological demixing is the decomposition of the spatiotemporal response of a structural system into two distinct contributions that are antithetical and complemen-tary in terms of their morphology. While the bulk of the dynamic response can be generally repre-sented in terms of smooth functions, the effect of nominally rare anomalies is captured through the projection of the data on a dictionary of sparse basis functions. The resulting sparse representa-tion de facto distills the part of the response carrying the signature of the defects, thus promoting their agile triangulation. It has been shown that the triangulation of rare and weak anomalies can be enhanced through the adoption of dictionaries whose atoms feature, in addition to the above mentioned sparsity requirements, other morphological attributes that are germane to the scattered fields observed in the neighborhood of localized defects. This additional capability introduced through morphologically germane dictionaries results in a more robust discrimination between the signatures of true defects and the spurious response features due to other (often benign) struc-tural elements. Our objective here is to explore the use of a dictionary constructed specifically to represent the anomalous response, whose atoms have morphological characteristics that mirror the structure produced in the response by a point excitation. The conclusions of our analysis are supported by numerical and laser-Acquired experimental data.