Beyond 48 Decibels: Visual Contrast Preserving Representation of High Dynamic Range Mathematical Functions Juha Jeronen Abstract When Gaussian distributed inputs, representing model parameters with some measurement error, are mapped through certain mechanical vibration models, the corresponding output probability distribution exhibits an approximately loga- rithmic data value distribution (in the histogram sense) with a high dynamic range (HDR). We look at applying tone mapping techniques from HDR photography to produce a low dynamic range, visual contrast preserving representation of such high dynamic range mathematical functions — thus enabling HDR plotting. This makes it possible to visualize HDR functions, displaying their structure in a clear manner on standard low dynamic range media such as computer screens and print. The ad- vantages over simple logarithmic scaling are the visual contrast preservation, and data adaptivity. Comparing to histogram equalization, the present approach has the advantage of not exaggerating small contrasts. Three methods are suggested, and demonstrated on two mechanical vibration problems: transverse waves in a classi- cal vibrating string, and the dynamic out-of-plane behaviour of an axially travelling panel submerged in axial potential flow. 1 Introduction What are high dynamic range mathematical functions, and where would one want to visualize them? The motivation for this study comes from physics and engineering problems, where model input is never exact. To obtain reliable analysis results, it is desirable to find out how stable the predictions of a given model are with respect to small perturbations in model input, and how large the expected range of output is. When input uncertainties are present, instead of a single solution, one obtains a solution set corresponding to the admissible inputs. Juha Jeronen Department of Mathematical Information Technology, University of Jyv¨ askyl¨ a, Mattilanniemi 2 (Agora), 40014 Jyv¨ askyl¨ a, Finland, e-mail: juha.jeronen@jyu.fi 1