## 2 Scale-Space Methodology  
Scale-space theory is a framework for representing signals on multiple scales, developed by the computer vision, image processing and signal processing communities [8]. A recent review of statistical scale-space methodology can be found in [9]. The goal of statistical scale-space methodology is to extract statistically significant features from noisy data at several scales, often corresponding to different levels of resolution in the underlying object of interest. The data could be a set of observed curves where features at different levels of resolution might be of interest. These curves could, for example, correspond to spectral signatures from fish being frozen for different numbers of days, as is the case in our real data application. One acquisition of data consists of a number of \(p\)-dimensional vectors with unknown distribution, each vector representing the spectral signature at a particular pixel in the hyperspectral image. Thus, in our application, \(p\) represents the number of frequency bands (spectral channels) in the spectral signatures. In Section 4.2 we analyze three different acquisitions from the frozen fish. Under the null hypothesis, the number of days is assumed the same and the distributions are therefore assumed identical. In our approach, we perform several tests to flag when a new acquisition differs significantly from several previous acquisitions of day 0. The outcome of the tests is presented as a scale-space map, described in more detail below. The core method of this paper is to test simultaneously for many different scales and positions (frequency bands). The scale \(s\) equals the number of different frequency bands being summed across. To be specific, this means that scale \(s=1\) corresponds to the situation where we test if the observed values at spectral frequency \(d\) are different between acquisitions of spectral signatures. At scale \(s=3\) and position \(d\), a smoothing in terms of a weighted average of the observed values for spectral frequencies \(d-1\), \(d\) and \(d+1\) are used to test whether the acquisitions are different. The weights are calculated from an Epanechnikov kernel function (i.e., parabolic function) [10], the same as in [7]. For other scales, completely analogous smoothing over the frequency bands are made and used to perform the tests. Note that by applying this smoothing, we are able to test for differences in the acquisitions at all locations for a large number of scales. In fact, the tests are performed at all \(p\) spectral frequencies for a total number of \(n_{s}\) different scales. Instead of looking at a single location or a single scale, the described scale-space approach can help detect changes that appear at several levels of smoothing, i.e., resolution. However, when testing for differences between spectral signature curves in different acquisitions it can be difficult to select the critical rejection thresholds due to multiple testing. One possibility is to use the Bonferroni correction method [11] designed to reduce false positives in testing multiple hypotheses. As an alternative, we also tried the statistical inference method described in [12] to find suitable critical rejection thresholds for the scale-space map. The critical values are used to test if a new acquisition differs from the existing acquisitions. The training procedure at a location \((d,s)\) is accomplished by comparing one acquisition to the others. To simplify the description we illustrate the methodology by testing for change in the sample mean, \(\bar{X}\), over the pixels in the image. This training-procedure is the core difference between the method presented here and in [7], where there is a more direct comparison between curve families. Also, instead of the non-parametric Andersson-Darling test combined with either Bonferroni or False Discovery Rate correction for multiple hypothesis testing employed in [7], our novel method uses the \(t\)-test either with a Bonferroni correction or the inference approach suggested in [12].