Inger Nicolette Fabris-Rotelli

LULU Operators and Discrete Pulse Transform for Multidimensional Arrays

The LULU operators for sequences, that is Ln, Un and their compositions, are extended to multidimensional arrays in a way which preserves their essential properties, e.g., consistent separation, total variation and shape preservation. The power of the operators is demonstrated by deriving the discrete pulse transform (DPT), which is a hierarchical decomposition of the input array into pulses. Similar to its 1-D counterpart this transform satisfies a basic consistency property.

Discrete Pulse Transform of Images

The Discrete Pulse Transform (DPT) of images is defined by using a new class of LULU operators on multidimensional arrays. This transform generalizes the DPT of sequences and replicates its essential properties, e.g. total variation preservation. Furthermore, the discrete pulses in the transform capture the contrast in the original image on the boundary of their supports. Since images are perceived via the contrast between neighbour pixels, the DPT may be a convenient new tool for image analysis.

The impact of the Canterbury Earthquakes on the temporal and spatial patterning of crime in Christchurch, New Zealand

The Canterbury Earthquakes struck the Canterbury region of the South Island of New Zealand between September 2010 and February 2011. The Earthquakes resulted in widespread structural damage to Christchurch, the main city of the region, and greatly impacted other aspects of society including crime. In this study, we adopt an exploratory approach to investigate the impact that these earthquakes have had on the temporal and spatial patterning of four types of crime in Christchurch: assault, domestic violence, burglary and arson. Overall crime has decreased in post-quake Christchurch with the notable exception of domestic violence. We found remarkably similar temporal signatures of crime for all crime types occurring across both the pre- and post-earthquake periods. Spatially, crime has increased in the majority of neighbourhoods in Christchurch post-quake despite overall crime levels being down. Explanations for this paradoxical and other finding are outlined in the context of a rebuilding and recovering city.

On the axiomatization of scale-space theory

Scale-spaces are a common method for analysis of images. It involves an operator depending on a parameter, called a scale, which acts in some space of real functions. Many different scale-spaces have been defined and used in practice. The most popular example is the Gaussian Scale-Space where the operator is defined as a convolution integral involving the Gaussian distribution function with the standard variation being the scale parameter. There have also been many attempts for axiomatization of scale-space theory. The novelty of our approach is that it does not assume integral representation of the scale-space operator. While the new theory is applicable to, and in fact represents and abstraction of the properties of all existing operators, most importantly that of causality, it uses as an essential example the operator and the respective scale-space provided by the Discrete Pulse Transform.