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In geometry processing and shape analysis, several applications
have been addressed through the properties of the Laplacian
spectral kernels and distances, such as commute time, biharmonic,
diffusion, and wave distances. Within this context, this book is
intended to provide a common background on the definition and
computation of the Laplacian spectral kernels and distances for
geometry processing and shape analysis. To this end, we define a
unified representation of the isotropic and anisotropic discrete
Laplacian operator on surfaces and volumes; then, we introduce the
associated differential equations, i.e., the harmonic equation, the
Laplacian eigenproblem, and the heat equation. Filtering the
Laplacian spectrum, we introduce the Laplacian spectral distances,
which generalize the commute-time, biharmonic, diffusion, and wave
distances, and their discretization in terms of the Laplacian
spectrum. As main applications, we discuss the design of smooth
functions and the Laplacian smoothing of noisy scalar functions.
All the reviewed numerical schemes are discussed and compared in
terms of robustness, approximation accuracy, and computational
cost, thus supporting the reader in the selection of the most
appropriate with respect to shape representation, computational
resources, and target application.
New data acquisition techniques are emerging and are providing fast
and efficient means for multidimensional spatial data collection.
Airborne LIDAR surveys, SAR satellites, stereo-photogrammetry and
mobile mapping systems are increasingly used for the digital
reconstruction of the environment. All these systems provide
extremely high volumes of raw data, often enriched with other
sensor data (e.g., beam intensity). Improving methods to process
and visually analyze this massive amount of geospatial and
user-generated data is crucial to increase the efficiency of
organizations and to better manage societal challenges. Within this
context, this book proposes an up-to-date view of computational
methods and tools for spatio-temporal data fusion, multivariate
surface generation, and feature extraction, along with their main
applications for surface approximation and rainfall analysis. The
book is intended to attract interest from different fields, such as
computer vision, computer graphics, geomatics, and remote sensing,
working on the common goal of processing 3D data. To this end, it
presents and compares methods that process and analyze the massive
amount of geospatial data in order to support better management of
societal challenges through more timely and better decision making,
independent of a specific data modeling paradigm (e.g., 2D vector
data, regular grids or 3D point clouds). We also show how current
research is developing from the traditional layered approach,
adopted by most GIS softwares, to intelligent methods for
integrating existing data sets that might contain important
information on a geographical area and environmental phenomenon.
These services combine traditional map-oriented visualization with
fully 3D visual decision support methods and exploit
semantics-oriented information (e.g., a-priori knowledge,
annotations, segmentations) when processing, merging, and
integrating big pre-existing data sets.
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