Discrete tomography focuses on the reconstruction of functions $f: A \to
\mathbb{R}$ from their line sums in a finite number $d$ of directions, where
$A$ is a finite subset of $\mathbb{Z}^2$. Consequently, the techniques of
discrete tomography often find application in areas where only a small number
of projections are available. In 1978 M.B. Katz gave a necessary and sufficient
condition for the uniqueness of the solution. Since then, several
reconstruction methods have been introduced. Recently Pagani and Tijdeman
developed a fast method to reconstruct $f$ if it is uniquely determined.
Subsequently Ceko, Pagani and Tijdeman extended the method to the
reconstruction of a function with the same line sums of $f$ in the general
case. Up to here we assumed that the line sums are exact. In this paper we
investigate the case where a small number of line sums are incorrect as may
happen when discrete tomography is applied for data storage or transmission. We
show how less than $d/2$ errors can be corrected and that this bound is the
best possible.
We consider a class of problems of Discrete Tomography which has been deeply
investigated in the past: the reconstruction of convex lattice sets from their
horizontal and/or vertical X-rays, i.e. from the number of points in a sequence
of consecutive horizontal and vertical lines. The reconstruction of the
HV-convex polyominoes works usually in two steps, first the filling step
consisting in filling operations, second the convex aggregation of the
switching components. We prove three results about the convex aggregation step:
(1) The convex aggregation step used for the reconstruction of HV-convex
polyominoes does not always provide a solution. The example yielding to this
result is called \textit{the bad guy} and disproves a conjecture of the domain.
(2) The reconstruction of a digital convex lattice set from only one X-ray can
be performed in polynomial time. We prove it by encoding the convex aggregation
problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that
the reconstruction of fat digital convex sets from their horizontal and
vertical X-rays can be solved in polynomial time. Fatness is a property of the
digital convex sets regarding the relative position of the left, right, top and
bottom points of the set. The complexity of the reconstruction of the lattice
sets which are not fat remains an open question.
A distance mean function measures the average distance of points from the
elements of a given set of points (focal set) in the space. The level sets of a
distance mean function are called generalized conics. In case of infinite focal
points the average distance is typically given by integration over the focal
set. The paper contains a survey on the applications of taxicab distance mean
functions and generalized conics' theory in geometric tomography: bisection of
the focal set and reconstruction problems by coordinate X-rays. The theoretical
results are illustrated by implementations in Maple, methods and examples as
well.
Schizophrenia (SZ) is a brain disorder leading to detached mind's normally
integrated processes. Hence, the exploration of the symptoms in relation to
functional connectivity (FC) had great relevance in the field. FC can be
investigated on different levels, going from global features to single edges
between regions, revealing diffuse and localized dysconnection patterns. In
this context, SZ is characterized by a diverse global integration with reduced
connectivity in specific areas of the Default Mode Network (DMN). However, the
assessment of FC presents various sources of uncertainty. This study proposes a
multi-level approach for more robust group-comparison. FC between 74 AAL brain
areas of 15 healthy controls (HC) and 12 SZ subjects were used. Multi-level
analyses and graph topological indexes evaluation were carried out by the
previously published SPIDER-NET tool. Robustness was augmented by bootstrapped
(BOOT) data and the stability was evaluated by removing one (RST1) or two
subjects (RST2). The DMN subgraph was evaluated, toegether with overall local
indexes and connection weights to enhance common activations/deactivations. At
a global level, expected trends were found. The robustness assessment tests
highlighted more stable results for BOOT compared to the direct data testing.
Conversely, significant results were found in the analysis at lower levels. The
DMN highlighted reduced connectivity and strength as well as increased
deactivation in the SZ group. At local […]