Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

Local congruence of chain complexes

The object of this paper is to transform a set of local chain complexes to a single global complex using an equivalence relation of congruence of cells, solving topologically the numerical inaccuracies of floating-point arithmetics. While computing the space arrangement generated by a collection of cellular complexes, one may start from independently and efficiently computing the intersection of each single input 2-cell with the others. The topology of these intersections is codified within a set of (0-2)-dimensional chain complexes. The target of this paper is to merge the local chains by using the equivalence relations of {\epsilon}-congruence between 0-, 1-, and 2-cells (elementary chains). In particular, we reduce the block-diagonal coboundary matrices [\Delta_0] and [\Delta_1], used as matrix accumulators of the local coboundary chains, to the global matrices [\delta_0] and [\delta_1], representative of congruence topology, i.e., of congruence quotients between all 0-,1-,2-cells, via elementary algebraic operations on their columns. This algorithm is codified using the Julia porting of the SuiteSparse:GraphBLAS implementation of the GraphBLAS standard, conceived to efficiently compute algorithms on large graphs using linear algebra and sparse matrices [1, 2].Comment: to submi

Chain-Based Representations for Solid and Physical Modeling

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness

Algebraic filtering of surfaces from 3d medical images with julia

In this paper we introduce a novel algebraic filter, based on algebraic topology methods, to extract and smooth the boundary surface of any subset of voxels arising from the segmentation of a 3D medical image. The input of the Linear Algebraic Representation (lar) Surface extraction filter (lar-surf) is defined as a chain, i.e., an element of a linear space of chains here subsets of voxels represented in coordinates as a sparse binary vector. The output is produced by a linear mapping between spaces of 3-and 2-chains, given by the boundary operator ∂3: C3 → C2. The only data structures used in this approach are sparse arrays with one or two indices, i.e., sparse vectors and sparse matrices. This work is based on lar algebraic methods and is implemented in Julia language, natively supporting parallel computing on hybrid hardware architectures

A global collaboRAtive study of CIC-rearranged, BCOR::CCNB3-rearranged and other ultra-rare unclassified undifferentiated small round cell sarcomas (GRACefUl)

[Background] Undifferentiated small round cell sarcomas (URCSs) represent a diagnostic challenge, and their optimal treatment is unknown. We aimed to define the clinical characteristics, treatment, and outcome of URCS patients.[Methods] URCS patients treated from 1983 to 2019 at 21 worldwide sarcoma reference centres were retrospectively identified. Based on molecular assessment, cases were classified as follows: (1) CIC-rearranged round cell sarcomas, (2) BCOR::CCNB3-rearranged round cell sarcomas, (3) unclassified URCSs. Treatment, prognostic factors and outcome were reviewed.[Results] In total, 148 patients were identified [88/148 (60%) CIC-rearranged sarcoma (median age 32 years, range 7–78), 33/148 (22%) BCOR::CCNB3-rearranged (median age 17 years, range 5–91), and 27/148 (18%) unclassified URCSs (median age 37 years, range 4–70)]. One hundred-one (68.2%) cases presented with localised disease; 47 (31.8%) had metastases at diagnosis. Male prevalence, younger age, bone primary site, and a low rate of synchronous metastases were observed in BCOR::CCNB3-rearranged cases. Local treatment was surgery in 67/148 (45%) patients, and surgery + radiotherapy in 52/148 (35%). Chemotherapy was given to 122/148 (82%) patients. At a 42.7-month median follow-up, the 3-year overall survival (OS) was 92.2% (95% CI 71.5–98.0) in BCOR::CCNB3 patients, 39.6% (95% CI 27.7–51.3) in CIC-rearranged sarcomas, and 78.7% in unclassified URCSs (95% CI 56.1–90.6; p < 0.0001).[Conclusions] This study is the largest conducted in URCS and confirms major differences in outcomes between URCS subtypes. A full molecular assessment should be undertaken when a diagnosis of URCS is suspected. Prospective studies are needed to better define the optimal treatment strategy in each URCS subtype.This work was supported by the Carisbo Foundation Call for Translational and Clinical Medical Research.Peer reviewe