Diffusion magnetic resonance imaging (dMRI) enables noninvasive mapping of tissue microstructure by probing water molecule diffusivity. While advanced multi-shell diffusion models offer improved sensitivity to cellular properties, their requirement for densely sampled q-space data leads to prohibitively long acquisition times. Current deep learning approaches for parameter estimation face three key limitations: (1) dependency on fixed acquisition protocols, (2) model-specific assumptions that constrain applicability, and (3) reliance on supervised learning paradigms that demand large labeled datasets and exhibit poor generalization to out-of-distribution cases. To address these challenges, we propose MINeR, a novel unsupervised subject-specific framework for reconstructing dense q-space data from highly undersampled acquisitions. Our method leverages direction-modulated implicit neural representation to flexibly sample diffusion signals across q-space, supporting the estimation of parameters for diverse diffusion models. Comprehensive evaluations demonstrate that MINeR maintains high fidelity in microstructural parameter estimation, particularly for advanced multi-shell diffusion models. The framework shows remarkable generalization capability, as evidenced by its robust performance on tumor data. Notably, MINeR effectively reconstructs high-quality diffusion signals by interpolating from 6 directions, significantly reducing acquisition time, while maintaining robust parameter estimation. This work presents a practical approach for enabling microstructural modeling from sparsely sampled q-space data, thereby improving the clinical applicability of diffusion MRI. The code is available at: https://github.com/AMRI-Lab/MINeR.

Abstract In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.

Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

Abstract We build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.