Slice Decomposition: Theory & Applications

• Slice Decomposition is a technique that partitions global structures into slice-indexed pieces with invariant properties, as seen in convex geometry, combinatorics, and medical imaging.
• It simplifies analysis by reducing heterogeneous problems to stable chambers or segments, allowing explicit algebraic and combinatorial formulations.
• Applications include optimizing polytope sections, performing efficient tensor decompositions in 3D imaging, and constructing adaptive representations in slice-regular function theory.

“Slice decomposition” denotes several non-equivalent constructions that share a common formal theme: an object, parameter space, or function class is partitioned into slice-indexed pieces on which structure becomes stable, computable, or recursively describable. In convex geometry, it is a finite partition of hyperplane-parameter space into “slicing chambers” within which the affine hyperplane sections of a convex polytope have invariant combinatorial type and chamberwise rational metric data (Brandenburg et al., 2023). In combinatorics, it is a bijective decomposition of planar hypermaps obtained by cutting along directed geodesics (Albenque et al., 8 Sep 2025). In medical imaging, it is a multi-slice low-rank tensor decomposition for 3D CT volumes that processes short windows of consecutive axial slices (Shi et al., 2021). In quaternionic and Clifford analysis, it refers to Almansi-type or adaptive decompositions of slice-regular functions into harmonic, monogenic, or orthonormal building blocks (Perotti, 2020, Binosi, 2022, Perotti, 2021, Jin et al., 2021). Other uses include presheaf-topos descriptions of slice categories of decomposition spaces (Kock et al., 2018), Alexander-primary decompositions in concordance theory (Kim et al., 2019, Cha, 2019), regular-versus-decomposable sliceness in symplectic topology (Breen, 2024), and model-theoretic stratifications of $2^\delta$ by increasing transitive models (Kostana et al., 2021).

1. Terminological scope

The term is therefore domain-specific rather than universal. The following representative meanings are explicit in the literature.

Domain	Object being decomposed	Output
Convex geometry	Hyperplane-parameter space of sections of a polytope	Slicing chambers with invariant combinatorial type and rational formulas (Brandenburg et al., 2023)
Planar hypermaps	Maps cut along directed geodesics	Sequences of elementary slices encoded by DSF walks (Albenque et al., 8 Sep 2025)
3D medical imaging	CT volumes partitioned into axial windows	Low-rank and sparse tensor components for atlas construction and MAS (Shi et al., 2021)
Slice-regular analysis	Quaternionic or Clifford-valued functions	Harmonic, monogenic, or adaptive kernel expansions (Perotti, 2020, Binosi, 2022, Perotti, 2021, Jin et al., 2021)
Higher-category theory	Slice category decomp/$D$	Presheaf topos $\mathrm{Psh}(\mathrm{tw}\,D)$ (Kock et al., 2018)
Topology and set theory	Concordance groups, Lagrangian sliceness, or $2^\delta$	Primary components, decomposable sliceness, or strictly increasing model slices (Kim et al., 2019, Breen, 2024, Kostana et al., 2021)

This suggests a common pattern: slice decomposition is typically introduced when a global object is too heterogeneous to analyze directly, but becomes tractable after passage to slices on which combinatorics, geometry, or operator theory stabilize.

2. Hyperplane-section slice decomposition for convex polytopes

In its most formal geometric sense, slice decomposition is defined for a full-dimensional convex polytope $P \subset \mathbb{R}^d$ and affine hyperplanes

$$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$

The parameter space of hyperplanes is taken modulo positive scaling, equivalently by unit normals $a\in S^{d-1}$ and offsets $b\in\mathbb{R}$, or by rays $\lambda(a,b)$ in $\mathbb{R}^{d+1}$, $D$0 (Brandenburg et al., 2023).

The paper develops two decompositions of the slice space. The first is the cocircuit-plus-central arrangement, described as “translate the rotation.” One considers a cocircuit arrangement in translation space $D$1,

$D$2

and, for $D$3 in a region $D$4, the central arrangement

$D$5

Its chambers $D$6 encode the directions $D$7 for which the central slice $D$8 intersects a fixed set of edges. The second is the sweep-plus-parallel arrangement, described as “rotate the translation.” It uses the sweep arrangement

$D$9

in normal space $\mathrm{Psh}(\mathrm{tw}\,D)$0, together with the parallel arrangement

$\mathrm{Psh}(\mathrm{tw}\,D)$1

whose chambers are the open intervals between consecutive support values (Brandenburg et al., 2023).

Within each slicing chamber, the sections intersect the same set of edges of $\mathrm{Psh}(\mathrm{tw}\,D)$2. In the central case they therefore have the same combinatorial type and admit the same triangulations. In the parallel case they are normally equivalent, so the normal fan is constant as well. In both settings, integrals of polynomials over the sections are chamberwise rational functions of the parameters (Brandenburg et al., 2023).

The paper records explicit boundary events. In the cocircuit-central decomposition, boundaries occur when $\mathrm{Psh}(\mathrm{tw}\,D)$3 or when $\mathrm{Psh}(\mathrm{tw}\,D)$4 crosses a cocircuit hyperplane. In the sweep-parallel decomposition, boundaries occur when $\mathrm{Psh}(\mathrm{tw}\,D)$5 or when $\mathrm{Psh}(\mathrm{tw}\,D)$6. These are polynomial equalities in the slice parameters (Brandenburg et al., 2023).

A central consequence is a finite, combinatorially controlled partition of hyperplane-parameter space. This is the sense in which slice decomposition functions as a classification device: the geometry of sections is constant on chambers, and all changes occur across explicitly described arrangement walls.

3. Optimization, bounds, and algorithmics in the polytope setting

Because combinatorial type and triangulations are constant within a slicing chamber, optimization over all sections reduces to chamber enumeration plus chamberwise algebraic optimization. The paper studies objectives including maximizing the number of $\mathrm{Psh}(\mathrm{tw}\,D)$7-faces, maximizing $\mathrm{Psh}(\mathrm{tw}\,D)$8-dimensional volume, and maximizing $\mathrm{Psh}(\mathrm{tw}\,D)$9 for a polynomial $2^\delta$0 (Brandenburg et al., 2023).

A principal complexity result is the bound that a $2^\delta$1-dimensional polytope with $2^\delta$2 vertices has at most $2^\delta$3 combinatorial types of hyperplane sections. The sweep decomposition yields the dominant bound: the sweep arrangement has at most $2^\delta$4 hyperplanes and contributes $2^\delta$5 cells, while the parallel arrangement contributes $2^\delta$6 chambers per sweep region (Brandenburg et al., 2023).

For fixed $2^\delta$7, the global optimization problems are polynomial-time solvable. The method is to enumerate the chambers, write the objective as a rational function with polynomial constraints, and solve the resulting semialgebraic optimization problem by quantifier-elimination techniques. The paper states that maximizing or minimizing the number of $2^\delta$8-faces, slice volume, integrals of polynomials, central halfspace integrals, and projection integrals is polynomial-time in fixed dimension. In variable dimension the situation changes sharply: maximal-volume slice is $2^\delta$9P-hard, weighted max $P \subset \mathbb{R}^d$0-face section is NP-hard, and the paper records further hardness conjectures (Brandenburg et al., 2023).

The chamberwise formulas are explicit. For an intersected edge $P \subset \mathbb{R}^d$1, the slice vertex in the translational setting is

$P \subset \mathbb{R}^d$2

Volumes are computed by simplex determinants, and polynomial integrals use the Lasserre–Baldoni formula together with monomial-to-linear-form decompositions (Brandenburg et al., 2023).

The examples illustrate the scope of the framework. The 3D permutahedron has an octagon max-volume slice of volume $P \subset \mathbb{R}^d$3, a min-volume central slice $P \subset \mathbb{R}^d$4 of volume $P \subset \mathbb{R}^d$5, and eight combinatorial types ranging from triangles to decagons. The paper also reports max-volume slices for Platonic solids and nine distinct types of 3D sections of the 4D cross-polytope (Brandenburg et al., 2023).

4. Directed-geodesic slice decomposition in planar hypermaps

In planar hypermaps, slice decomposition is a bijective method rather than a chamber decomposition. A planar hypermap is a connected planar map whose faces are properly bicolored white and black, and each edge receives a canonical orientation so that the face on its right is white and the face on its left is black. This orientation forces a directed notion of distance and geodesic, and the decomposition cuts along directed leftmost geodesics (Albenque et al., 8 Sep 2025).

A slice is a planar hypermap with one boundary and three distinguished outer corners $P \subset \mathbb{R}^d$6 in counterclockwise order. Its boundary splits into a left boundary from $P \subset \mathbb{R}^d$7 to $P \subset \mathbb{R}^d$8, which is a directed geodesic; a right boundary from $P \subset \mathbb{R}^d$9 to $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$0, which is the unique directed geodesic; and a base, which is a directed path from $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$1 to $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$2 in type A or from $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$3 to $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$4 in type B. Writing $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$5, the slice increment is

$$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$6

Elementary slices have a base consisting of a single edge and are denoted $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$7 or $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$8 according to type and increment $$H(a,b)=\{x\in\mathbb{R}^d:a^\top x=b\}, \qquad S(a,b)=P\cap H(a,b).$$9 (Albenque et al., 8 Sep 2025).

The key enumerative mechanism is the encoding by downward skip-free walks. If $a\in S^{d-1}$0 is a DSF walk, there is a weight-preserving bijection between slices of type A with base condition $a\in S^{d-1}$1 and $a\in S^{d-1}$2-tuples of elementary A-slices whose increments are the successive step increments $a\in S^{d-1}$3; an analogous statement holds for type B. This yields Laurent series

$a\in S^{d-1}$4

with

$a\in S^{d-1}$5

over slices of fixed base length $a\in S^{d-1}$6 (Albenque et al., 8 Sep 2025).

Removing the base edge of an elementary slice gives the recursive system

$a\in S^{d-1}$7

or equivalently

$a\in S^{d-1}$8

Under bounded face degrees, $a\in S^{d-1}$9 and $b\in\mathbb{R}$0 become Laurent polynomials and the system becomes finite algebraic (Albenque et al., 8 Sep 2025).

This slice language supports several families of bijective decompositions. Increment $b\in\mathbb{R}$1 slices wrap to pointed disks; nonzero increments wrap to trumpets and cornets; cylinders are decomposed by an innermost minimal counterclockwise separating cycle; and disks with a Dobrushin boundary condition are expressed by a blob decomposition and one-way cylinders. The generating functions then admit explicit formulas, and under bounded face degrees the resolvents satisfy the rational parametrization

$b\in\mathbb{R}$2

The paper presents this as a combinatorial explanation of the algebraicity and rational parametrizability of hypermap generating functions (Albenque et al., 8 Sep 2025).

The directed character is the essential difference from the classical slice decomposition of maps. The right boundary geodesic is unique, the step set is downward skip-free, and the resulting A/B asymmetry is needed to recover formulas associated with the two-matrix model and the Ising model on random maps (Albenque et al., 8 Sep 2025).

5. Multi-slice low-rank tensor decomposition in medical imaging

In pathological liver CT segmentation, slice decomposition is a data-batching strategy for tensor robust principal component analysis. A 3D CT volume is modeled as a tensor $b\in\mathbb{R}$3, with frontal slices $b\in\mathbb{R}$4. Instead of decomposing an entire repository tensor at once, each training case $b\in\mathbb{R}$5 is partitioned into $b\in\mathbb{R}$6 consecutive axial segments of fixed length $b\in\mathbb{R}$7,

$b\in\mathbb{R}$8

and, for segment $b\in\mathbb{R}$9,

$\lambda(a,b)$0

Segments are processed with overlap by one neighboring slice at each end, and overlapping reconstructions are averaged. The smallest possible $\lambda(a,b)$1 is $\lambda(a,b)$2; in the reported experiments, $\lambda(a,b)$3 was selected (Shi et al., 2021).

Each segment-level repository tensor is decomposed by Tensor Principal Component Pursuit,

$\lambda(a,b)$4

where the tensor nuclear norm is defined under the $\lambda(a,b)$5 framework using an invertible transform $\lambda(a,b)$6. The paper chooses the DCT as $\lambda(a,b)$7, reporting that it reduced intensity standard deviation and entropy most and was the fastest among FFT, DWT, and DCT. The decomposition is solved by ADMM with tensor singular value thresholding; the stopping criterion is based on infinity norms with $\lambda(a,b)$8, and the reported parameters are $\lambda(a,b)$9, $\mathbb{R}^{d+1}$0, and $\mathbb{R}^{d+1}$1 (Shi et al., 2021).

The purpose of the multi-slice scheme is explicitly comparative. Per-slice matrix SVD ignores inter-slice correlations, whereas full-volume decomposition is memory-heavy and less stable because the low-rank assumption weakens when slices vary rapidly through the volume. Multi-slice LRTD uses short windows of consecutive slices to preserve local low-rankness while still coupling information across slices in the DCT transform domain. The low-rank component $\mathbb{R}^{d+1}$2 is described as smoother, homogeneous, and vessel-preserving, while the sparse component $\mathbb{R}^{d+1}$3 isolates hypodense and hyperdense lesions (Shi et al., 2021).

The decomposition is embedded in a multi-atlas segmentation pipeline. Training images are clustered into $\mathbb{R}^{d+1}$4 clusters by spectral clustering on combined intensity and shape affinities. Within each cluster, a template is initialized by the image with smallest tumor burden, non-rigidly registered by cubic B-spline FFD optimized on normalized mutual information, and updated as the mean of low-rank images for $\mathbb{R}^{d+1}$5 iterations. Tumor-free atlases are then produced by LRTD in the template space and mapped back to original space. For a test image $\mathbb{R}^{d+1}$6, the cluster with largest NCC to the warped probabilistic atlas is selected, a repository tensor

$\mathbb{R}^{d+1}$7

is formed, and multi-slice LRTD yields a tumor-free test image $\mathbb{R}^{d+1}$8. Pairwise registration is performed with elastix, labels are propagated, and Joint Label Fusion is followed by Otsu thresholding and morphological opening and closing (Shi et al., 2021).

The reported empirical results are tied directly to the slice decomposition design. On 3Dircadb1, conventional MAS achieved DC $\mathbb{R}^{d+1}$9, JI $D$00, ASD $D$01 mm; LRMD-based MAS achieved DC $D$02, JI $D$03, ASD $D$04 mm; and LRTD-based MAS achieved DC $D$05, JI $D$06, ASD $D$07 mm, with improvements over both baselines statistically significant at $D$08. Registration quality by NMI improved to $D$09 versus $D$10 and $D$11. On LiTS2017-Test, the method achieved DC $D$12, JI $D$13, ASD $D$14 mm, while training on only 20 SLIVER07-Train cases. The end-to-end runtime was reported as approximately 95 minutes per test case, dominated by non-rigid registrations and JLF (Shi et al., 2021).

In this setting, “slice decomposition” therefore does not mean a rank prior defined per slice in the spatial domain. The data explicitly distinguishes the notion from classical tensor slice viewpoints: it is a multi-slice batching strategy, with the rank prior defined via transform-domain $D$15-SVD (Shi et al., 2021).

6. Analytic, categorical, and topological variants

In Clifford and quaternionic analysis, slice decomposition usually means a representation theorem for slice-regular functions. For slice-regular functions on Clifford algebras, the Almansi-type decomposition states that for a slice-regular $D$16 on a circular domain there exist unique zonal $D$17-harmonic slice functions $D$18 and $D$19 such that

$D$20

with

$D$21

For polynomials $D$22, this becomes

$D$23

where $D$24 are zonal $D$25-harmonic polynomials (Perotti, 2020).

For slice regular functions of several quaternionic variables, the paper proves $D$26 distinct and unique decompositions indexed by subsets $D$27. If $D$28 is fixed, then

$D$29

where $D$30. For slice-regular $D$31, the components are circular in the variables of $D$32 and harmonic in each selected variable. The ordered case $D$33 leads to mean value and Poisson formulas, another proof of Fueter’s theorem in $D$34, and separate biharmonicity $D$35 for $D$36 (Binosi, 2022).

A distinct but related quaternionic result decomposes a slice-regular function on an axially symmetric set into two axially monogenic functions. Under the hypothesis that each connected component of the stem domain is simply connected, a slice-regular $D$37 can be written uniquely as

$D$38

with $D$39. This decomposition is then used to prove a local Cauchy-type boundary integral formula on bounded open sets with rectifiable boundary and with no requirement of axial symmetry of the integration domain (Perotti, 2021).

In the slice Hardy space over the quaternionic unit ball, the phrase acquires an adaptive approximation meaning. For $D$40 and $D$41,

$D$42

where $D$43 is the slice normalized Szegő kernel, $D$44 is the slice Blaschke factor, and $D$45 is the slice hyperbolic backward shift. Iterating this identity with a maximal selection principle produces the slice adaptive Fourier decomposition and the adaptive slice Takenaka–Malmquist orthonormal system (Jin et al., 2021).

Outside analysis, the phrase appears in categorical, topological, and set-theoretic forms. For a decomposition space $D$46, the slice category of decomposition spaces over $D$47 via CULF maps satisfies

$D$48

so “decomposition-space slices are toposes.” The twisted arrow category $D$49 is obtained by edgewise subdivision, and the equivalence identifies each slice category with a presheaf topos (Kock et al., 2018).

In symplectic topology, the paper explicitly formulates a “slice decomposition philosophy”: if a Legendrian knot is regularly slice, then it is once-stably decomposably slice. The main theorem states that $D$50 implies

$D$51

so regular sliceness implies once-stably decomposable sliceness after one stabilization (Breen, 2024).

In smooth concordance theory, “slice decomposition” is used for Alexander-primary decompositions. Kim–Kim–Kim prove that $D$52 has infinite rank and that for infinitely many

$D$53

the quotient $D$54 has infinite rank (Kim et al., 2019). Cha gives a broader formalism of left and right primary decomposition for the smooth concordance group of topologically slice knots, and proves that a large subgroup $D$55 decomposes as a direct sum of $D$56-primary parts indexed by $D$57 (Cha, 2019).

Finally, in set theory, the slicing axioms $D$58 assert the existence of a $D$59-increasing sequence of transitive models $D$60 such that for every $D$61,

$D$62

and the inclusions $D$63 are strict for $D$64. The paper proves that $D$65 implies that $D$66 is regular and $D$67, that $D$68 implies $D$69, and that $D$70 is consistent with $D$71 (Kostana et al., 2021).

These usages do not collapse to a single definition. What they share is a precise replacement of a global problem by slice-wise pieces with controlled interaction: chambers in convex geometry, geodesic components in map enumeration, local tensor windows in imaging, harmonic or monogenic components in function theory, presheaf fibers in higher-category theory, and primary or model-theoretic strata in topology and set theory.