A Sliced Coprocessor for Native Clifford Algebra Operations

Franchini, Silvia; Gentile, Vincenzo; Grimaudo,; Hung, Jui-Chung; Impastato,; Sorbello,; Vassallo, Raquel Frizera; Vitabile, Salvatore

doi:10.1109/dsd.2007.4341505

Cited by 26 publications

(27 citation statements)

References 3 publications

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“…Initial attempts include Cliffosor [19] and its successor S-Cliffosor ( [20]. However, these systems suffered from a number of architectural bottlenecks or only supporting limited GA operations (e.g., low dimensionality, restricted operator subset).…”

Section: Related Workmentioning

confidence: 99%

FPGA-accelerated color edge detection using a Geometric-Algebra-to-Verilog compiler

Stock

Koch

Hildenbrand

2013

2013 International Symposium on System on Chip (SoC)

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Abstract-Geometric Algebra (GA) is a branch of mathematics that generalizes complex numbers and quaternions. One of the advantages of the framework is, that it allows intuitive description and manipulation of geometric objects. While even complex operations can be described concisely, the actual evaluation of these GA expressions is extremely compute intensive. However, it has significant fine-grained parallelism, which makes it a profitable target for hardware implementation. In this paper, we present the automatic acceleration of a color edge-detection algorithm from a GA description. Using our Gaalop GA compiler with its Verilog back-end, we can show speed-ups of over 1000x even compared to a recent GA processor ASIC.

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Section: Related Workmentioning

confidence: 99%

FPGA-accelerated color edge detection using a Geometric-Algebra-to-Verilog compiler

Stock

Koch

Hildenbrand

2013

2013 International Symposium on System on Chip (SoC)

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“…The current literature on geometric algebra proposes very few examples of hardware implementations of the algebra [13], [21], [22], [23].…”

Section: Related Workmentioning

confidence: 99%

“…Gaalop decomposes a generic geometric algebra algorithm into basic arithmetic operations and generates an ultra-specialized hardware which executes these basic operations. Conversely, the systems presented in [13], [21] are complete coprocessors designed for the native execution of 4D geometric algebra operations.…”

Section: Related Workmentioning

confidence: 99%

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Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Franchini

Gentile

Sorbello

et al. 2010

Adv. Appl. Clifford Algebras

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Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruplebased Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23× speedup for Clifford products and a 33× speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor.Mathematics Subject Classification (2010). Primary 15A66; Secondary 68U05; Secondary 68W30.

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“…[20], [21], [22]). The prototype was implemented on a Field Programmable Gate Array and initial tests showed a 4× speedup for Clifford products over the analogous operations in GAIGEN.…”

Section: Introductionmentioning

confidence: 99%

Reductions in Computational Complexity Using Clifford Algebras

Schott

Staples

2008

AACA

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A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ k , where Λ is an appropriate nilpotent adjacency matrix, the kcycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph 1 . Mathematics Subject Classification (2000). 68Q15, 05C50, 05C38, 60B99, 81P68.

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A Sliced Coprocessor for Native Clifford Algebra Operations

Cited by 26 publications

References 3 publications

FPGA-accelerated color edge detection using a Geometric-Algebra-to-Verilog compiler

FPGA-accelerated color edge detection using a Geometric-Algebra-to-Verilog compiler

Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Reductions in Computational Complexity Using Clifford Algebras

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