Farey Sequences and the Planar Euclidean Medial Axis Test Mask

Hulin, Jérôme; Thiel, Edouard

doi:10.1007/978-3-642-10210-3_7

Cited by 2 publications

(1 citation statement)

References 8 publications

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“…Hence, there have been studies related to their uneven distributions [26]. In a recent work, a relation between the Euclidean test mask and the Farey sequence has been established to detect the Euclidean medial axis of any discrete shape [10]. Farey sequences can also be related to Farey graphs, which were introduced in 1970's [15].…”

Section: Introductionmentioning

confidence: 99%

On Farey table and its compression for space optimization with guaranteed error bounds

Paria¹,

Pratihar²,

Bhowmick³

2017

Math. Appl.

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Abstract. Farey sequences, introduced by such renowned mathematicians as John Farey, Charles Haros, and Augustin-L. Cauchy over 200 years ago, are quite wellknown by today in theory of fractions, but its computational perspectives are possibly not yet explored up to its merit. In this paper, we present some novel theoretical results and efficient algorithms for representation of a Farey sequence through a Farey table. The ranks of the fractions in a Farey sequence are stored in the Farey table to provide an efficient solution to the rank problem, thereby aiding in and speeding up any application frequently requiring fraction ranks for computational speed-up. As the size of the Farey sequence grows quadratically with its order, the Farey table becomes inadvertently large, which calls for its (lossy) compression up to a permissible error. We have, therefore, proposed two compression schemes to obtain a compressed Farey table (CFT). The necessary analysis has been done in detail to derive the error bound in a CFT. As the final step towards space optimization, we have also shown how a CFT can be stored in a 1-dimensional array. Experimental results have been furnished to demonstrate the characteristics and efficiency of a Farey table and its compressed form.

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Section: Introductionmentioning

confidence: 99%

On Farey table and its compression for space optimization with guaranteed error bounds

Paria¹,

Pratihar²,

Bhowmick³

2017

Math. Appl.

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Optimum design of chamfer masks using symmetric mean absolute percentage error

Maiseli

2019

J Image Video Proc.

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Distance transform, a central operation in image and video analysis, involves finding the shortest path between feature and non-feature entries of a binary image. The process may be implemented using chamfer-based sequential algorithms that apply small-neighborhood masks to estimate the Euclidean metric. Success of these algorithms depends on the cost function used to optimize chamfer weights. And, for years, mean absolute error and mean squared error have been used for optimization. However, studies have revealed weaknesses of these cost functions-sensitivity against outliers, lack of symmetry, and biasedness-which limit their application. In this work, we have proposed a robust and a more accurate cost function, symmetric mean absolute percentage error, which attempts to address some weaknesses. The proposed function averages the absolute percentage errors in a set of measurements and offers interesting mathematical properties (smoothness, differentiability, boundedness, and robustness) that allow easy interpretation and analysis of the results. Numerical results show that chamfer masks designed under our optimization criterion generate lower errors. The present work has also proposed an automatic algorithm that converts coefficients of the designed real-valued masks into integers, which are preferable in most practical computing devices. Lastly, we have modified the chamfer algorithm to improve its speed and then embedded the proposed weights into the algorithm to compute distance maps of real images. Results show that the proposed algorithm is faster and uses fewer number of operations compared with those consumed by the classical chamfer algorithm. Our results may be useful in robotics to address the matching problem.

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Farey Sequences and the Planar Euclidean Medial Axis Test Mask

Cited by 2 publications

References 8 publications

On Farey table and its compression for space optimization with guaranteed error bounds

On Farey table and its compression for space optimization with guaranteed error bounds

Optimum design of chamfer masks using symmetric mean absolute percentage error

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