• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.397-403
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• 2004

Cutler showed some duality results about Hausdorff and packing dimensions of a measure on a compact set in Euclidean space if its s-dimensional Hausdorff measure or packing measure is positive. We show that the similar results in a perturbed Cantor set hold according to its quasi s-dimensional Hausdorff measure or packing measure and we find concrete measures in this case while Cutler showed the existence of such measures. Finally under some strong condition, we give a concrete measure whose Hausdorff and packing dimensions are the same as those of the perturbed Cantor set without the condition that it has positive s-dimensional Hausdorff or packing measures.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.81-93
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• 2002

We consider the Hausdorff measure for Brownian motion(BM) in ι$_2$. Several path properties of BM in ι$_2$ are used to show the upper bound of Hausdorff measure. We also show the lower bound of it applying a law of iterated logarithm for the occupation time of BM in ι$_2$.

• Journal of the Korean Institute of Telematics and Electronics S
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• v.36S no.11
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• pp.60-67
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• 1999

This paper proposes a robust oriented Hausdorff measure (ROHM) for 20D object matching. The ROHM is introduced by replacing the distance concept of the conventional Hausdorff distance (HD) algorithm by the accumulation scheme of the Hough transform (HT). The proposed algorithm can be considered as the modified directed HT using the distance transform (DT). The orientation information at each pixel is also used for removing incorrect correspondences. Experiments with various test images show that the performance of the proposed algorithm is better than that of conventional HD algorithms considered.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.241-246
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• 2007

We investigate the relation between Hausdorff(packing) measure and lower(packing) Cantor measure on a deranged Cantor set. If the infimum of some distortion of contraction ratios is positive, then Hausdorff(packing) measure and lower(packing) Cantor measure of a deranged Cantor set are equivalent except for some singular behavior for packing measure case. It is a generalization of already known result on the perturbed Cantor set.

• Kyungpook Mathematical Journal
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• v.49 no.4
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• pp.619-621
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• 2009

We exhibit an example that the scaling property of Hausdorff measure on $\mathbb{R}^{\infty}$ does not hold.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.38 no.2
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• pp.56-62
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• 2015

The Hausdorff distance is commonly used as a similarity measure between two-dimensional binary images. Since the document images may be contaminated by a variety of noise sources during transmission, scanning or conversion to digital form, the measure should be robust to the noise. Original Hausdorff distance has been known to be sensitive to outliers. Transforming the given image to grayscale image is one of methods to deal with the noises. In this paper, we propose a Hausdorff distance applied to grayscale images. The proposed method is tested with synthetic images with various levels of noises and compared with other methods to show its robustness.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.213-230
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• 2019

We construct new metric outer measures (multifractal analogues of the Hewitt-Stromberg measure) $H^{q,t}_{\mu}$ and $P^{q,t}_{\mu}$ lying between the multifractal Hausdorff measure ${\mathcal{H}}^{q,t}_{\mu}$ and the multifractal packing measure ${\mathcal{P}}^{q,t}_{\mu}$. We set up a necessary and sufficient condition for which multifractal Hausdorff and packing measures are equivalent to the new ones. Also, we focus our study on some regularities for these given measures. In particular, we try to formulate a new version of Olsen's density theorem when ${\mu}$ satisfies the doubling condition. As an application, we extend the density theorem given in [3].

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.447-454
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• 2004

We study the Hausdorff and packing dimensions of subsets composing multifractal spectrum generated by a quasi-self-similar probability measure on a perturbed Cantor set.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.459-471
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• 2018

We obtain a relation between generalized Hausdorff and packing multifractal premeasures and generalized Hausdorff and packing multifractal measures. As an application, we study a general formalism for the multifractal analysis of one probability measure with respect to an other.

• Korean Journal of Mathematics
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• v.26 no.2
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• pp.271-283
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• 2018

In this paper, we generelize the Olsen's density theorem to any measurable set, allowing us to extend the main results of H.K. Baek in (Proc. Indian Acad. Sci. (Math. Sci.) Vol. 118, (2008), pp. 273-279.). In particular, we tried through these results to improve the decomposition theorem of Besicovitch's type for the regularities of multifractal Hausdorff measure and packing measure.