Shape Preserving Digitization of Binary Images After Blurring

2013

J Math Imaging Vis

Local algorithms are common tools for estimating intrinsic volumes from blackand-white digital images. However, these algorithms are typically biased in the design based setting, even when the resolution tends to infinity. Moreover, images recorded in practice are most often blurred grey-scale images rather than black-and-white. In this paper, an extended definition of local algorithms, applying directly to grey-scale images without thresholding, is suggested. We investigate the asymptotics of these new algorithms when the resolution tends to infinity and apply this to construct estimators for surface area and integrated mean curvature that are asymptotically unbiased in certain natural settings.

Section: Introductionmentioning

confidence: 99%

Estimation of Intrinsic Volumes from Digital Grey-Scale Images

2013

J Math Imaging Vis

“…For instance, Jürgen Kampf has shown in a yet unpublished paper (see [5]) that without the isotropy of the lattice, there are no local estimators for V 0 that are asymptotically unbiased for all polyconvex sets. On the other hand, it is well known that there exists a local algorithm for V 0 , which is asymptotically unbiased on the class of so-called r-regular sets; see, for example, the discussion in [15]. We shall assume throughout this section that X is a compact full-dimensional C 2 manifold, which is slightly stronger than r-regularity.…”

Section: Euler Characteristic In the Design-based Settingmentioning

confidence: 97%

“…On the other hand, it is well-known that there exists a local algorithm for V 0 which is asymptotically unbiased on the class of so-called r-regular sets, see e.g. the discussion in [14]. We will assume throughout this section that X is a compact full-dimensional C 2 manifold, which is slightly stronger than r-regularity.…”

Section: Euler Characteristic In the Design Based Settingmentioning

confidence: 99%

Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

2014

Adv. Appl. Probab.

In order to estimate the specific intrinsic volumes of a planar Boolean model from a binary image, we consider local digital algorithms based on weighted sums of 2 × 2 configuration counts. For Boolean models with balls as grains, explicit formulas for the bias of such algorithms are derived, resulting in a set of linear equations that the weights must satisfy in order to minimize the bias in high resolution. These results generalize to larger classes of random sets, as well as to the design-based situation, where a fixed set is observed on a stationary isotropic lattice. Finally, the formulas for the bias obtained for Boolean models are applied to existing algorithms in order to compare their accuracy.

“…For instance, Jürgen Kampf has shown in a yet unpublished paper (see [5]) that without the isotropy of the lattice, there are no local estimators Local digital estimators of intrinsic volumes SGSA • 49 for V 0 that are asymptotically unbiased for all polyconvex sets. On the other hand, it is well known that there exists a local algorithm for V 0 , which is asymptotically unbiased on the class of so-called r-regular sets; see, for example, the discussion in [15]. We shall assume throughout this section that X is a compact full-dimensional C 2 manifold, which is slightly stronger than r-regularity.…”

Section: Euler Characteristic In the Design-based Settingmentioning

confidence: 99%

Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

Advances in Applied Probability

2014

In order to estimate the specific intrinsic volumes of a planar Boolean model from a binary image, we consider local digital algorithms based on weighted sums of 2×2 configuration counts. For Boolean models with balls as grains, explicit formulas for the bias of such algorithms are derived, resulting in a set of linear equations that the weights must satisfy in order to minimize the bias in high resolution. These results generalize to larger classes of random sets, as well as to the design-based situation, where a fixed set is observed on a stationary isotropic lattice. Finally, the formulas for the bias obtained for Boolean models are applied to existing algorithms in order to compare their accuracy.