The correlation coefficient r is a measure of similarity used to compare regions of interest in image pairs. In fluorescence microscopy there is a basic tradeoff between the degree of image noise and the frequency with which images can be acquired and therefore the ability to follow dynamic events. The correlation coefficient r is commonly used in fluorescence microscopy for colocalization measurements, when the relative distributions of two fluorophores are of interest. Unfortunately, r is known to be biased understating the true correlation when noise is present. A better measure of correlation is needed. This article analyses the expected value of r and comes up with a procedure for evaluating the bias of r, expected value formulas. A Taylor series of so-called invariant factors is analyzed in detail. These formulas indicate ways to correct r and thereby obtain a corrected value free from the influence of noise that is on average accurate (unbiased). One possible correction is the attenuated corrected correlation coefficient R, introduced heuristically by Spearman (in Am. J. Psychol. 15:72-101, 1904). An ideal correction formula in terms of expected values is derived. For large samples R tends towards the ideal correction formula and the true noise-free correlation. Correlation measurements using simulation based on the types of noise found in fluorescence microscopy images illustrate both the power of the method and the variance of R. We conclude that the correction formula is valid and is particularly useful for making correct analyses from very noisy datasets.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Digital Khalimsky Manifolds2009In: Journal of Mathematical Imaging and Vision, ISSN 0924-9907, E-ISSN 1573-7683, Vol. 33, no 3, p. 267-280Article in journal (Refereed)

Abstract [en]

We consider different possibilities to define digital manifolds that are locally homeomorphic to Khalimsky n-space. We prove existence and non-existence of certain types of Khalimsky manifolds. An embedding theorem is proved. We introduce the join operator and use it to analyze the structure of adjacency neighborhoods and of intersections of neighborhoods in ℤ^{ n }.

Let X be a smallest-neighborhood space, sometimes called an Alexandrov space. We demonstrate that the graph of a Khalimsky-continuous mapping X->Z is a surface having a Jordan--Brouwer type separation property. We study infima and suprema of families of such continuous mappings, a study that naturally leads to the introduction of an extended Khalimsky line. Moreover, we show that the boundary of a connected subset, U, of the Khalimsky plane is connected precisely when the complement of U is connected.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computerized Image Analysis and Human-Computer Interaction.

Normand, Nicolas

Univ Nantes, LS2N, UMR, CNRS 6004, Nantes, France.

In this paper, we present a general framework for digital distance functions, defined as minimal cost paths, on the square grid. Each path is a sequence of pixels, where any two consecutive pixels are adjacent and associated with a weight. The allowed weights between any two adjacent pixels along a path are given by a weight sequence, which can hold an arbitrary number of weights. We build on our previous results, where only two or three unique weights are considered, and present a framework that allows any number of weights. We show that the rotational dependency can be very low when as few as three or four unique weights are used. Moreover, by using n weights, the Euclidean distance can be perfectly obtained on the perimeter of a square with side length 2n. A sufficient condition for weight sequences to provide metrics is proven.