Least-square Ellipse Fitting Research Articles

Microscale modelling of fruit tissue using Voronoi tessellations

Microstructural properties are the core in defining and modelling macroscopic properties of fruits. In this regard, realistic microstructures that could be validated against microscopic images of fruit tissues were generated. Microscopic images of tissues of different apple cultivars (Greenstar, Kanzi, Pinova and Cameo) were digitized using an image analysis software program. The geometrical cell structure was described by means of statistical distributions of the cell area, cell aspect ratio and cell orientation. The cell orientation and cell aspect ratio were calculated based on the moments of inertia (area moments) and least-square ellipse fitting, respectively. The statistical geometrical properties of different cultivars were compared. These quantitative descriptors were then used in conjunction with a Poisson Voronoi tessellation algorithm to produce virtual cell tissue with the same statistical properties as the real tissue. The geometrical models will be used in a multiscale modelling approach to investigate water and gas transport in fruits.

Direct least square fitting of ellipses

This paper presents a new efficient method for fitting ellipses to scattered data. Previous algorithms either fitted general conics or were computationally expensive. By minimizing the algebraic distance subject to the constraint 4ac-b/sup 2/=1 the new method incorporates the ellipticity constraint into the normalization factor. The new method combines several advantages: 1) it is ellipse-specific so that even bad data will always return an ellipse; 2) it can be solved naturally by a generalized eigensystem, and 3) it is extremely robust, efficient and easy to implement. We compare the proposed method to other approaches and show its robustness on several examples in which other nonellipse-specific approaches would fail or require computationally expensive iterative refinements.

Analysing error of fit functions for ellipses

We describe several established error of fit (EOF) functions for use in the least square fitting of ellipses, and introduce a further four new EOFs. Four measures are used for assessing the suitability of such EOFs, quantifying their linearity, curvature bias, asymmetry, and overall goodness. These measures enable a better understanding to be gained of the individual merits of the EOF functions.

Assessing Error of Fit Functions for Ellipses

A variety of error of fit (EOF) functions have been proposed for use in the least-square fitting of ellipses. We describe four measures for assessing the suitability of such EOFs, quantifying their linearity, curvature bias, asymmetry, and overall goodness. These measures enable a better understanding to be gained of the individual merits of the EOF functions.

A note on the least squares fitting of ellipses

The characteristics of two normalisations for the general conic equation are investigated for use in least squares fitting: either setting F = 1 or A + C = 1. The normalisations vary in three main areas: curvature bias, singularities, transformational invariance. It is shown that setting F = 1 is the more appropriate for ellipse fitting since it is less heavily curvature biased. Setting A + C = 1 produces more eccentric conics, resulting either in over-elongated ellipses or hyperbolae. Although the F = 1 normalisation is less well suited than the A + C = 1 normalisation with respect to singularities and transformational invariance both these problems are solved by normalising the data, shifting it so that it is centred on the origin before shifting fitting, and then re-expressing the fit in the original frame of reference.