Reconstructing Ellipsoids from Projections

Karl, W.C.; Verghese, George C.; Willsky, Alan S.

doi:10.1006/cgip.1994.1012

Cited by 30 publications

(5 citation statements)

References 31 publications

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“…In Section 3.1.1 , it was shown that there exists a direct relation between the lengths of the principal axes of an ellipse and the covariance matrix

of the pixel coordinates. In [ 30 ], it is shown how to obtain the variances of the ellipsoid projections using its own variances.…”

Section: Materials and Methodsmentioning

confidence: 99%

“…Examples of fruits that approximate these shapes are also shown in Figure 6. An interesting property of ellipsoids, and spheroids in particular, is that the shapes of their orthogonal projections are ellipses [30]. Given the size of the fruits and the typical height of the camera (about 1 m), perspective effects are negligible and a parallel camera can be assumed locally for each fruit [31].…”

Section: Modeling the 3d Shape Of The Fruitsmentioning

confidence: 99%

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Fast 3D Rotation Estimation of Fruits Using Spheroid Models

Albiol

Merás³

2021

Sensors

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Automated fruit inspection using cameras involves the analysis of a collection of views of the same fruit obtained by rotating a fruit while it is transported. Conventionally, each view is analyzed independently. However, in order to get a global score of the fruit quality, it is necessary to match the defects between adjacent views to prevent counting them more than once and assert that the whole surface has been examined. To accomplish this goal, this paper estimates the 3D rotation undergone by the fruit using a single camera. A 3D model of the fruit geometry is needed to estimate the rotation. This paper proposes to model the fruit shape as a 3D spheroid. The spheroid size and pose in each view is estimated from the silhouettes of all views. Once the geometric model has been fitted, a single 3D rotation for each view transition is estimated. Once all rotations have been estimated, it is possible to use them to propagate defects to neighbor views or to even build a topographic map of the whole fruit surface, thus opening the possibility to analyze a single image (the map) instead of a collection of individual views. A large effort was made to make this method as fast as possible. Execution times are under 0.5 ms to estimate each 3D rotation on a standard I7 CPU using a single core.

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“…In Section 3.1.1 , it was shown that there exists a direct relation between the lengths of the principal axes of an ellipse and the covariance matrix

of the pixel coordinates. In [ 30 ], it is shown how to obtain the variances of the ellipsoid projections using its own variances.…”

Section: Materials and Methodsmentioning

confidence: 99%

Section: Modeling the 3d Shape Of The Fruitsmentioning

confidence: 99%

Fast 3D Rotation Estimation of Fruits Using Spheroid Models

Albiol

Merás³

2021

Sensors

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“…Computation methods for the evaluation of the cross-section of the ellipsoid have been studied by other authors. 8,9 For completeness, the the main points of these calculations are summarized here. A 3D ellipsoid centered at the origin is given by…”

Section: Cross-sectional Ellipsementioning

confidence: 99%

Developing a reconstruction algorithm for 3D birefringence from tomographic polarimetry

Seigo,

Fukui,

Kawano

et al. 2023

Polarization Science and Remote Sensing XI

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Stress-induced birefringence is created by the manufacturing process of injection-molded lenses. Characterizing the degradation in optical performance is important information to guide manufacturing improvements. The 3D birefringence distribution is proportional to the inhomogenous stress field. Birefringence is an anistropic property therefore retardance measurements change if the optical component is rotated. Reconstruction of a 3D birefringence distribution from a series of retardance measurements poses a challenge due to the extension of tomographic algorithms to tensor-valued quantities. Our approach is a closed-form forward model to linearly relate the polarimetric measurements to a line projection through an discrete arrangement of index ellipsoids. The rank of this linear system is investigated for varying arrangements of index ellipsoids. For example, given an homogenous birefringence distribution, only three non-parallel line projections are required for a full-rank operator that reconstructs the index ellipsoid of a planar slice. These three line projections are full-rank for some arrangements of three unique index ellipsoids and rank deficient for others. This mathematical framework is developed to design a tomographic polarimeter for inspecting the stress-induced 3D birefringence distribution of injection-molded optics. INTRODUCTIONInjection-molded lenses have an inhomogeneous stress-induced birefringence that can degrade optical performance. Many studies have been conducted to investigate residual stress and resulting birefringence caused by molding conditions. 1, 2 These studies have contributed to improved performance and higher yield of injection molded optical elements. Numerical modeling software such as Moldflow 3 can be used to establish the relationship between molding conditions and birefringence distribution and thereby identify process improvements for reducing birefringence. 4 The magnitude of stress-induced birefringence is also dependent on the mechanical properties of the material. 5 Commercially available analyzers like Photonic Lattice's 2D birefringence analyzer 6 are widely used to analyze birefringence distributions. However, such instruments are limited to measuring 2D birefringence. This paper presents a new approach for measuring and analyzing anisotropic inhomogeneous samples. A tomographic reconstruction of the 3D birefringence distribution is developed from a linear line projection relationship between the spatially varying index ellipsoids and tomographic polarimetric observations. To simplify the geometry of the line projection, the refractive index gradients are assumed to be small enough that light propagates in a straight line within the optic. This forward representation allows tomographic inverse solutions to be applied to quantify the 3D birefringence distribution of an inhomogeneous and anisotropic sample. TOMOGRAPHIC POLARIZATION Index EllipsoidAn index ellipsoid defines the polarization-dependent index of refraction (e.g. birefringence) of a material for all dire...

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“…Mathematically we seek a vector to satisfy the following equation (4) III. PARAMETERIZATION OF GEOMETRY Many shape parameterization options are available in literature such as Fourier descriptors [25], Lagrange interpolation [26] and parametrically defined shapes [27], [28] based on global representation of the geometry, as well as B-splines [29]- [31] where the control points determine the properties of the curve. We choose B-splines to model the irregular transition layer between adipose and fibroglandular layers where the control points determine the shape of boundary.…”

Section: Problem Statementmentioning

confidence: 99%

A Shape-Based Inversion Algorithm Applied to Microwave Imaging of Breast Tumors

Firoozabadi

Miller

2011

IEEE Trans. Antennas Propagat.

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We propose a new shape-based inversion algorithm to identify an anomaly embedded in an inhomogeneous layered geometry. We apply our approach to microwave breast imaging where the geometry consists of several inhomogeneous layers and the potential tumor is embedded in the innermost layer. In addition to the tumor identification, we estimate the irregular transition layer between the breast inner layers. Our inversion algorithm is based on a low-dimensional parametric form of the relevant geometries, and uses multiple-frequency multi-source data to estimate the unknowns. Several numerical examples are provided to evaluate the effectiveness of our approach and demonstrate its robustness both to the initial choice of parameter values and uncertainty in the complex dielectric properties of the media even with the complex geometry and the simplified inverse model.

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Reconstructing Ellipsoids from Projections

Cited by 30 publications

References 31 publications

Fast 3D Rotation Estimation of Fruits Using Spheroid Models

Fast 3D Rotation Estimation of Fruits Using Spheroid Models

Developing a reconstruction algorithm for 3D birefringence from tomographic polarimetry

A Shape-Based Inversion Algorithm Applied to Microwave Imaging of Breast Tumors

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