Closed-form solution of absolute orientation using orthonormal matrices
01 Jul 1988-Journal of The Optical Society of America A-optics Image Science and Vision (Optical Society of America)-Vol. 5, Iss: 7, pp 1127-1135
TL;DR: In this paper, a closed-form solution to the least square problem for three or more points is presented, which requires the computation of the square root of a symmetric matrix, and the best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids.
Abstract: Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.
Summary (3 min read)
Jump to: [1. Introduction] – [2.1. Governing equations] – [2.3. Characterisation of air flow] – [3.1. Potential flow equations] – [4. Scaling of air compression] – [4.2. Scaling recommendations] – [5. Validation and example applications] – [5.1. Validation] – [5.2.1. OWC experiments in UWA [24]] – [5.2.2. Pico power plant [30, 31, 33]] and [6. Discussion and recommendations for use]
1. Introduction
- The oscillating water column (OWC) is arguably the most successful wave energy conversion (WEC) device, as its design and installation are relatively simple, and its maintenance is not demanding [1].
- Given the assumptions above, the method can give predictions of the e ciency of the system, but not of hydrodynamic loads.
- Nevertheless, the study does give useful insights for the causes of air compression in prototype OWC, demonstrating that these are more likely to25 appear when the height of the OWC is increased.
- A recent example of such work is [9], where the introduction and influence of wave collectors in front of an OWC structure is studied, considering incompressible flow.
- An additional goal is to provide some theoretical background for validating numerical models that include the compressibility of the air.
2.1. Governing equations
- Figure 1 shows a sketch of the domain in the vicinity of the OWC.
- Subsequently, the total volume of air inside the OWC chamber is: V (t) = Z Z b 0 dV = AOWCh Z Z b 0 ⌘dxdz = AOWC(h ⌘̄(t)) (2) where the double-integrals are over the uniform cross-section of the OWC,160 AOWC is the OWC cross-sectional area, ⌘̄(t) is the spatially averaged water level inside the OWC.
- Any additional energy losses caused by compressible throttling e↵ects can be nevertheless included in the resistance layer coe cients,185 as long as relevant empirical relations are provided.
- Coe cients K1 and K2 are primarily representative of the PTO re- sistance, meaning the pressure drop induced by the flow rate passing205 through the PTO.
2.3. Characterisation of air flow
- The compression number ⌦ can be considered as a parameter that measures the relative importance of air compressibility, with respect to the OWC characteristics.
- For ⌦ >> 0 (compressible behaviour), there is less volume flux passing through the resistance layer than that displaced by the water surface.
- The concept of the compression number and its importance for characterising air flow inside the OWC chamber can be also assessed by comparing it to the non-dimensional number derived by [16], defined according to the following:310 = !.
3.1. Potential flow equations
- The geometry of the domain around the OWC device is shown in Figure 5,350 where the subdomains for the air and water and their boundaries are identified.
- In addition, the analysis above allows us to consider only the incompressible e↵ects, by using the scattering wave field that excites the incompressible mode435 and by considering the incompressible radiation equation.
- It can be shown that ⌦ 0 opt < 1 from equation (48) by considering K = Kopt, K 0 = K 0opt and multiplying both parts with495 !h/ po. Equation (56) suggests that by taking into account the compressibility e↵ects, the optimal PTO resistance coe cients is increased compared to the one calculated by [5], by assuming incompressible flow.
4. Scaling of air compression
- The authors investigate the influence on their mathematical model of applying Froude scaling to the description of the water flow.
- By applying these relations to definition (14), the authors find that the compression numbers for the prototype and model scale di↵er in the ratio:510 ⌦M ⌦P = 1/sF . (58) Equation (58) makes clear that using a global Froude scale incorrectly scales compressibility, as ⌦ changes with scale.
- The potential flow equations can be used to inform us of the implications of using Froude scaling, as they are satisfied in all scales.
- In the latter525 the authors observe that the scattering potential obeys Froude scaling laws, which is expected, as this component concerns the interaction with the OWC without air pressurisation.
4.2. Scaling recommendations
- As the authors have demonstrated in (48 and 49), it is possible to manipulate the scattering and incompressible radiation equations in order to obtain the same as those encountered in [5], by using the concept of the scattering potential that excites the incompressible mode and the equivalent resistance.
- The authors can use equations (74)-(75) to scale the air pressure and hydrodynamic pressures inside the OWC, assuming that they are proportional to the amplitude.
- If the waves have a relatively narrow band of frequencies, then the authors may choose a value of ! in the centre of the band.
5. Validation and example applications
- The method is validated using numerical and experimental data originally presented in [26, 29, 23] and subsequently, example applications680 are presented from other physical model tests e.g. [24] and prototype OWC structures, such as the Pico Power Plant [30, 31].
- For the purposes of all calculations, it is assumed that po=140kPa.
5.1. Validation
- The experimental configuration of the physical model tests in the Grosse685 Welle Kanal (GWK) is presented in [26, 29, 23] in detail.
- A set regular and random wave conditions were tested combined with di↵erent orifice diameters varying from 0.05 m to 0.3 m, to investigate the e↵ect of di↵erent PTO configurations.
- The compression number ⌦ was also calculated theoretically using equation715 14.
- When the pressure drop is related to the velocity quadratically, a calcula- tion of the vertical velocities in the OWC chamber is necessary to obtain740 a value for K.
- As ⌦ is part of the calculation procedure.
5.2.1. OWC experiments in UWA [24]
- The experimental tests were performed in a 50m long and 1.5m wide wave flume at the University of Western Australia (UWA).
- Both openings practically extend at the full width of the flume, thus making the configuration two-dimensional.
- The wave energy captured at the PTO is calculated as EOWC = nEw, where n is taken from the derived e ciency curve in [24] and Ew is calculated from the incident wave field using equation 50.
5.2.2. Pico power plant [30, 31, 33]
- The OWC structure from the Pico power plant in Azores, Portugal is used as an example [30].
- The linear damping coe cient with respect to the air flow discharge was set to 120 Pa · s/m3.
6. Discussion and recommendations for use
- The authors propose the use of a compression number800 ⌦, for the air flow characterisation inside the OWC.
- The authors have shown that when the compression number is su ciently805 small (e.g. ⌦ 0.1) then the air flow can be considered as incompressible, whereas when ⌦ >> 10 1, air compressibility is significant.
- These e↵ects can only be investigated by considering the coupled problem of810 wave interaction with the OWC structure and response of the PTO.
- Therefore, the air compressibility is introduced to the potential flow equations for the water phase through the air-water dynamic boundary condition inside the OWC.
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This book presents a coherent approach to the fast-moving field of computer vision, using a consistent notation based on a detailed understanding of the image formation process. It covers even the most recent research and will provide a useful and current reference for professionals working in the fields of machine vision, image processing, and pattern recognition.
An outgrowth of the author's course at MIT, Robot Vision presents a solid framework for understanding existing work and planning future research. Its coverage includes a great deal of material that is important to engineers applying machine vision methods in the real world. The chapters on binary image processing, for example, help explain and suggest how to improve the many commercial devices now available. And the material on photometric stereo and the extended Gaussian image points the way to what may be the next thrust in commercialization of the results in this area.
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