Leaching (chemistry)

In computer vision, the trifocal tensor (also tritensor) is a 3×3×3 array of numbers (i.e., a tensor) that incorporates all projective geometric relationships among three views. It relates the coordinates of corresponding points or lines in three views, being independent of the scene structure and depending only on the relative motion (i.e., pose) among the three views and their intrinsic calibration parameters. Hence, the trifocal tensor can be considered as the generalization of the fundamental matrix in three views. It is noted that despite that the tensor is made up of 27 elements, only 18 of them are actually independent.

Correlation slices

The tensor can also be seen as a collection of three rank-two 3 x 3 matrices ${\mathbf{T}}_1,{\mathbf{T}}_2,{\mathbf{T}}_3$ known as its correlation slices. Assuming that the projection matrices of three views are $\mathbf{P}=[\mathbf{I}|\mathbf{0}]$, ${\mathbf{P}}^{\text{'}}=[\mathbf{A}|{\mathbf{a}}_4]$ and ${\mathbf{P}}^{{\text{'}}^{\prime }}=[\mathbf{B}|{\mathbf{b}}_4]$, the correlation slices of the corresponding tensor can be expressed in closed form as ${\mathbf{T}}_i={\mathbf{a}}_i{\mathbf{b}}_4^t-{\mathbf{a}}_4{\mathbf{b}}_i^t,i=1\dots 3$, where ${\mathbf{a}}_i,{\mathbf{b}}_i$ are respectively the i^th columns of the camera matrices. In practice, however, the tensor is estimated from point and line matches across the three views.

Trilinear constraints

One of the most important properties of the trifocal tensor is that it gives rise to linear relationships between lines and points in three images. More specifically, for triplets of corresponding points $\mathbf{x}\leftrightarrow {\mathbf{x}}^{\text{'}}\leftrightarrow {\mathbf{x}}^{{\text{'}}^{\prime }}$ and any corresponding lines $\mathbf{l}\leftrightarrow {\mathbf{l}}^{\text{'}}\leftrightarrow {\mathbf{l}}^{{\text{'}}^{\prime }}$ through them, the following trilinear constraints hold:

$({\mathbf{l}}^{\text{'}t}[{\mathbf{T}}_1,{\mathbf{T}}_2,{\mathbf{T}}_3]{\mathbf{l}}^{{\text{'}}^{\prime }})[\mathbf{l}{]}_{\times }={\mathbf{0}}^t$

${\mathbf{l}}^{\text{'}t}\left(\sum _i{x}_i{\mathbf{T}}_i\right){\mathbf{l}}^{{\text{'}}^{\prime }}=0$

${\mathbf{l}}^{\text{'}t}\left(\sum _i{x}_i{\mathbf{T}}_i\right)[{\mathbf{x}}^{{\text{'}}^{\prime }}{]}_{\times }={\mathbf{0}}^t$

$[{\mathbf{x}}^{\text{'}}{]}_{\times }\left(\sum _i{x}_i{\mathbf{T}}_i\right){\mathbf{l}}^{{\text{'}}^{\prime }}=\mathbf{0}$

$[{\mathbf{x}}^{\text{'}}{]}_{\times }\left(\sum _i{x}_i{\mathbf{T}}_i\right)[{\mathbf{x}}^{{\text{'}}^{\prime }}{]}_{\times }={\mathbf{0}}_{3\times 3}$

where $[\cdot {]}_{\times }$ denotes the skew-symmetric cross product matrix.

Transfer

Given the trifocal tensor of three views and a pair of matched points in two views, it is possible to determine the location of the point in the third view without any further information. This is known as point transfer and a similar result holds for lines.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• Visualization of trifocal geometry (originally by Sylvain Bougnoux of INRIA Robotvis, requires Java)