CGAL users discussion list

[杨成林 <>]

PCA in CGAL is TLS, I think.

2010/1/15, Pierre Alliez 
<>:
> hi,
>
> for PCA in CGAL it is the default orthogonal projection.
>
> pierre
>
> Pierre Alliez
> INRIA Sophia Antipolis - Mediterranee
> Project-team GEOMETRICA
> http://www-sop.inria.fr/members/Pierre.Alliez/
> Tel: +33 4 92 38 76 77
> Fax: +33 4 97 15 53 95
>
>
>
> ganders a écrit :
>> Hi Pierre,
>>
>> thanks for your reply, but I read this manual and it confuses me a bit and
>> thus I had - and still have - this question. Looking at figure 1 on page 3
>>
>> I. Markovsky and S. Van Huffel, Overview of total least squares methods,
>> Signal Processing, Volume 87, pages 2283-2302, 2007.
>> ftp://ftp.esat.kuleuven.ac.be/pub/SISTA/markovsky/reports/05-34.pdf
>>
>> the difference between LS and TLS is clearly described and obvious. The
>> right TLS picture has a orthogonal projection, the left LS one does not.
>> This CGAL manual is mentioning a LS - but not explicitely a TLS - with
>> orthogonal projection. Interpreting your answer, I guess it is a TLS,
>> right?
>> Thanks in advance.
>>
>> Cheers,
>> Gerd
>>
>>
>> Pierre Alliez wrote:
>>
>>> hi Gerd,
>>>
>>> the documentation of the PCA component (user manual
>>> <http://www.cgal.org/Manual/last/doc_html/cgal_manual/Principal_component_analysis/Chapter_main.html>)
>>>
>>> says:
>>> Given a set of objects, /linear least squares fitting/ amounts to
>>> finding the linear sub-space which minimizes the sum of squared
>>> distances from all points composing the objects of the set, to their
>>> projection onto this linear sub-space. Such linear sub-space is obtained
>>> by so-called principal component analysis (PCA). PCA is defined as a
>>> transformation that transforms the objects to a new coordinate system
>>> such that the greatest variance by orthogonal projection of the objects
>>> comes to lie on the first coordinate (called the first principal
>>> component), the second greatest variance on the second coordinate, and
>>> so on.
>>>
>>> pierre
>>>
>>> Pierre Alliez
>>> INRIA Sophia Antipolis - Mediterranee
>>> Project-team GEOMETRICA
>>> http://www-sop.inria.fr/members/Pierre.Alliez/
>>> Tel: +33 4 92 38 76 77
>>> Fax: +33 4 97 15 53 95
>>>
>>>
>>>
>>> ganders a écrit :
>>>
>>>> Dear CGAL-team,
>>>>
>>>> I am searching for a method that calculates a fitting line in terms of a
>>>> total-least-squares (TLS) approach for 3D points. As input I would like
>>>> to
>>>> provide a matrix with the x/y/z coordinates and as output I would like
>>>> to
>>>> receive two points describing the straight line. It looks that
>>>> linear_least_squares_fitting_3 is exactly that what I am looking for.
>>>> But
>>>> I
>>>> am not sure if it really calculates the TLS or the different
>>>> linear-least-squares (LLS) solution. May I ask you to tell me what
>>>> approach
>>>> is implemented (TLS or LLS)? Thanks a lot in advance.
>>>>
>>>> Cheers,
>>>> Gerd
>>>>
>>>>
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>>
>>
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-- 
杨成林
Yang Chenglin