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Subject: Re: 3D curve fitting with Taylor Expansion
Date: Wed, 22 Apr 2009 22:20:02 +0000 (UTC)
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"Jade Forest" <sllinios@pchome.com.tw> wrote in message <gsmajj$72g$1@fred.mathworks.com>...
> I am now study the SIFT(scale invariant feature transformation) and try to implement in Matlab.  
> However, I am stuck on the step: "Localization", this include a 3D curve fitting technique to find the local max & min in an image.   
> I tried to find books and documents that talk about this subject, but didn't find any.   
> The 3D quadratic Taylor Expansion is like below: 
> D(x) = D + (&#8706;D[T]/&#8706;X)X + (1/2)X[T](&#8706;2D[T]/&#8706;X2)X

  Except to refer you to websites such as

 http://en.wikipedia.org/wiki/Scale-invariant_feature_transform

I cannot help you with the main thrust of your question.  I am not familiar with the subject.

  For an interpretation of the Taylor expansion you quote if that puzzles you, I refer you to the site:

 http://en.wikipedia.org/wiki/Taylor_expansion

in the section called "Taylor series in several variables", (in your case three variables.)  What you write as (dD[T]/dX)X stands for a row vector multiplied by a column vector:

 [Dx,Dy,Dz]*[x;y;z]

where Dx, Dy, and Dz are the first partial derivatives of D(x,y,z) with respect to x, y, and z, respectively, evaluated at some "keypoint" and x, y, and z are the three coordinate differences from these keypoint values.  The next term

 (1/2)X[T](d2D[T]/dX2)X

signifies half the product of a row vector, a 3 x 3 matrix, and a column vector:

 1/2 * [x,y,z] * [Dxx,Dxy,Dxz;Dyx,Dyy,Dyz;Dzx,Dzy,Dzz] * [x;y;z]

where Dxx, Dxy, Dxz, Dyx, etc. signify the second partial derivatives of D with respect to x and x, with respect to x then y, etc., all evaluated at the keypoint, and again x, y, and z are coordinate differences from those of the keypoint.  That is the second order Taylor expansion in three variables about the point called here the keypoint.

Roger Stafford