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Wed, 22 Apr 2009 05:40:03 +0000
3D curve fitting with Taylor Expansion
http://www.mathworks.com/matlabcentral/newsreader/view_thread/249643#644558
Jade Forest
Hi <br>
<br>
This is Jade Forest. It's good to meet all of you on this site. <br>
<br>
I am now study the SIFT(scale invariant feature transformation) and try to implement in Matlab. <br>
<br>
However, I am stuck on the step: "Localization", this include a 3D curve fitting technique to find the local max & min in an image. <br>
<br>
I tried to find books and documents that talk about this subject, but didn't find any. <br>
<br>
The 3D quadratic Taylor Expansion is like below: <br>
<br>
D(x) = D + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂X2)X<br>
<br>
Is there anyone could telll me where I can find information about this subject(books or documents) and is there anyone available to share a code about this?<br>
<br>
Or, anybody available to offer complete SIFT codes for my reference? <br>
<br>
tks<br>
<br>
Jade Forest <br>
<br>

Wed, 22 Apr 2009 22:20:02 +0000
Re: 3D curve fitting with Taylor Expansion
http://www.mathworks.com/matlabcentral/newsreader/view_thread/249643#644771
Roger Stafford
"Jade Forest" <sllinios@pchome.com.tw> wrote in message <gsmajj$72g$1@fred.mathworks.com>...<br>
> I am now study the SIFT(scale invariant feature transformation) and try to implement in Matlab. <br>
> However, I am stuck on the step: "Localization", this include a 3D curve fitting technique to find the local max & min in an image. <br>
> I tried to find books and documents that talk about this subject, but didn't find any. <br>
> The 3D quadratic Taylor Expansion is like below: <br>
> D(x) = D + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂X2)X<br>
<br>
Except to refer you to websites such as<br>
<br>
<a href="http://en.wikipedia.org/wiki/Scaleinvariant_feature_transform">http://en.wikipedia.org/wiki/Scaleinvariant_feature_transform</a><br>
<br>
I cannot help you with the main thrust of your question. I am not familiar with the subject.<br>
<br>
For an interpretation of the Taylor expansion you quote if that puzzles you, I refer you to the site:<br>
<br>
<a href="http://en.wikipedia.org/wiki/Taylor_expansion">http://en.wikipedia.org/wiki/Taylor_expansion</a><br>
<br>
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:<br>
<br>
[Dx,Dy,Dz]*[x;y;z]<br>
<br>
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<br>
<br>
(1/2)X[T](d2D[T]/dX2)X<br>
<br>
signifies half the product of a row vector, a 3 x 3 matrix, and a column vector:<br>
<br>
1/2 * [x,y,z] * [Dxx,Dxy,Dxz;Dyx,Dyy,Dyz;Dzx,Dzy,Dzz] * [x;y;z]<br>
<br>
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.<br>
<br>
Roger Stafford

Fri, 05 Mar 2010 04:06:05 +0000
Re: 3D curve fitting with Taylor Expansion
http://www.mathworks.com/matlabcentral/newsreader/view_thread/249643#723593
nethaji anandhavalli
"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <gso56i$d88$1@fred.mathworks.com>...<br>
> "Jade Forest" <sllinios@pchome.com.tw> wrote in message <gsmajj$72g$1@fred.mathworks.com>...<br>
> > I am now study the SIFT(scale invariant feature transformation) and try to implement in Matlab. <br>
> > However, I am stuck on the step: "Localization", this include a 3D curve fitting technique to find the local max & min in an image. <br>
> > I tried to find books and documents that talk about this subject, but didn't find any. <br>
> > The 3D quadratic Taylor Expansion is like below: <br>
> > D(x) = D + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂X2)X<br>
> <br>
> Except to refer you to websites such as<br>
> <br>
> <a href="http://en.wikipedia.org/wiki/Scaleinvariant_feature_transform">http://en.wikipedia.org/wiki/Scaleinvariant_feature_transform</a><br>
> <br>
> I cannot help you with the main thrust of your question. I am not familiar with the subject.<br>
> <br>
> For an interpretation of the Taylor expansion you quote if that puzzles you, I refer you to the site:<br>
> <br>
> <a href="http://en.wikipedia.org/wiki/Taylor_expansion">http://en.wikipedia.org/wiki/Taylor_expansion</a><br>
> <br>
> 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:<br>
> <br>
> [Dx,Dy,Dz]*[x;y;z]<br>
> <br>
> 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<br>
> <br>
> (1/2)X[T](d2D[T]/dX2)X<br>
> <br>
> signifies half the product of a row vector, a 3 x 3 matrix, and a column vector:<br>
> <br>
> 1/2 * [x,y,z] * [Dxx,Dxy,Dxz;Dyx,Dyy,Dyz;Dzx,Dzy,Dzz] * [x;y;z]<br>
> <br>
> 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.<br>
> <br>
> Roger Stafford