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Thread Subject:
3D curve fitting with Taylor Expansion

Subject: 3D curve fitting with Taylor Expansion

From: Jade Forest

Date: 22 Apr, 2009 05:40:03

Message: 1 of 3

Hi

This is Jade Forest. It's good to meet all of you on this site.

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 + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂X2)X

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?

Or, anybody available to offer complete SIFT codes for my reference?

tks

Jade Forest
    
      

Subject: 3D curve fitting with Taylor Expansion

From: Roger Stafford

Date: 22 Apr, 2009 22:20:02

Message: 2 of 3

"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 + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂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

Subject: 3D curve fitting with Taylor Expansion

From: nethaji anandhavalli

Date: 5 Mar, 2010 04:06:05

Message: 3 of 3

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <gso56i$d88$1@fred.mathworks.com>...
> "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 + (∂D[T]/∂X)X + (1/2)X[T](∂2D[T]/∂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

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