Path: news.mathworks.com!not-for-mail From: <HIDDEN> Newsgroups: comp.soft-sys.matlab Subject: Re: 3D curve fitting with Taylor Expansion Date: Wed, 22 Apr 2009 22:20:02 +0000 (UTC) Organization: The MathWorks, Inc. Lines: 32 Message-ID: <gso56i$d88$1@fred.mathworks.com> References: <gsmajj$72g$1@fred.mathworks.com> Reply-To: <HIDDEN> NNTP-Posting-Host: webapp-05-blr.mathworks.com Content-Type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: 8bit X-Trace: fred.mathworks.com 1240438802 13576 172.30.248.35 (22 Apr 2009 22:20:02 GMT) X-Complaints-To: news@mathworks.com NNTP-Posting-Date: Wed, 22 Apr 2009 22:20:02 +0000 (UTC) X-Newsreader: MATLAB Central Newsreader 1187260 Xref: news.mathworks.com comp.soft-sys.matlab:534817 "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