Savitzky-Golay Smoothing and Differentiation Filter

Savitzky-Golay smoothing and differentiation filter.

Description

Function:
        Savitzky-Golay Smoothing and Differentiation Filter
        The Savitzky-Golay smoothing/differentiation filter (i.e., the polynomial smoothing/differentiation filter,
        or the least-squares smoothing/differentiation filters) optimally fit a set of data points to polynomials
        of different degrees.
        See for detail in Matlab Documents (help sgolay). The sgolay function in Matlab can deal with only
        symmetrical and uniformly spaced data of even number.
        This function presented here is a general implement of the sgolay function in Matlab. The Savitzky-Golay filter
        coefficients for even number, nonsymmetrical and nonuniformly spaced data can be obtained. And the filter coefficients
        for the initial point or the end point can be obtained too.In addition, either numerical results or symbolical
        results can be obtained.
 Usage:
        fc = sgsdf(x,n,dn,x0,flag)
        x = the original data point, e.g., -5:5
        n = polynomial order, default=1
        dn = differentation order (0=smoothing), default=0
        x0 = estimation point, default=0
        flag = numerical(0) or symbolical(1), default=0
        fc = filter coefficients obtained.
 Notes:
 1. x can be arbitrary, e.g., odd number or even number,symmetrical or nonsymmetrical, uniformly spaced
           or nonuniformly spaced, etc.
 2. x0 can be arbitrary, e.g., the initial point,the end point,etc.
 3. Either numerical results or symbolical results can be obtained.
 Example:
        sgsdf([-3:3],2,0,0,0)
        sgsdf([-3:3],2,0,0,1)
        sgsdf([-3:3],2,0,-3,1)
        sgsdf([-3:3],2,1,2,1)
        sgsdf([-2:3],2,1,1/2,1)
        sgsdf([-5:2:5],2,1,0,1)
        sgsdf([-1:1 2:2:8],2,0,0,1)
 Author:
        Jianwen Luo <[e-mail], [e-mail]> 2003-10-05
        Department of Biomedical Engineering, Department of Electrical Engineering
        Tsinghua University, Beijing 100084, P. R. China
 Reference
 [1]A. Savitzky and M. J. E. Golay, "Smoothing and Differentiation of Data by Simplified Least Squares Procedures," Analytical Chemistry, vol. 36, pp. 1627-1639, 1964.
 [2]J. Steinier, Y. Termonia, and J. Deltour, "Comments on Smoothing and Differentiation of Data by Simplified Least Square Procedures,"
    Analytical Chemistry, vol. 44, pp. 1906-1909, 1972.
 [3]H. H. Madden, "Comments on Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data," Analytical Chemistry, vol. 50, pp. 1383-1386, 1978.
 [4]R. A. Leach, C. A. Carter, and J. M. Harris, "Least-Squares Polynomial Filters for Initial Point and Slope Estimation," Analytical Chemistry, vol. 56, pp. 2304-2307, 1984.
 [5]P. A. Baedecker, "Comments on Least-Square Polynomial Filters for Initial Point and Slope Estimation," Analytical Chemistry, vol. 57, pp. 1477-1479, 1985.
 [6]P. A. Gorry, "General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method," Analytical Chemistry, vol. 62, pp. 570-573, 1990.
 See also:
        sglay, sgsdf_2d2

• Acknowledgements: This submission has inspired the following:; z-transform of 1D & 2D Savitzky-Golay Smoothing and Differentiation Filter, fSGolayFilt, Savitzky-Golay smooth/differentiation filters and filter application
• MATLAB release: MATLAB 6.5 (R13)

[YangQuan Chen]

Please also check my similar contribution

http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3514&objectType=FILE

File Name: Savitzky-Golay Smoothing Filter
  Author: YangQuan Chen
  Summary: FIR filter design via local moving window LS fitting - A magic smooth and derivative formula
  Submitted: 2003-05-29
  Downloads: 479
 
  Description: Check more information at

http://www.csois.usu.edu/people/yqchen/dd/index.html
   

File Details
 
  File Id: 3514 Average review: 5.0
  MATLAB Release: R13 # of reviews: 1
  Size: 1 KB Subscribers: 0
  Keywords: Integer-Order Digital Differentiator, FIR,

[Grant Murphy]

Love your work :-)
It saved me doing this very thing ..
I like that it can be used to generate moving extraploation filters.

[Jianwen Luo]

errata:

change the last line in this function

fc=sum(h);

to

if size(h,1)>1
    fc=sum(h);
else
    fc=h;
end

for consideration of the case when n=0

[Alpay Koc]

 

[Chunsheng Wu]

Good!

[Jianwen Luo]

See also at http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=5556&objectType=FILE

[Juha Abolmaesumi]

Excellent.

[Jianwen Luo]

Recently we have got our manuscript on the proprieties of Savitzky-Golay differentiation filters accepted for publication in Digital Signal Processing. The abstract is as below.

The Savitzky-Golay (SG) filters are generally used for smoothing and differentiation in many fields. The properties of the SG smoothing filters have been well studied. However, the properties of the SG differentiation filters or SG digital differentiators (SGDD, for the first order differentiation) are not developed well somehow, although they have been widely used. In the paper, the properties of the SGDD are discussed in detail. The effects of the SGDD on a noise-free single Gaussian line and a doublet are studied via simulation. The results indicate that the contrast and resolution loss of the SGDD depends on the ratio of the width of signal derivative and filter length. The linear SGDD is not preferable for preserving small details of signal derivative from this study. The cubic and quintic SGDD with the choice of optimal filter length are recommended in order to maintain the resolution of signal derivative.

An electrical copy would be sent on request via E-mail. ([e-mail] or [e-mail]). Thank you.

[Jianwen Luo]

Our work on the Savitzky-Golay differentiation filter has been published, as

Luo J W, Ying K, He P, Bai J. Properties of Savitzky-Golay Digital Differentiators, Digital Signal Processing, 2005, 15(2): 122-136.

The paper is available at:
http://www.cs.wright.edu/~phe/Research/Journal.html

Thank you very much for your interest.

[sigit ariatmoko]

 

[DON NGUYEN]

 

[newmouse newmouse]

 

[senuma senuma]

 

[Duane Hanselman]

Posting of .exe files is not acceptable since they do not run on all MATLAB supported platforms. In addition, it is easy to hide spyware or a virus in an executable file. What happens when this exe file is run???

[Linnea Goodman]

very helpful!

[aungsoe khaing]

 

[Olivier Planchon]

A good job, I will be using it for the unsymetric case. I have a single comment. I suggest changing line 85 from :
h=inv(A'*A)*A';
to :
h=M\eye(n) ;
Note : n is size(A,2)

This comment is worth a line of justification:
line 85 is intended to solve the following equation of unknown h : M*h = I ; (M, rectangular, and I = identity). Mathlab standards commands to use the \ operator instead of the classical inv(A'*A) stuff.

Beyond correctness, the gain is in speed, precision, and robustness.

[Hyeong-yong Jeon]

Thank you for sharing your codes. It's very good! Got will be blessing you. Good-Luck!

[Bob Marley]

This algorithm can be made much more concise.
Using the author's notation:

N = length(x);
x=x(:);
a_matrix = cumprod([ones(N,1) ...
                   x*ones(1,n)],2)\eye(N);
for i = 0:n
const(i+1) = ...
              x0^(i-dn)*prod(i:-1:i-dn+1);
end
fc = sum((const'*ones(1,N)).*a_matrix)';

[Pavel Holoborodko]

Alternative approach for noise-robust differentiation is described here:
http://www.holoborodko.com/pavel/?page_id=245
On the contrary to Savitzky-Golay filters it has nice property of guaranteed noise suppression.

[Roman Balabin]

 

[Khawaja Asim]

I am new user of matlab.. I need to use Savitzky-Golay smoothing on image for surface estimation. Can someone give me tips, that are helpful in my work.. Thanks