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MaxPol Smoothing and Differentiation Package

version 1.0 (12.2 MB) by Mahdi S. Hosseini
MaxPol is a numerical differentiation toolbox for signal and image processing


Updated 26 Nov 2017

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What is MaxPol Package?
MaxPol is an open source code written in MATLAB delivers a comprehensive tool for numerical differentiation. The toolbox renders variety of finite impulse response (FIR) filter kernels in closed form that can be used to approximate numerical derivatives of a given discrete signals and images.

MaxPol V1.0 supports a number of utilities including:

(1) Cutoff (lowpass) design with no side-lob artifacts (for noise-robust case)
(2) Fullband finite difference formulation
(3) Derivative matrix with four design cases of boundary fomulations
(4) Arbitrary order of differentiation
(5) Arbitrary polynomial accuracy
(5) 2D Derivative Kernels with Steering moments
(6) Intuitive examples in Signal and Image processing
Which disciplinary fields can use this package?

MaxPol package provides comprehensive numerical solutions for uniform discrete differentiation that can be of interest to broad audiences in engineering and science. In the context of image processing, the package can be used in diverse imaging applications. In general if you need to compute discrete derivatives of a uniform signal or image samples in the form of gradient, Hessian, or even high order tensor, this package can be easily used with high accuracy approximations.

In the context of image processing MaxPol can be used to estimate high order feature moments applied in edge detection, curvature estimation, image enhancement, and image sharpening. MaxPol can be also used in inverse imaging problems for image restoration such as in gradient surface reconstruction, image stitching, image diffusion, variational regularization problems applied in image in-painting, enhancement/de-noising, deconvolution problems, and many more.

The package can be also utilized as a numerical solver to the PDE problems applied in fluid mechanics, acoustics, and wave equations. It can be of interest in dynamic control systems to directly estimate state variables to avoid noise robust numerical integration.


[1] Hosseini, Mahdi S., and Konstantinos N. Plataniotis. "Derivative kernels: Numerics and applications." IEEE Transactions on Image Processing 26.10 (2017): 4596-4611.

Publisher link:

[2] Hosseini, Mahdi S., and Konstantinos N. Plataniotis. "Finite Differences in Forward and Inverse Imaging Problems: MaxPol Design." SIAM Journal on Imaging Sciences 10.4 (2017): 1963-1996.

Publisher link:

For more information please also refer to the MaxPol user' guide inside the package.

Cite As

Mahdi S. Hosseini (2021). MaxPol Smoothing and Differentiation Package (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (5)

Mahdi S. Hosseini

@Alessandro: thank you for the comment. If you read the toolbox guideline you will notice it says is works for both symbolic and numerical calculations. In particular, we use symbolic for both Lowpass/Fullband filters, and numerical calculation only for Fullband filters!


It should be mentioned that his works only with the symbolic toolbox.

Usman Ali



Worked better than any other available off-the-shelf toolboxes on zero-phase filtering for me.

MATLAB Release Compatibility
Created with R2015b
Compatible with any release
Platform Compatibility
Windows macOS Linux

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MaxPol Package/

MaxPol Package/demo examples/FIR frequency responses/

MaxPol Package/demo examples/directional (steerable) 2D derivatives/

MaxPol Package/demo examples/forward signal and imaging problems/Canny edge detection/

MaxPol Package/demo examples/forward signal and imaging problems/image sharpening (unsharp masking)/

MaxPol Package/demo examples/forward signal and imaging problems/one dimensional decomposition/

MaxPol Package/demo examples/forward signal and imaging problems/three dimensional decomposition/

MaxPol Package/demo examples/forward signal and imaging problems/two dimensional decomposition/

MaxPol Package/demo examples/inverse imaging problems/

MaxPol Package/demo examples/spectral analysis of derivative matrix/

MaxPol Package/third party codes/

MaxPol Package/third party codes/Fornberg/

MaxPol Package/third party codes/SI/

MaxPol Package/third party codes/Savitzky Golay/

MaxPol Package/utilities/