File Exchange

image thumbnail

Zernike Moments

version 1.5 (15.1 KB) by Amir Tahmasbi
MATLAB Code for the Fast Calculation of Zernike Moments of order n and repetition m on NxN images.


Updated 20 Nov 2014

View Version History

View License

This submission includes 3 mfiles and 6 image files:
1- Zernike_main.m (The main script that takes care of everything)
2- Zernikmoment.m (Calculates the Zernike moments for an NxN ROI)
3- radialpoly.m (Calculates the radial polynomials which are prerequisites for calculating Zernike moments)
4- Six .png files to test the code.
When you run the Zernike_main.m, it will calculate the Zernike moments of order n=4 and repetition m=2 for the input images. Since the first row images are just the rotated versions of a unique object (oval), the magnitudes of the Zernike moments for these three images are the same. In addition, the differences between the phases of the moments are proportional to the rotation angles of the images. Expectedly, the Zernike moments of two different shapes (e.g. oval and rectangle) are totally different. The reason of this behavior is the ability of Zernike moments in describing the shape of objects.

License agreement: To acknowledge the use of the code please cite the following papers:

A. Tahmasbi, F. Saki, S. B. Shokouhi, Classification of Benign and Malignant Masses Based on Zernike Moments, Comput. Biol. Med., vol. 41, no. 8, pp. 726-735, 2011.

F. Saki, A. Tahmasbi, H. Soltanian-Zadeh, S. B. Shokouhi, Fast opposite weight learning rules with application in breast cancer diagnosis, Comput. Biol. Med., vol. 43, no. 1, pp. 32-41, 2013.

Cite As

Amir Tahmasbi (2021). Zernike Moments (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (73)

起灵 张

您好 初学冒昧的问一下
我想问一下怎么对解决图像旋转 缩小后zernike矩阵保证一致呢

Balasaheb Patre

I want to use this code on MXN image. I made required changes in code but I have a query.
How will it map MXN image onto unit circle ?


I'm trying to use this code for my own image (dog sillohoute) and I get the following error. Can anyone help ?

Index in position 2 exceeds array bounds (must not exceed 446).

Error in Zernikmoment (line 47)
Product = p(x,y).*Rad.*exp(-1i*m*Theta);

Error in Zernike_main (line 36)
[~, AOH, PhiOH] = Zernikmoment(p,n,m); % Call Zernikemoment fuction


Mahsa Abdollahi

can this code be used for NxM images?
also none of the N and M are even.


When I try to obtain momnets with n=2 and n=! an error appears, only with this moment

Error using factorial (line 20)
N must be an array of real non-negative integers.

Error in radialpoly (line 30)
c = (-1)^s*factorial(n-s)/(factorial(s)*factorial((n+abs(m))/2-s)*...

Error in Zernikmoment (line 45)
Rad = radialpoly(R,n,m); % get the radial polynomial

Error in Zernike_main (line 42)
[~, AOH, PhiOH] = Zernikmoment(p,n,m); % Call Zernikemoment fuction


It is very useful, thank you !

Kavitha N

Hi Amir,
Can you give a suggestion to display the moments of the image.
n=4, creates 3 moments. Can you let me know how many features are there in these 3 moments.
six or more


Hi Amir,thank you so much for share this code.I had a question.Can this be used for image reconstruction and how?Otherwise,please tell me how to extract feature from Remote images using zernike features? Thank you very much!

Aminul huq

Hello Amir, thank you so much for the code. I am using this in my work. I had a question. In any type of object recognition can both AOH and PhiOH be used as feature or just AOH should be used as feature ?

Anselm Griffin


Elif Nur Korkmaz

Hi Amir,
first of all thank you for this code n i asked one thing im student and make a proje copy-rotate-move forgery and im working on MxN image and LxL block n i have to use zernike at the all blocks but you did ;

x = 1:N; y = x;
[X,Y] = meshgrid(x,y);
R = sqrt((2.*X-N-1).^2+(2.*Y-N-1).^2)/N; % R=yarıçap yani x^2 + y^2 nin kökü
Theta = atan2((N-1-2.*Y+2),(2.*X-N+1-2));
R = (R<=1).*R;
Rad = radialpoly(R,nMax,m); % get the radial polynomial
Product = Block(x,y).*Rad.*exp(-1i*m*Theta);
Z = sum(Product(:));

n i wonder am i using this code for all blokcs? meshgrid is it any problem to block ? please answer me thank you.
(my english is so bad i know i hope i can say what i want)

Aminul huq

Can this be used for image reconstruction ?

mary stephy

please tell me how to extract feature from xray images using zernike features?

mary stephy


Hi Amir,
I've tried your code. It is really good but when I tried it shows these error
Index exceeds matrix dimensions.

Error in Zernikmoment (line 47)
Product = p(x,y).*Rad.*exp(-1i*m*Theta);

Error in process_predict_leaf (line 80)
[mom, amplitude, angle] =
Zernikmoment(b,4,0); % Call
Zernikemoment fuction n=4, m=0

Can you explain about the error? I really need help as I'm new in Zernike moment.
Thank you


Hi Amir
I have question
how can I get more value of Zernike moments?
thank you

Krzysztof Magda


Thank you very much for the detailed answer. I understand now how it works in terms of optical aberrations. There is only one thing that remains unclear to me now, i.e. is there any way to predict which value of m would yield the best results for a particular application? Are there any rules of thumb? Or is calculating moments for different m's and comparing errors the only way of predicting the best value of m?


Amir Tahmasbi

Hi Kpu,

Yes, it should and, hopefully, it will.

Amir Tahmasbi

Hi Krzysztof,

Thanks for your comment. As you may know, the Zernike moments were first developed to study optical aberrations. Though they are currently heavily used in image processing applications. Speaking of optical aberrations, please take a look at the following image

You will see the pupil functions associated with different aberrations on the left-hand side and their corresponding spatial images on the right-hand side. The spatial images are just the Fourier transform (FT) of the pupil functions since a lens takes an FT.

The important point is that different combinations of order (n) and repetition (m) of the Zernike moments represent different unique optical aberrations. For example, let's focus on the tip and tilt aberrations. For both of those, the order n is 1. In this case, the repetition (m) changes the rotation of the pupil function, where m=0 leads to tip and m=1 leads to tilt. Note that the tip and tilt aberrations correspond to a horizontal shift and vertical shift in the position of the image, respectively.

The take home message is that the repetition parameter typically only rotates the aberration pattern. Hope this answers your question.

Please feel free to contact me if you have any other concerns.


kpu kk

Will it run on any order with appropriate value of m (repeatn)?

Krzysztof Magda

Hello Amir,

Thank you for the piece of code that you've shared with us. I am new to the whole Zernike polynomials idea and I don't understand what is the 'repetition' parameter and how I could predict it's value for a particular application. I've read several articles about ZM algorithms and no one really explains that, and this is a key definition. Could you explain it to me? I think many people would be grateful for that (including me of course).

Thank you for your time.



I'm trying to use your code for 1024*1024 images and I keep getting the following error,

Error using .*
Integers can only be combined with integers of the same class, or
scalar doubles.
Error in Zernikmoment (line 47)
Product = p(x,y).*Rad.*exp(-1i*m*Theta);

Can you possibly explain me what causes the error?


mohammed Abed

Tatiara Régis

I have been reading the article you sent me.

Thank you.


Swati Vyas

Very informative...I Love the code..

Amir Tahmasbi

Hi Tatiara,

Here is the main publication I was referring to:

Hope this helps.


Tatiara Régis

Dear Tahmasbi,

I believe have located the article in:


Tatiara Régis

Dear Tahmasbi,
good afternoon!
The link sends detailed study on the frequency of Zernike Moments mainly seemed to be broken.
Please could you tell me the title of the publication?
I'm trying to implement the Zernike moments for fingerprints of microorganisms. I read some of his works and certainly I will quote them in my works. For then enrich my knowledge on the subject.

Rashi BN

good one

Vijayalakshmi Mahesh

Hi Amir,

Can you please help me in constructing the Zernike moment based kernels . These kernels when convolved with an image should detect the edge of it.

Thanks in advance


Can you provide me the help how to reconstruct the signal back after applying Zernike.

Sumeet padhi

sir please tell me how can i count the zernike moment at some predefined points using ur code.thank u

Meghana Dinesh


I want to use your code in my application to match templates. I have explained my issue here:

I am not very familiar with Image Moments so this might seem very trivial to you. The outputs A and phi are almost similar for images with and without noise (shown in screenshot on that link). Is this beyond the scope of your code? Is there anything I can alter so it can solve my issue?


Meghana Dinesh

Meghana Dinesh

* 'A' seems very similar


Hi Amir,

This code is excellent.
My doubt is how to use zernike moment for classification in back propagation algorithm. please help

Amir Tahmasbi

Hi Yijun,

Thanks for your feedback. I think the code should work for MxN images with minor modifications. I will try to incorporate this feature into the next update of the code.

Unfortunately, I don't have the code for Pseudo-Zernike moments.


Yijun Yan

Hi Amir,

This code is excellent.
But I wonder if it works well for the image with size M*N?
And do you happen to have the code for Pseudo-Zernike moment?


Amir Tahmasbi

Hi Satya,

Can you please be a little more specific about what you mean by the "kernel"? My understanding is that you what to plot the radial polynomials. Is this correct?


satya singh

dear amir,
can you please provide codes for plotting the kernel

Amir Tahmasbi

Hi Frederik,

Thank you for your interest in my code. I have addressed your concerns as follows:

1- Regarding “Product = p(x,y).*Rad.*exp(-1i*m*Theta);” I should say this is actually correct. Remember that the Zernike moments are calculated by multiplying the input image p(x,y) and the complex conjugate of 2-D Zernike basis functions (typically denoted by V_n,m = R_n,m(\rho) . exp(j m \theta). In other words, we have “Product = p(x,y) V*_m,n = p(x,y) . R_n,m(\rho) . exp( - j m \theta)” .The negative sign that you are refereeing to is, therefore, added due to the complex conjugate operator. For a clearer explanation please see my paper, p. 730, eq. 11 (available on-line at ). Also you can take a look at the classic paper on Zernike moments by Dr. Khotanzad, p. 490, eq. 5 (available on-line at ). You can further verify this using eq. 49 in the web page you sent me.
2- You are right. The mapping we have used in the code, loses the information of the corners of the image. This is, however, a common mapping strategy from the image space into the unit disk that researchers typically use for the calculation of the Zernike moments (see e.g. Dr. Khotanzad’s paper above). From a practical point of view, it usually does not cause any problem since the object of interest is scaled such that it is circumscribed by the unit disc.
3- The code can easily be generalized to the non-square shaped ROIs. Since this is a questions that many other people have also asked, I will update the code on the MATLAB central page for rectangular shaped images when I get a chance (probably in a few weeks). You can also do it yourself.
4- I can make some MxN test image sets, but I will need some time to do this. You can also make those images using the square images I put on the MATLAB central page (just add say 5 rows above and below the ROI). Then, you can rotate the object within the same ROI and verify if the magnitudes of the moments remain the same.

Frederik Kratzert

Hello Amir,
first of all, thanks for sharing your code.
secondly i would like to ask a question, in which i dont know if i make a mistake or you did in your code.

Product = p(x,y).*Rad.*exp(i*m*Theta)
to calculate the complex polynominal in all the formulars i found
(e.g: )
its written that j = sqrt(-1);
you wrote in your code -1*i instead, which is not the same. because sqrt(-1) is just the definition of the complex number to be i. so am i wrong or should it be:

Product = p(x,y).*Rad.*exp(i*m*Theta)

in your code, without the -1?

i hope its understandable what i want to say.

tanks again, frederik



Thank you

Dimitris Tsolakidis

Hello Amir. Thank you for your code. I have used your code to calculate zernike moments from an image, but i'm having problem reconstruct the image from it's moments. Could you provide me some help please?

mariem farhat

Another question:

1. your code assume that the image is square; and if it wasn't?

2. I don't understand Product = p(x,y).*Vnm? why not Product = p.*Vnm;

mariem farhat

Thank you for the code.
But why in the image mapping you use (2.*X-N-1) and not (2.*X-N)?

Thank you!

mariem farhat


Actually want to find the global features with zernike moments of the image but dono how to proceed with it??


can Any one say how to find the Zernike moments for rgb image...Thanks in advance

dana Wang

Hi Tahmasbi,
Thank you for your code. can u please hepl me to modify your code substituding p(x/a+x1,y/a+x2) to p(x,y),where p(x,y) is original image, x1 and x2 are the centroid of p(x,y),x1=m10/m00,x2=m01/m00,a=sqrt(β/m00),β is a predetermined value. In fact, this is doing scale and translation normalization. Thank you very much.

Amir Tahmasbi

Hi Mitra,

1- It mainly depends on your application and you need some experience to pick the appropriate order (n) and repetition (m) for your moments. However, the rule of thumb is that the lower orders provide less information (details) but are more robust to noise. Higher orders, on the other hand, provide more information about the details of the object but are more sensitive to the measurement noise. An efficient way is to calculate the Zernike moments for a variety of m and n values. You may then apply a feature selection algorithm and remove the correlated features (moments). You can finally use a group of say 20 moments, which are reasonably uncorrelated, to pass to your classifier.

2- I think with appropriate preprocessing steps, the Zernike moments should be useful for Farsi digit recognition.


mitra Isaei

oh, excuse me. I understand my answer.
but I have other questions:
1.How do I understand What values are appropriate for Assignment to n and m?
I'm using zernike moment for handwritten farsi digits recognition.
2. Are features that obtain from Zernike moments, useful for classification in handwritten farsi digits recognition?

mitra Isaei

hi sir
your code calculates one Zernike moment and its magnitude. how to change this code to calculate more Zernike moments.

Amir Tahmasbi

Hi Shriniwas,

Yes, the code is capable of calculating the Zernike moments of any order and repetition. All you need to do is change m (repetition) and n (order). The default ones are n=4 and m=2. Hope this helps.


Shriniwas Chavan

Sir I am using zernike moment for handwritten character recognition, I have applied zernike moment in terms of geometric moment, but it is upto 3 and 4 order, i want to extend the code can your files can extend it or extract zernike moment


Hi Amir
thanks for replay
I know what you say
now almost I have identification for some plamprints
I used high zernike order
and better database
also i maked unit disk for pic
I will make some addition and see what happen
Thanks you

Amir Tahmasbi

Hi Reyadh,

In principle, the Zernike moments can be extracted from any shape. However, in your case, there could be a variety of potential issues. First of all, you need to make sure the objects (palm prints) are of equal size within the ROI. If not, you need to equalize the size of your objects in all ROIs.

The other thing is that the palm prints are more complicated object than simple shapes, e.g. oval and circle. Thus, you might need to extract a set of higher order Zernike moments. You can then use these moments to classify the palm prints of different people more reliably.

FYI, the magnitude of the Zernike moments of two similar shapes might be slightly vary due to the pixelation and noise. But the difference should be insignificant.




hello Tamanna
I used your zernike to find zernike moment for palmprint
but there are some problem
when use for that A=0,Theta=0
I make some changed to the code
I replaced p = logical(not(p));
to p=im2double(p);
and it is work after that with palmprint and your pics nad values equal with yours
but the problem the values does not equal for same person palmprint
any help?

Amir Tahmasbi

Hi Tareq

Thanks! The input arguments "m" and "n" are scalars. This means that if you would like to extract say 32 moments, you need to have a "for loop" in which you change "n" and "m", and call the Zernikemoment(p, n, m). Depending on your application you can either change "n" from y to 32+y or use different combinations of "m" and "n". For more information, see Table 1 in page 731 of this article:

Hope this helps.


Tareq Al-Zubaidi

nice job,,, one question, if we say 32 moments so i expect when I call [ZOH AOH PhiOH] = Zernikmoment(p,n,m);

the ZOH to be an array of 32 ? or what 32 moments mean here ?

Amir Tahmasbi

Hi Tamanna,

The Zernike moments are rotation-invariant, no question on it! So, if you use the sample pictures included in the package, you will see this feature.

The reason that you are getting different results for the abs of Zernike moments is explained as follows. The MATLAB function "imrotate" does not preserve the size of an object in the ROI. Please note that the ROI size will be the same but the original image will be shrank in the new ROI. Thus, you are changing the object size that alters the abs of the Zernike moments. Hope this helps.



hey..can u please help me to identify that how this code is invariant to rotation..

because if i apply 'imrotate' with 30 or 45 degree on an image, then the result is different.

Amir Tahmasbi

Hi Anandh,

Yes, that is correct. By changing 'n' you can change the order of the Zernike moments to generate a set of say 32 features. However, you should keep in mind that also the variable 'm' (i.e. the repetition of moments) plays an important rule in the behavior of the moments. Hence, 'm' should be selected adaptively by changing 'n'. To find a suitable repetition for your proposed order, please see:

Hope this helps! Let me know if you have any other concerns.



Hi Amir,

Am new to this Zernike Moments. I tried to understand your code. I have few doubts. How to find 32 order features for a given image? Is it by varying the number 'n'? Please reply.

Cobi Biton

Amir Tahmasbi

Thanks guys! Chris: I refer you to one of our papers in which we normalize the ROIs before extracting the Zernike moments (i.e. we remove the dependency of Zernike moments on the translation and scaling of object in the preprocessing step). Here it is:

Another way is to use the Zernike moment invariants explained nicely here:


Good job!And, how is the zernike moment invariants

Fatemeh Saki

Nice job!! Thank you so much.

MATLAB Release Compatibility
Created with R2010b
Compatible with any release
Platform Compatibility
Windows macOS Linux

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!