[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86161-en":3,"doc-seo-86161-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86161,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","An Edge-Based Formulation for the Exact Computation of High-Order Zernike Moments of 2D Shapes and Images","Zernike moments serve as rotation-invariant descriptors for 2D shape and image analysis, yet conventional computation relies on pixel-centered quadrature that creates spatial aliasing growing with moment order. This work introduces an edge-based formulation that applies Green’s theorem to convert the 2D area integral defining a Zernike moment into boundary sums of one-dimensional integrals. The method supports polygonal shapes and various image types, derives recurrence relations for radial primitives, and enables exact polynomial evaluation via Clenshaw–Curtis quadrature.","An Edge-Based Formulation for the Exact Computation of High-Order Zernike Moments of  \n2D Shapes and Images  \nPatrice Koehl and Stephan Tillmann  \narXiv :2607 . 11158v1 [math .NA] 13 Jul 2026  \nAbstract—Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green’s theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw–Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixelbased approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.  \nIndex Terms—2D Zernike moments, Raster images, shape contours.  \nI. INTRODUCTION  \nMoment-based shape descriptors have played a central role in image analysis and pattern recognition for more than six decades. The idea of using algebraic moments of image intensity functions was pioneered by Hu [1], who showed that a small set of moment invariants under rotation, translation, and scaling encodes powerful geometric information about a planar shape. Shortly thereafter, Teague [2] proposed replacing ordinary algebraic moments with projections onto a complete orthogonal basis defined over the unit disk—the Zernike polynomials originally introduced by Frits Zernike in the context of optical aberration theory [3] . The resulting Zernike moments inherit the classical desirable properties of classical moment invariants, including rotation invariance, information compactness, and robustness to noise, while adding a crucial practical advantage: the orthogonality of the basis guarantees  \nP. Koehl is with the Department of Computer Science and Genome Center, University of California, Davis, CA, 95616 .  \nE-mail: [koehl@cs.ucdavis.edu](koehl@cs.ucdavis.edu)  \nS. Tillmann is with the School of Mathematics and Statistics, The University of Sydney, Sydney, NSW, Australia, 2007 .  \nE-mail: [stephan.tillmann@sydney.edu.au](stephan.tillmann@sydney.edu.au)  \nthat individual moments can be computed and truncated independently, with no information leakage between coefficients.  \nThe Zernike polynomials Vmn(r,θ) = Rn|m| (r) eimθ are defined for non-negative integer order n and integer repetition m satisfying |m| ≤ n and n − |m| ≡ 0 (mod 2) . The radial part Rn|m| (r) is a polynomial of degree n in r that vanishes outside the unit disk [3], [4]:  \n(n−|m|)/2  \nRn|m| (r) = X (−1)kF (m, n, k)rn−2k , (1)  \nk=0  \nwhere  \nF (n, m, k) =  (n − k)!  . (2)  \nk! 􀀐 n+2|~~ ~~m~~ ~~| − k􀀑 ! 􀀐 n−2|~~ ~~m~~ ~~| − k􀀑 ! The Zernike moment of order (n, m) of a function f supported on (a subset of) the unit disk is  \nZmn = rn~~ ~~~~ ~~1 ZZD f (r,θ)(Vmn)∗ (r,θ)r dr dθ . (3)  \nSince Teague’s foundational work, Zernike moments have been applied across a broad spectrum of scientific and engineering disciplines. In optical engineering, they remain the standard language for describing wavefront aberrations an","cbCairNQ9tFnpUeZ","https://ap.wps.com/l/cbCairNQ9tFnpUeZ","pdf",3812711,4,1,17,"English","en",105,"# Abstract\n# Introduction\n## Moment-based shape descriptors\n## Zernike polynomials and moments\n## Applications and motivation\n# Proposed edge-based computation (high-level)","[{\"question\":\"What problem does the standard pixel-based computation of high-order Zernike moments have?\",\"answer\":\"It uses pixel-based quadrature treating each pixel as a point mass at its center, which introduces spatial aliasing that increases with moment order, degrading reconstruction and the discriminative power of high-order moments.\"},{\"question\":\"How does the proposed edge-based method compute Zernike moments more accurately?\",\"answer\":\"It applies Green’s theorem to transform the defining 2D area integral into a sum of one-dimensional integrals along image boundaries, eliminating the discretization error source inherent to pixel-based quadrature.\"},{\"question\":\"For which data types does the edge-based framework work?\",\"answer\":\"The formulation applies to polygonal shapes and to binary, grayscale, and color images, enabling exact computation without spatial discretization errors while remaining practical for high-order moments.\"}]",1784208993,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"an-edge-based-formulation-for-the-exact-computation-of-high-order-zernike-moments-of-2d-shapes-and-images","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/an-edge-based-formulation-for-the-exact-computation-of-high-order-zernike-moments-of-2d-shapes-and-images/86161/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the standard pixel-based computation of high-order Zernike moments have?","Question",{"text":75,"@type":76},"It uses pixel-based quadrature treating each pixel as a point mass at its center, which introduces spatial aliasing that increases with moment order, degrading reconstruction and the discriminative power of high-order moments.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed edge-based method compute Zernike moments more accurately?",{"text":80,"@type":76},"It applies Green’s theorem to transform the defining 2D area integral into a sum of one-dimensional integrals along image boundaries, eliminating the discretization error source inherent to pixel-based quadrature.",{"name":82,"@type":73,"acceptedAnswer":83},"For which data types does the edge-based framework work?",{"text":84,"@type":76},"The formulation applies to polygonal shapes and to binary, grayscale, and color images, enabling exact computation without spatial discretization 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