+ +

Using Artificial Intelligence and Novel Polynomials to Predict Subjective Refraction. Rampat R, Debellemanière G, Malet J, Gatinel D. Sci Rep 10, 8565 (2020)

Here we present a synopsis of our study, where machine learning using wavefront data processed with a new polynomial decomposition basis is leading to accurate and precise prediction of spectacle correction.  

The paper is available here: https://www.nature.com/articles/s41598-020-65417-y

Optical wavefront sensing using aberrometry is an objective method that allows mathematical reconstruction and analysis of lower and higher-order (degree) monochromatic aberrations of the eye. We believe that this objective method had the potential to be the new standard for optimizing the correction of refractive errors by converting aberrometry data to accurate sphero-cylindrical refractions.

The output of a typical wavefront analysis is a list of coefficients, each weighing a specific aberration mode. It is expected that the taxonomy of the higher-order aberrations is clinically relevant and equipped with interesting mathematical properties.

When a wavefront decomposition is performed, each mode is given a coefficient value. The higher the value of the coefficient, the higher the « weight » or effect of the considered mode on the total wavefront error. Wavefront reconstruction over a circular pupil can be achieved by adding some Zernike modes, each weighted by a coefficient whose value is specific to the wavefront of interest.

Various metrics can be derived from the wavefront coefficients analysis:

– The low degree component of the wavefront should correspond to or be highly correlated with the spectacle correction of the measured eye.

– The higher degree component should predict the residual optical defects that persist after correcting that eye with the best spectacles’ correction.


In general, there is a difference between the subjective refraction and the objective refraction predicted by the analysis of the wavefront from the value of the coefficients weighting the low-degree spherical and cylindrical defocus modes. Part of this discrepancy may be related to the influence of high degree aberrations on the subjective perception of the effect of a given low degree correction. The study of the interactions between the best subjective refraction and high degree aberrations is interesting for more than one reason: beyond the prediction of subjective refraction, it also aims to improve the design of personalized refractive corrections and adjust the component low degree refractive with respect to the possible presence of high degree aberrations. Current analyzes are hampered by the fact that the Zernike classification is not suitable for this situation, since certain so-called high-degree aberration modes contain terms of low degree.

Machine learning is already in use in Ophthalmology for image analysis in medical retina and glaucoma as well as regression tasks, notably in IOL calculations.

Our aim was to build and evaluate a set of predictive machine learning models to accurately and precisely objectively refract a patient using wavefront aberrometry with a new aberration decomposition and to evaluate the relative importance of each aberration in the prediction process for each vector.

The accurate distinction between lower and higher wavefront components is mandatory to accurately predict spectacle refraction (as well as accurate retinal image metrics, which was not investigated in this study).

However, Zernike polynomials have one major drawback for describing the ocular wavefront in an ophthalmic context: these polynomials do not accurately distinguish between Lower and Higher Order Aberrations. Lower order terms are present within the expression of some Higher Order coefficients so we can’t accurately predict sphero-cylindrical refraction or accurately define the effects Higher-Order Aberrations on the retinal image viewed.

Thus, the influence of the Zernike spherical aberration Z(4,0) on the spherical cylindrical refraction is essentially linked to the low degree component (defocus) contained in this mode. These characteristics do not allow us to isolate well the specific influence of the high degree component in r4.

The Zernike spherical aberration mode contains not only a high degree term in r^4 but a low degree term in r^2 which is equivalent to defocus. Each micron of Zernike spherical aberration Z40 contains nearly 4 microns of defocus of opposite sign. If an ocular wavefront is exclusively a r ^4 phase aberration, its reconstruction in Zernike polynomials should automatically include an artifactual defocus component (low degree) to compensate for the defocus necessarily introduced by the Zernike Z(4,0) mode. This mixture does not allow us to simply predict the subjective refraction and study the specific influence of the high degree term in r^4 from a Zernike expansion.


The same goes for high-degree astigmatism Z(4,+/-2), which contains low-degree astigmatism:

Comparison of the modes aimed at modeling the low and higher degree astigmatism wavefront error. Note the tormented aspect of the higher-order Zernike modes (bottom left) which is caused by the presence of lower-order astigmatism within their analytical expression
Better reconstruction of low order astigmatism would be expected if higher-order astigmatism was devoid of low order astigmatism (bottom right). In eyes with high order astigmatism, this would also allow us to better estimate the specific influence of pure high-order astigmatism on the subjective low-order astigmatism correction.


We have developed a solution to these problems based on new polynomial functions to better distinguish between low degree (LD) and high degree (HD) aberrations: the LD/HD modes expansion. The core of this work has recently been published in the Journal of Optical Society of America (JOSA A), and some relevant clinical examples in the Journal of Refractive Surgery.

We have created new orthogonal and Normalized modes G(n,m) to replace the high order Zernike « impure » modes. Their analytical expression no longer contains low order terms (2 or less), and their central portion is flat.

The LD/HD aberration classification proposes a clear cut between low and high-order (degree) aberrations. The higher-degree modes are “paraxially flat” ( cross-sections, left). There is no orthogonality between the LD and HD spaces.
However, orthogonality is maintained between the modes comprised within the low degree and the high degree space. The rupture of orthogonality echoes the clinical approach, which «decouples » the low (spectacle-corrected) and high-order aberrations. It is possible to calculate RMS coefficients of grouped modes within the low and within the high-degree modes. The total RMS cannot be calculated directly from the low degree RMS and high degree RMS values using Pythagorean calculations.

A wavefront error can be decomposed in the LD/HD classification as a sum of weighted G(n,m) modes. The g(n,m) coefficients can be directly computed from the Zernike coefficients z(n,m) without the need for a new wavefront acquisition.

The low order modes G(2,0), G(2-2), and G(2,2) are orthogonal and their respective coefficients can be easily converted in the 3D dioptric vector space components of the refraction M, J0, and J45.

In the LD/HD decomposition, no low degree term is present in the high degree modes. All the terms corresponding to 2nd degree spherical and cylindrical defocus are gathered in the low degree component (LD). The low order coefficients of the 2nd radial degree mode of the LD/HD classification can be converted in the vector components of the refraction (r is the radius of the pupil). This comprehensive 2nd degree component is equivalent to an osculating quadric to the wavefront and can serve to compute an objective “paraxial refraction”. However, this objective refraction, although close to the subjective refraction, often differs slightly from this one on an individual basis.

The remaining high degree component is devoid of low degree terms e.g. defocus and 2nd-degree astigmatism. This allows unequivocal analysis of the influence of the high-order aberrations on the subjective spectacle low order correction.

A total of 2890 electronic medical records of patients (6397 eyes) evaluated for refractive surgery at the Laser Vision Institute Noémie de Rothschild (Foundation Adolphe de Rothschild Hospital, Paris) were retrieved and analyzed. For each eye, the following data was available:

Non-cycloplegic subjective refractions: Each refraction in Sphere S, Cylinder C, and axis A format was transformed into 3D dioptric vector space (M, J0, J45) where the three components are orthogonal.

Wavefront analysis: obtained using the OPD-Scan III (Nidek, Gamagori, Japan) aberrometer specially configured to run using beta-software incorporating the new series of LD/HD polynomials up to the 6th order.

A machine learning approach using LD/HD polynomials was developed for predicting the results of conventional, sphero-cylindrical refraction from wavefront aberrations.

Diagram summarizing the study protocol: the machine learning algorithm was trained to predict subjective refraction components using low +/- high degree G(n,m) wavefront coefficients from the LD/HD expansion.

XGBoost is an implementation of gradient boosted trees focused on performance and computer efficiency. It can perform both regression and classification tasks. It was chosen because of its recognized performances and its resistance to overfitting.

Feature importance analysis was achieved via the calculation of SHAP (SHapley Additive exPlanations) values for each model in order to determine the most influential polynomials.

Read this paper to learn how the best precision and accuracy were obtained, when all the novel polynomials coefficients were used as predictors, demonstrating the significant influence higher-order aberrations have on spectacle correction.


I wish to thank the co-authors who have contributed to this work: Guillaume Debellemanière (MD), Radhika Rampat (MD), and my Ph.D. directors: Laurent Dumas (Ph.D.), and Jacques Malet (Ph.D.).

I also wish to thank the Nidek company and its R&D team for implementing this new method in a beta-software, now able to process wavefront data obtained with our OPD-Scan III wavefront sensor.


More pages on this website about wavefront sensing and analysis:

Wavefront sensing explained

Zernike polynomials

Novel wavefront decomposition method

Novel wavefront decomposition method: clinical analysis

Aberrations in keratoconus

Pre and post LASIK aberrations

Post Presby LASIK aberrations

14 réponses à “Using Artificial Intelligence and Novel Polynomials to Predict Subjective Refraction. Rampat R, Debellemanière G, Malet J, Gatinel D. Sci Rep 10, 8565 (2020)”

  1. Recovery help dit :


    “Unfortunately, The first mistake was getting your funds stolen (crypto scam, trading/Fx investment scam,romance scam etc.)”
    Let The most tactical funds recovery team, Assetsretrievalboard. com save you from making the second mistake. Do not seek help from any of these fake recovery emails (scammers) on this very platform or elsewhere , it will land you in more loss.
    Sending any message to emails here that claim to be hackers.?
    There is more to being a “Real Hacker” (or recovery agent) than just using the words “hack and recovery” to create an email to lure victims to contact them.
    📝1: You need a certified and effective team to help recover your misappropriated funds.
    📝2: you need at-least, a 15years hacking or recovery experienced and oriented dialectical team to enable that your funds recovery is a success, and therefore we at assetsretrievalboard. com is the team to render this service, we work so hard to enable that, the right hacker (recovery agent) is assigned to you after reviews/investigations have been carried out and confirmed that you are a victim of stolen funds. For more informations and detailed proof of works(testimonials)
    Visit our website now and let’s have your funds retrieved.
    assetsretrievalboard. com

    assetsretrievalboard. com
    ©️ 2022 signed.

  2. daniella nelson dit :

    Get your ex lover back no matter why he left you.
    My boyfriend that left me a few months ago just came back to me last night crying for me to take him back. After a 3 year relationship with my boyfriend, he changed suddenly and stopped contacting me regularly, he would come up with excuses of not seeing me all the time. He stopped answering my calls and my sms and he stopped seeing me regularly. I then started seeing him with different girls several times but every time he would say that he loved me and that he needed some time to think about our relationship. But I couldn’t stop thinking about him so I decided to go online and I saw so many good talks about this spell caster called Dr Kadiri and I contacted him and explained my problems to him. He cast a love spell for me. and After 11hours, my boyfriend came back to me and started begging me to forgive him. We moved in together after a few months and he was more open to me than before and he started spending more time with me than his friends. We eventually got married and now have been married happily with a son. Ever since Dr Kadiri helped me, my boyfriend is very stable, faithful and closer to me than before, You can also contact this spell caster and get your relationship fixed .Here’s his contact, WhatsApp him: +2348102430456 or Email him at: drkadirilovespellhome@gmail.com

  3. OLIVIA EMMA dit :

    reetings to you all out there. I was scammed by a fake online crypto investor, I got a message from a trader on my Facebook who told me about how high their profit was after investing with them, I decided to invest in the online trade because I wanted to make more profit to set up a business because my job wasn’t enough to take care of myself and my family, invested $35,000.00 worth of bitcoin. I was promised that I will be able to withdraw my profit after one week of investment. when it was time for me to withdraw the profit. I tried to make a withdrawal and it failed. I informed the trader about it and he asked me to pay more money as the withdrawal charges. was a bit strange to me. I wasn’t informed earlier that I have to pay extra money to be able to withdraw my winning profit. I did as he instructed and made another deposit of $5,000.00, thinking I will be able to finally withdraw my profit. but he kept asking for more money until I realized that I was being scammed. I couldn’t let go because I invested all the money that I was saving to set up a new business. I was confused and lost for days because the world was against me. I had to do research to see if there is anything I can do to recover my money back and I came across the comment about MICHAEL CALCE WIZARD after so many searches. I read several good reviews about them from various scam victims and how MICHAEL CALCE WIZARD helped them in recovering their stolen crypto. I decided to contact them for help. I was opportune to get my money back from the scam brokers through the help of MICHAEL CALCE WIZARD, Are you a victim of any kind of scam? MICHAEL CALCE WIZARD is capable of helping you get back whatever you may have lost to internet Scammers. Reach them through email address at: michaelcalcewizard78@gmail.com or WhatsApp Number: +447375542502

  4. Maria Turk dit :

    I rarely write reviews, but I had to for this! Because I tried several apps to help with no luck. Back then, I was very confused and always felt awful about my partner’s cheating attitude. I really wanted to track and catch him red-handed. I downloaded this app hoping it would help but I wasn’t satisfied with its features as I needed to monitor my partner closely. I then spoke with a trusted colleague of mine at work and she gave me a genuine recommendation about an ethical private investigator named Michael Calce Wizard, I wrote to his email at michaelcalcewizard78@gmail.com explaining how I wanted to have complete access to his call log, iMessage/SMS, social media activities, gallery, texts, deleted or not, his precise location day in day out. I paid for his services and in about 2-3 hours, I had complete access to his phone. I got concrete evidence. it was unbelievable to see the evidence of cheating on me. Who likes to live with a cheater, I used the evidence I got to file for a divorce. I’m thankful to Michael Calce Wizard for helping me out of my dilemma and suspicions. His services are highly rated and affordable. If you are having trust issues and need valid evidence, how about you contact michaelcalcewizard78@gmail.com. Thank you Michael Calce Wizard for an incredible job !! I highly recommend his services, simply the best.
    Contact : michaelcalcewizard78@gmail.com
    WhatsApp Number : +447375542502

  5. Cynthia Murk dit :

    Hello everyone, my name is Cynthia Murk and I’m going to share with you all how I got scammed after falling victim to a cryptocurrency investment scam, I was introduced to this cryptocurrency scam by a guy I met on Instagram and after investing all my life savings, they swindled me off $960,000. It was the saddest moment of my life because I couldn’t believe what had happened, I was bankrupt and my credit was damaged. I reported to the authorities and there was nothing they could do to get these scammers or return my money. I thought all hope was lost and was going into depression until I came across an article online about a hacker who could help me recover my funds, at first I thought it was impossible to do that but I realize I had no other choice but to give it a try, to my surprise MICHAEL CALCE WIZARD HACKING GROUP was able to recover all my funds and upgrade my credit score, I wanted to put this out there for anyone who might be going through the same situation and have been swindled by this scammers, you can contact MICHAEL CALCE WIZARD HACKING GROUP to help you recover your stolen funds. Contact them at (michaelcalcewizard78@gmail.com) and on WhatsApp at +44 (737) 554-2502

  6. DatsameBoi dit :

    My advice to everyone out there, please do deep research before investing in these online cryptocurrency trading platforms. My brother introduced me to one such platform called BITbyBIT TRADING where he had invested $300,000 which would generate $500,000 in 2 weeks. After much convincing from my brother, I saw it as a lucrative opportunity and so I jumped on it and invested my $100,000 to earn $105,000 within the same period. I started noticing red flags when they asked my bro to pay for such charges to facilitate the withdrawal process which was never successful. The same happened to me when my 2 weeks was due, By the time we realized it was a scam, they had already robbed us a total of $850,000. We opted to hire an investigator and found MICHAEL CALCE WIZARD SOFTWARE HACKING GROUP who took our case, and in three days, they had tracked the scammers and refunded our money. Contact them on michaelcalcewizard78@gmail.com or send them a message on WhatsApp via: +44 (737) 554-2502

  7. Winner Rita dit :

    Using this opportunity I would love to extend my gratitude to Mr. Michael Calce Wizard whom I contacted to report my issues. I consulted Michael Calce Wizard when I was involved in a bitcoin trading scam. It was unknown to me that there were systematic ways to retrieve scammed funds. I followed all the directives Mr. Michael Calce Wizard gave to me and it was like a dream when I recovered all my money. You can reach out to him if you had the same experience as well on: michaelcalcewizard78@gmail.com
    / WhatsAPP +447375542502

  8. Nelson Star dit :

    I fully recommend you to contact him MICHAEL CALCE WIZARD SOFTWARE HACKING GROUP ★ ★, he will help you recover all data and bitcoin , funds recovering if you have lost a phone he will help in tracking the phone till it will be found, Spy on mobile , mobile cloning , Facebook hack , Instagram hack , any type of social media hack , tracer etc. contact him (michaelcalcewizard78@gmail.com)) WhatsApp : ( +447375542502 )

  9. Clement George dit :

    Hello Everyone, please contact MICHAEL CALCE WIZARD SOFTWARE HACKING GROUP for the following services:





    ★ You can also contact them for other Cyber Attacks And Hijackings, they do all Hacking Services

    ★You can contact them via:

    Email: michaelcalcewizard78@gmail.com

    via WhatsApp: +447375542502

  10. Alicia Bradley dit :

    I was so impatient to carry out the necessary research, but I really wanted to jump on the crypto trading and investment buzz. Unfortunately for me, I invested 90,700 USD worth of bitcoin with a fraudulent company. I was happy to watch my account grow to 340,575 USD within a couple of weeks. But I didn’t realize I was dealing with a scam company, until I tried to make an attempt to withdraw. I made a withdrawal request, and noticed that my account was suddenly blocked for no apparent reason. I tried contacting customer support, but all to no avail. I needed my money back at all costs, because I could not afford to let it go. So I tried all possible means to make sure I recovered my scammed bitcoin. I did a lot of online search for help, and tried to see if there were other people who had any similar experience. I stumbled upon a cryptocurrency forum were a couple of people mentioned that they had been through the same process but were able to recover their lost cryptocurrency, funds with the help of MICHAEL CALCE WIZARD SOFTWARE HACKING GROUP . So I filed a report and he was able to help me get back all my lost funds within two hours. I feel indebted to him. Apart from trying to express my gratitude to them once again using this medium, I will recommend anyone who wants to recover scammed bitcoin, stolen cryptocurrency, funds lost to binary options forex, investment and any other form of online scam, reach out to michaelcalcewizard78@gmail.com or via WhatsApp:+447375542502

  11. Sandra Williams dit :

    This is for everyone seeking recovery of stolen crypto coins, recovery of any lost funds to online scams , recovery of cryptocurrency lost to reapers in forex trades and binary options
    This is for persons seeking to recover all of their lost funds to online scams, contact address–> michaelcalcewizard78@gmail.com for successful recovery and thank me later. I had my blockchain wallet spoofed by merciless scams, due to lack of 2FA authorization, they were able to get away with 4.0147BTC from my wallet , this made me very sad and depressed as i was desperately in need of help , i made my research online and came across a very credible and reliable recovery agent on the internet with this information michaelcalcewizard78@gmail.com. The hack agency helped me recover all I lost and also revealed the identity of the perpetrators , that’s why I’m most appreciative and also sharing their contacts info for anyone in a similar situation seeking to recover funds lost to online scams , contact the recovery specialist at:


    Business whatsapp:+447375542502

  12. Gina Rineheart dit :

    Good News From Martins Blank ATM Cards And Bitcoin Investments:( martinshackers22@gmail.com )
    Have you been trying to get a real blank ATM Card or bitcoin and it has been a problem trying to get one? Here is Martins Blank ATM Card easy and affordable to get and it can be delivered to you waiting 24hrs after you have made your order from me at: martinshackers22@gmail.com

  13. Manuel Franco dit :

    I just want to say Thank You to everyone who supported me through the years. My name is Manuel Franco, New Berlin, Wisconsin. My story of how I won the Powerball lottery of $768.4M is a bit of a tale. I have been playing Powerball tickets for 6 years now since I turned 18. I bought my first ticket on my 18 birthday. I was feeling very lucky that day because I had contacted Dr. Michael to help me with the winning Powerball numbers. I really had that great great feeling that I looked at the camera wanting to wink at it. I only did a tiny part of it and trusted him. He gave me the numbers after I played a couple other tickets along with it for $10. I checked my ticket after the winnings came online and saw the numbers were correct including the Power play. I screamed for about 10 minutes because it felt like a dream. I had won $768.4M. You can check my winning testimony with the lottery officials just with my name search. Thank you Dr Michael. Well, his email is odungaspelltemple@gmail.com and you can also call or Whats-app him at +2348167159012 so you guys can contact him

  14. Wagner Clerk dit :

    Tous mes remerciements à M. Anderson Carl pour m’avoir aidé avec mes bénéfices et rendu possible mon cinquième retrait. Je suis ici pour partager avec vous une incroyable opportunité qui change la vie. c’est ce qu’on appelle les options de trading Bitcoin / Forex. c’est une entreprise très lucrative qui peut vous rapporter jusqu’à 2 570 $ en une semaine à partir d’un investissement initial de seulement 200 $. Je suis la preuve vivante de cette formidable opportunité commerciale. Si quelqu’un est intéressé par le trading sur bitcoin ou toute autre crypto-monnaie et souhaite un échange réussi sans perdre, informez-en M. Anderson Carl maintenant sur Whatsapp : +1(252)285-2093 Email : andersoncarlassettrade@gmail.com

Laisser un commentaire

Vous pouvez poser des questions ou commenter ce contenu : pour cela, utilisez le formulaire "commentaires" situé ci-dessous. Seront traitées et publiées les questions et commentaires qui revêtent un intérêt général, et éclairent ou complètent les informations délivrées sur les pages concernées.

Votre adresse e-mail ne sera pas publiée. Les champs obligatoires sont indiqués avec *