METHOD AND APPARATUS OPTIMIZING SPECTACLE LENSES FOR WEARERS OF IMPLANTED INTRAOCULAR LENSES

Abstract

Optimizing a spectacle lens for a wearer of implanted intraocular lenses. The method includes providing individual refraction data on the at least one eye of the spectacle wearer; defining an individual eye model in which at least a shape and/or power of a cornea, in particular a corneal front surface, of a model eye, a cornea-lens distance, parameters of the lens of the model eye, and a lens-retina distance are defined as parameters of the individual eye model. Here, defining the parameters of the individual eye model takes place on the basis of data on visual acuity correction of the at least one eye having the intraocular lens and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data.

Claims

1-22. (canceled)

23. A computer-implemented method for identifying relevant individual parameters of at least one eye of a spectacle wearer for the calculation or optimization of a spectacle lens for the at least one eye of the spectacle wearer, wherein an intraocular lens was implanted in the at least one eye of the spectacle wearer as part of surgery, comprising: providing individual refraction data on the at least one eye of the spectacle wearer; defining an individual eye model in which at least: a shape and/or power of a corneal front surface of a model eye; a cornea-lens distance; parameters of the lens of the model eye; and a lens-retina distance are defined as parameters of the individual eye model, wherein defining the parameters of the individual eye model takes place on the basis of data on visual acuity correction of the at least one eye having the intraocular lens and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data.

24. The computer-implemented method according to claim 23, wherein defining the parameters of the individual eye model takes place on the basis of intraocular lens data and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data, wherein defining the parameters of the lens of the model eye takes place on the basis of the intraocular lens data.

25. The computer-implemented method according to claim 23, wherein a lens-retina distance of the eye of the spectacle wearer is identified, and defining the parameters of the individual eye model takes place on the basis of the identified lens-retina distance and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data, wherein the lens-retina distance of the model eye is defined by the identified lens-retina distance of the eye of the spectacle wearer.

26. The computer-implemented method according to claim 23, wherein defining the lens-retina distance takes place by measuring and/or calculating.

27. The computer-implemented method according to claim 24, wherein the intraocular lens data comprises at least a defocus of the front surface of the intraocular lens, a defocus of the back surface of the intraocular lens, and a thickness of the intraocular lens; and/or wherein the intraocular lens data comprises at least a defocus of the refractive power of the intraocular lens, and/or wherein the intraocular lens data includes a specification of the A constant.

28. The computer-implemented method according to claim 24, wherein the intraocular lens data is provided on the basis of type or serial number information.

29. The computer-implemented method according to claim 1, further comprising: carrying out a consistency check of the defined eye model, and solving any inconsistencies with the aid of analytical and/or numerical and/or probabilistic methods.

30. The computer-implemented method according to claim 29, wherein any inconsistencies are solved by: adapting one or more parameters of the eye model, wherein several parameters of the eye model are adapted and the adaptation is divided among the several parameters of the eye model; and/or adding at least one new parameter to the eye model and defining it such that the eye model becomes consistent; and/or adapting a target power of the ophthalmic lens.

31. The computer-implemented method according to claim 23, wherein the parameters of the eye model are determined with the aid of probabilistic methods using Bayesian statistics and/or a maximum likelihood algorithm.

32. The computer-implemented method according to claim 23, further comprising: providing an initial distribution of parameters of the eye model and individual data on properties of the at least one eye; and determining the parameters of the individual eye model on the basis of the initial distribution of parameters of the eye model and the individual data using probability calculations.

33. The computer-implemented method according to claim 23, wherein the provided individual refraction data on the at least one eye of the spectacle wearer is individual post-surgery refraction data of the at least one eye of the spectacle wearer, and wherein the individual eye model is a post-surgery eye model.

34. The computer-implemented method according to claim 33, further comprising: providing individual pre-surgery refraction data of the at least one eye of the spectacle wearer, wherein determining a lens-retina distance of the eye of the spectacle wearer is based on an individual pre-surgery eye model using the provided individual pre-surgery refraction data.

35. The computer-implemented method according to claim 34, wherein in the pre-surgery eye model at least: a shape and/or power of a corneal front surface of a model eye of the pre-surgery eye model; a cornea-lens distance of the model eye of the pre-surgery eye model; parameters of the lens of the model eye of the pre-surgery eye model; and a lens-retina distance of the model eye of the pre-surgery eye model are defined on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual pre-surgery refraction data such that the model eye has the provided individual pre-surgery refraction data, wherein at least defining the lens-retina distance takes place by measuring and/or calculating.

36. A computer-implemented method for calculating or optimizing a spectacle lens for at least one eye of a spectacle wearer, comprising: a method for identifying relevant individual parameters of the at least one eye of the spectacle wearer according to claim 23; specifying a first surface and a second surface for the spectacle lens to be calculated or optimized; identifying the course of a main ray through at least one visual point of at least one surface of the spectacle lens to be calculated or optimized into the model eye; evaluating an aberration of a wavefront resulting from a spherical wavefront incident on the first surface of the spectacle lens along the main ray on an evaluation surface compared to a wavefront converging in one point on the retina of the eye model; and iteratively varying the at least one surface of the spectacle lens to be calculated or optimized until the evaluated aberration corresponds to a predetermined target aberration.

37. A method for producing a spectacle lens, comprising: calculating or optimizing a spectacle lens according to the method for calculating or optimizing a spectacle lens according to claim 36; and manufacturing the thus-calculated or optimized spectacle lens.

38. A device for identifying relevant individual parameters of at least one eye of a spectacle wearer for the calculation or optimization of a spectacle lens for the at least one eye of the spectacle wearer, the at least one eye of the spectacle wearer having an implanted intraocular lens, comprising: at least one data interface configured to provide individual refraction data on the at least one eye of the spectacle wearer; and a modeling module configure to define an individual eye model, which at least defines a shape and/or power of a corneal front surface of a model eye; a cornea-lens distance; parameters of the lens of the model eye; and a lens-retina distance as parameters of the individual eye model, wherein defining the parameters of the individual eye model takes place on the basis of data on visual acuity correction of the at least one eye having the intraocular lens and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data.

39. The device according to claim 38, wherein defining the parameters of the individual eye model takes place on the basis of intraocular lens data and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data, and wherein defining the parameters of the lens of the model eye takes place on the basis of the intraocular lens data.

40. The device according to claim 38, wherein a lens-retina distance of the eye of the spectacle wearer is identified by the modeling module and defining the parameters of the individual eye model takes place on the basis of the identified lens-retina distance and further on the basis of individual measurement values for the eye of the spectacle wearer and/or standard values and/or on the basis of the provided individual refraction data such that the model eye has the provided individual refraction data, and wherein the lens-retina distance of the model eye is defined by the identified lens-retina distance of the eye of the spectacle wearer.

41. A device for calculating or optimizing a spectacle lens for at least one eye of a spectacle wearer, comprising: a device configured to identify relevant individual parameters of the at least one eye of the spectacle wearer according to claim 38; a surface model database configured to specify a first surface and a second surface for the spectacle lens to be calculated or optimized; a main ray identification module configured to identify the course of a main ray through at least one visual point of at least one surface of the spectacle lens to be calculated or optimized into the model eye; an evaluation module configured to evaluate an aberration of a wavefront resulting from a spherical wavefront incident on the first surface of the spectacle lens along the main ray on an evaluation surface compared to a wavefront converging in one point on the retina of the eye model; and an optimization module configured to iteratively vary the at least one surface of the spectacle lens to be calculated or optimized until the evaluated aberration corresponds to a predetermined target aberration.

42. A device for producing an ophthalmic lens, comprising: a calculator or optimizer configured to calculate or optimize the spectacle lens according to a method for calculating or optimizing a spectacle lens according to claim 36; and a machining configured to machine the spectacle lens in accordance with the result of the calculation or optimization.

43. A non-transitory computer program product including program code configured to, when loaded and executed on a computer, perform a method for identifying relevant individual parameters of at least one eye of a spectacle wearer according to claim 23.

44. A non-transitory computer program product including program code configured to, when loaded and executed on a computer, perform a method for calculating or optimizing a spectacle lens according to claim 36.

45. A spectacle lens produced by a method according to claim 37.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0251] Preferred embodiments and examples of the invention will be explained below by way of example, at least in part with reference to the accompanying drawings, which show:

[0252] FIG. 1 a schematic representation of the physiological and physical model of a spectacle lens and an eye together with a beam path in a predetermined wearing position; and

[0253] FIG. 2 a graph with an exemplary dependency of P.sup.mess on S.sub.IOL and L.sub.2,IOL to illustrate and explain a method for determining parameters under constraint conditions according to a preferred embodiment of the present invention.

DETAILED DESCRIPTION

[0254] FIG. 1 shows a schematic representation of the physiological and physical model of a spectacle lens and an eye in a predetermined wearing position together with an exemplary beam path on which an individual spectacle lens calculation or optimization according to a preferred embodiment of the invention is based.

[0255] Here, only a single ray is preferably calculated for each visual point of the spectacle lens (the main ray 10, which preferably runs through the ocular center of rotation Z), but also the derivatives of the vertex depths of the wavefront according to the transverse coordinates (perpendicular to the main ray). These derivatives are taken into account up to the desired order, the second derivatives describing the local curvature properties of the wavefront and the higher derivatives being related to the higher-order aberrations.

[0256] Upon light tracing through the spectacle lens up to the eye 12 according to the individually provided eye model, the local derivatives of the wavefronts are ultimately identified at a suitable position in the beam path in order to compare them there with a reference wavefront that converges in a point on the retina of the eye 12. In particular, the two wavefronts (i.e. the wavefront coming from the spectacle lens and the reference wavefront) are compared with one another in an evaluation surface.

[0257] “Position” does not simply mean a certain value of the z-coordinate (in the direction of light), but such a coordinate value in combination with the specification of all surfaces through which refraction took place before reaching the evaluation surface. In a preferred embodiment, refraction takes place through all refractive surfaces including the lens back surface. In this case, the reference wavefront is preferably a spherical wavefront whose center of curvature is located on the retina of the eye 12.

[0258] It is particularly preferred not to propagate any further after this last refraction, so that the radius of curvature of this reference wavefront does correspond to the distance between the lens back surface and the retina. In a further preferred embodiment, propagation is still carried out after the last refraction, preferably up to the exit pupil AP of eye 12. This is, for example, at a distance d.sub.AR=d.sub.LR.sup.(b)=d.sub.LR−d.sub.LR.sup.(a)>d.sub.LR in front of the retina and thus even in front of the lens back surface, so that the propagation in this case is a back propagation (the designations d.sub.LR.sup.(a), d.sub.LR.sup.(b) will be described below in the listing of steps 1-6). In this case, too, the reference wavefront is spherical with the center of curvature on the retina, but has a radius of curvature 1/d.sub.AR.

[0259] To this end, it is assumed that a spherical wavefront w.sub.0 emanates from the object point and propagates up to the first spectacle lens surface 14. There it is refracted and then it propagates to the second lens surface 16, where it is refracted again. The wavefront w.sub.g1 exiting the spectacle lens then propagates along the main ray in the direction of the eye 12 (propagated wavefront w.sub.g2) until it hits the cornea 18, where it is refracted again (wavefront w.sub.c). After further propagation within the anterior chamber of the eye up to the eye lens 20, the wavefront is also refracted again by the eye lens 20, whereby the resulting wavefront w.sub.e arises on the back surface of the eye lens 20 or on the exit pupil of the eye, for example. This is compared with the spherical reference wavefront w.sub.s and the deviations are evaluated for all visual points in the target function (preferably with corresponding weightings for the individual visual points).

[0260] Thus, the vision disorder is no longer described only by a thin sphero-cylindrical lens, as was customary in many conventional methods, but rather the corneal topography, the eye lens, the distances in the eye, and the deformation of the wavefront (including the low-order aberrations—i.e. sphere, cylinder and cylinder axis—as well as preferably including the higher-order aberrations) are directly taken into account in the eye. Here, the vitreous body length d.sub.LR is calculated individually in the eye model according to the invention.

[0261] An aberrometer measurement preferably provides the individual wavefront deformations of the real eye having the visual defect for distance and near (deviations, no absolute refractive indices) and the individual mesopic and photopic pupil diameters. From a measurement of the corneal topography (surface measurement of the corneal front surface), an individual real corneal front surface is preferably obtained, which generally makes up almost 75% of the total refractive power of the eye. In a preferred embodiment, it is not necessary to measure the corneal back surface. It is preferably described by an adaptation of the refractive index of the cornea and not by a separate refractive surface due to the small refractive index difference compared to the aqueous humor and because of the small cornea thickness in a good approximation.

[0262] In general, in this description, bold lowercase letters are intended to denote vectors and bold capital letters are intended to denote matrices, such as the (2×2) vergence matrices or refractive index matrices

[00002]$\begin{array}{cccc}S=\left(\begin{array}{cc}{S}_{\mathrm{xx}}& S_{\mathrm{xy}}\\ S_{\mathrm{xy}}& S_{\mathrm{yy}}\end{array}\right),& C\left(\begin{array}{cc}{C}_{\mathrm{xx}}& C_{\mathrm{xy}}\\ C_{\mathrm{xy}}& C_{\mathrm{yy}}\end{array}\right),& L=\left(\begin{array}{cc}{L}_{\mathrm{xx}}& L_{\mathrm{xy}}\\ L_{\mathrm{xy}}& L_{\mathrm{yy}}\end{array}\right),& 1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\end{array}$

and italics like d are intended to denote scalar quantities.

[0263] Furthermore, bold, italicized capital letters are intended to denote wavefronts or surfaces as a whole. For example, S is the vergence matrix of the wavefront S of the same name, except that S besides the 2nd order aberrations summarized in S, also comprises the entirety of all higher-order aberrations (HOA) of the wavefront. From a mathematical point of view, S stands for the set of all parameters necessary to describe a wavefront (with sufficient accuracy) in relation to a given coordinate system. Preferably, S stands for a set of Zernike coefficients with a pupil radius or a set of coefficients of a Taylor series. Particularly preferably, S stands for the set of a vergence matrix S for describing the wavefront properties of the 2nd order and a set of Zernike coefficients (with a pupil radius), which is used to describe all remaining wavefront properties except the 2nd order, or a set of coefficients according to a Taylor decomposition. Analogous statements apply to surfaces instead of wavefronts.

[0264] Among other things, the following data can be measured directly in principle: [0265] The wavefront S.sub.M, which is generated by the laser spot on the retina and the passage through the eye (from aberrometric measurement)— [0266] Shape of the corneal front surface C (through corneal topography)— [0267] Distance between cornea and lens front surface d.sub.CL (by pachymetry). This variable can also be determined indirectly by measuring the distance between the cornea and the iris; correction values can be applied if necessary here. Such corrections can be the distance between the lens front surface and the iris from known eye models (e.g. literature values). [0268] Curvature of the lens front surface in a direction L.sub.1xx (by pachymetry) In this case, without restricting generality, the x-plane can be defined exemplarily such that this section lies in the x-plane. If the coordinate system is defined such that this plane is inclined, the derivative must be supplemented by the functions of the corresponding angle. It is not required that this be a main section. For so example, it can be the section in the horizontal plane.

[0269] Furthermore, the following data—depending on the embodiment—can either be measured or taken from the literature: [0270] Thickness of the lens d.sub.L [0271] Curvature of the lens back surface in the same direction as the lens front surface L.sub.2,xx (by pachymetry)

[0272] Thus, there are the following options for the lens back surface: [0273] Measurement of L.sub.2,xx (L.sub.2,M) and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M and L.sub.2,xy=L.sub.2,yx=0 [0274] Taking L.sub.2,xx from the literature (L.sub.2,Lit) and assumption of a rotational symmetry L.sub.2,xx=L.sub.2,yy=L.sub.2=L.sub.2,M und L.sub.2,xy=L.sub.2,yx=0 [0275] Taking the complete (asymmetrical) form L.sub.2 from the literature (L.sub.2,Lit) [0276] Measurement of L.sub.2,xx (L.sub.2,M) and assumption of a cylinder or an otherwise specified asymmetry a.sub.Lit from the literature L.sub.2,xx=L.sub.2,M und L.sub.2,xy=L.sub.2,yx=f(L.sub.2,xx,a.sub.Lit) and L.sub.2,yy=g(L.sub.2,xx,a.sub.Lit)

[0277] The following data can be found in the literature: [0278] Refractive indices n.sub.CL of the cornea and anterior chamber of the eye as well as the aqueous humor n.sub.LR and that of the lens n.sub.L

[0279] This leaves in particular the distance d.sub.LR between the lens back surface and the retina and the components L.sub.1,yy and L.sub.1,xy=L.sub.1,yx of the lens front surface as unknown parameters. To simplify the formalism, the former can also be written as a vergence matrix D.sub.LR=D.sub.LR.Math.1 with D.sub.LR=n.sub.LR/d.sub.LR. Furthermore, the variable τ is generally used, which is defined as τ=d/n (whereby for the refractive index, always the corresponding index must be used as n, as for d and τ, e.g. as τ.sub.LR=d.sub.LR/n.sub.LR, τ.sub.CL=d.sub.CL/n.sub.CL).

[0280] The modeling of the passage of the wavefront through the eye model used according to the invention, i.e. after passage through the surfaces of the spectacle lens, can be described as follows in a preferred embodiment in which the lens is described via a front and a back surface, with the transformations the vergence matrices is explicitly being specified: [0281] 1. Refraction of the wavefront S with the vergence matrix S on the cornea C with the surface power matrix C to the wavefront S′.sub.C with vergence matrix S′.sub.C=S+C [0282] 2. Propagation around the anterior chamber depth d.sub.CL (distance between cornea and lens front surface) to wavefront S.sub.L1 with vergence matrix S.sub.L1=S′.sub.C/(1−τ.sub.CL.Math.S′)

[00003]$S_{L1}=\frac{S_C^{\prime }}{\left(1-{\tau }_{\mathrm{CL}}.Math.S_C^{\prime }\right)}$ [0283] 3. Refraction on the lens front surface L.sub.1 with the surface power matrix L.sub.1 to the wavefront S′L.sub.1 with the vergence matrix S′.sub.L1=S.sub.L1+L.sub.1 [0284] 4. Propagation around the lens thickness d.sub.L to the wavefront S.sub.L2 with vergence matrix S.sub.L2=S′.sub.L1/(1−τ.sub.L.Math.S′.sub.L1) [0285] 5. Refraction on the lens back surface L.sub.2 with the surface power matrix L.sub.2 to the wavefront S′.sub.L2 with vergence matrix S′.sub.L2=S.sub.L2−L.sub.2 [0286] 6. Propagation around the distance between lens and retina d.sub.LR to wavefront S.sub.R with the vergence matrix SS.sub.R=S′.sub.L2/(1−τ.sub.LR.Math.S′.sub.L2)

[0287] Each of the steps 2, 4, 6, in which the distances τ.sub.CL, τ.sub.CL, and τ.sub.CL are propagated, can be divided into two partial propagations 2a, b), 4a, b) or 6a, b) according to the following scheme, which explicitly reads for step 6a, b): [0288] 6a. Propagation around the distance d.sub.LR.sup.(a) between the lens and the intermediate plane to the wavefront S.sub.LR with the vergence matrix S.sub.LR=S′.sub.L2/(1−τ.sub.LR.sup.(a)S′.sub.L2) [0289] 6b. Propagation around the distance d.sub.LR.sup.(b) between the intermediate plane and the retina to the wavefront S.sub.R with the vergence matrix S.sub.R=S.sub.LR/(1−τ.sub.LR.sup.(b)S.sub.LR)

[0290] Here, τ.sub.LR.sup.(a)=d.sub.LR.sup.(a)/n.sub.LR.sup.(a) and τ.sub.LR.sup.(b)=d.sub.LR.sup.(b)/n.sub.LR.sup.(b) can be positive or negative, whereby it should always hold that n.sub.LR.sup.(a)=n.sub.LR.sup.(b)=n.sub.LR and τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b)=τ.sub.LR. In any case, steps 6a and 6b can be combined again by S.sub.R=S′.sub.L2/(1−(τ.sub.LR.sup.(a)+τ.sub.LR.sup.(b))S′.sub.L2)=S′.sub.L2/(1−τ.sub.LRS′.sub.L2). The division in steps 6a and 6b offers advantages, however, and the intermediate plane can preferably be placed in the plane of the exit pupil AP, which preferably is located in front of the lens back surface. In this case τ.sub.LR.sup.(a)<0 and τ.sub.LR.sup.(b)>0.

[0291] The division of steps 2, 4 can be analogous to the division of step 6 in 6a, b).

[0292] Decisive for the choice of the evaluation surface of the wavefront is not only the absolute position in relation to the z-coordinate (in the direction of light), but also the number of surfaces through which refraction took place up to the evaluation surface. Thus, one and the same level can be passed several times. For example, the plane of the AP (which normally is located between the lens front surface and the lens back surface) is formally passed by the light for the first time after an imaginary step 4a, in which propagation takes place from the lens front surface by the length τ.sub.L.sup.(a)>0. The same plane is reached for the second time after step 6a when, after the refraction through the lens back surface, propagation back to the AP plane takes place, i.e. τ.sub.LR.sup.(a)=−τ.sub.L+τ.sub.L.sup.(a)=−τ.sub.L.sup.(b)<0, which is synonymous with τ.sub.LR.sup.(a)=τ.sub.LR.sup.(b)<0. In the case of the wavefronts S.sub.AP that refer to the AP in the text, the wavefront S.sub.AP=S.sub.LR that is the result of step 6a should preferably always be meant (unless explicitly stated otherwise).

[0293] Reference will be made to these steps 1 to 6 again in the further course of the description. They describe a preferred relationship between the vergence matrix S of a wavefront S on the cornea and the vergence matrices of all intermediate wavefronts resulting therefrom at the refractive intermediate surfaces of the eye, in particular the vergence matrix S′.sub.L2 of a wavefront S′.sub.L2 after the eye lens (or even a wavefront S.sub.R on the retina). These relationships can be used both to calculate parameters that are not known a priori (e.g. d.sub.LR or L.sub.1) and thus to assign values to the model either individually or generically, and to simulate the propagation of the wavefront in the eye for optimizing spectacle lenses with the models that underwent assignment.

[0294] In a preferred embodiment, the surfaces and wavefronts are treated up to the second order, for which a representation by means of vergence matrices is sufficient. Another preferred embodiment described later takes into account and also uses higher orders of aberrations.

[0295] In a description in the second order, the eye model, in a preferred embodiment, has twelve parameters as degrees of freedom of the model that have to undergo assignment. These preferably include the three degrees of freedom of the surface power matrix C of the cornea C, the three degrees of freedom of the surface power matrices L.sub.1 and L.sub.2 for the front and back surfaces of the lens, as well as respectively one for the length parameters anterior chamber depth d.sub.CL, lens thickness d.sub.L, and the vitreous body length d.sub.LR.

[0296] In principle, these parameters can undergo assignment in several ways: [0297] i. Directly, i.e. individual measurement of a parameter [0298] ii. Value of a parameter given a priori, e.g. as a literature value or from an estimate, for example due to the presence of a measurement value for another variable, which correlates in a known manner with the parameter to be determined on the basis of a previous population analysis [0299] iii. Calculation from consistency conditions, e.g. compatibility with a known refraction

[0300] The total number df.sub.2 of degrees of freedom of the eye model in the second order (df stands for ‘degree of freedom’, index ‘2’ for second order) is thus composed of:

df.sub.2=df.sub.2(i)+df.sub.2(ii)+df.sub.2(iii)

[0301] For example, if there are direct measurement values for all twelve model parameters, then df.sub.2(i)=12, df.sub.2(ii)=0 and df.sub.2(iii)=0, which will be expressed by the notation df.sub.2=12+0+0 in the following for the sake of simplicity. In such a case, the objective refraction of the relevant eye is also defined, so that an objective refraction determination no longer has to be carried out in addition.

[0302] A central aspect of the invention relates precisely to the airm of not having to measure all parameters directly. In particular, it is significantly easier to measure the refraction of the relevant eye or to determine it objectively and/or subjectively than to measure all parameters of the model eye individually. Preferably, there is thus at least one refraction, i.e. measurement data for the wavefront S.sub.M of the eye up to the second order, which corresponds to the data on the vergence matrix S.sub.M. If the eye model undergoes assignment on the basis of purely objectively measurement data, these values can be taken from aberrometric measurements or autorefractometric measurements or, according to (ii), be assigned with other given data. Consideration of subjective methods (i.e. subjective refraction), be it as a replacement for the objective measurement of the refraction or by combining both results, will be described later. The three conditions of conformance with the three independent parameters of the vergence matrix S.sub.M thus make it possible to derive three parameters of the eye model, which corresponds to df.sub.2(iii)=3 in the notation introduced above.

[0303] The invention therefore makes use of the possibility, in cases in which not all model parameters are accessible to direct measurements or in which these measurements would be very complex, to assign the missing parameters in a useful way. For example, if direct measurement values (df.sub.2(i)≤9) are available for a maximum of nine model parameters, then the refraction conditions mentioned can be used to calculate three of the model parameters (df.sub.2(iii)=3). If exactly df.sub.2(i)=9, then all twelve model parameters are uniquely determined by the measurements and the calculation, and it holds that (df.sub.2(ii)=0). If, on the other hand, df.sub.2(i)<9, then df.sub.2(ii)=9−df.sub.2(i)>0, i.e. the model is underdetermined in the sense that df.sub.2(ii) parameters must be defined a priori.

[0304] With the provision of an individual refraction, i.e. measurement data on the wavefront S.sub.M of the eye, in particular up to the second order, the necessary data on the vergence matrix S.sub.M are correspondingly available. According to a conventional method described in WO°2013/104548°A1, in particular the parameters {C, d.sub.CL, S.sub.M} are measured. In contrast, the two length parameters d.sub.L and d.sub.LR (or D.sub.LR) are conventionally defined a priori (e.g. by literature values or estimates). In WO°2013/104548°A1, a distinction is in particular made between the two cases in which either L.sub.2 is defined a priori and L.sub.1 is calculated therefrom, or vice versa. The aforementioned laid-open publication discloses Eq. (4) and Eq. (5) as a calculation rule. For both cases, it holds that df.sub.2=4+5+3.

[0305] In the terminology used in steps 1 to 6 above, L.sub.1 is adapted to the measurements in particular by calculating the measured vergence matrix S.sub.M by means of steps 1, 2 through the likewise measured matrix C and propagating it up to the object-side side of the lens front surface. On the other hand, a spherical wave is calculated back to front from an imaginary point-like light source on the retina by means of steps 6, 5, 4 carried out backward by refracting this spherical wave at the previously defined surface power matrix L.sub.2 of the lens back surface and propagating the then-obtained wavefront from the lens back surface up to the image-side side of the lens front surface. The difference of the thus-determined vergence matrices S.sub.L1 and S′L.sub.1, which are located on the object side or on the image side of the lens front surface, must have been caused by the matrix L.sub.1, since in the aberrometric measurement the measured wavefront emerges from a wavefront that emanates from a point pm the retina, and therefore, due to the reversibility of the beam paths, is identical to the incident wavefront (S=S.sub.M) that converges on this point of the retina. This leads to equation (4) in the laid-open publication mentioned:

[00004]$\begin{array}{cc}{L}_1\left(D_{\mathrm{LR}}\right)=\frac{D_{\mathrm{LR}}.Math.1-L_2}{1+{\tau }_L.Math.\left(D_{\mathrm{LR}}.Math.1-L_2\right)}-\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}& \left(1a\right)\end{array}$

[0306] The other case in the cited laid-open publication relates to the adaptation of the matrix L.sub.2 to the measurements after the matrix L.sub.1 has been defined. The only difference now is that the measured wavefront S.sub.M is subjected to steps 1, 2, 3, 4 and the assumed wavefront from the point-like light source only to step 6, and the missing step to be carried out to adapt the lens back surface L.sub.2 is now step 5, according to equation (5) of the above-mentioned laid-open publication:

[00005]$\begin{array}{cc}{L}_2=D_{\mathrm{LR}}-\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right){\left(1-{\tau }_L\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right)\right)}^{-1}& \left(1b\right)\end{array}$

[0307] The central idea of the invention is to calculate at least the length parameter d.sub.LR (or D.sub.LR) from other measurement data and a priori assumptions about other degrees of freedom and not to assume it a priori itself, as is conventional. In the context of the present invention, it turned out that this brought about a remarkable improvement in the individual adaptation with comparatively little effort since the wavefront tracing turned out to be very sensitive to this length parameter. This means that it is an advantage according to the invention if at least the length parameter d.sub.LR belongs to the df.sub.2(iii)=3 parameters that are calculated. In particular, it is difficult to measure this parameter directly, it varies more strongly between different test subjects, and these variations have a comparatively strong influence on the imaging of the eye.

[0308] The data on the vergence matrix S.sub.M and particularly preferably also the data on C are preferably available from individual measurements. In a further preferred aspect, which is preferably also taken into account in the following embodiments, in the assumption of data on the lens back surface, a spherical back surface, i.e. a back surface without astigmatic components is taken as a basis.

[0309] In a preferred embodiment of the invention, measurement data up to the second order are available for the cornea C, which corresponds to the data on the surface power matrix C. Although these values can be taken from topographical measurements, the latter are not necessary. Rather, topometric measurements are sufficient. This situation corresponds to the case df.sub.2=3+6+3, with the anterior chamber depth d.sub.CL in particular being one of the six parameters to be defined a priori.

[0310] If no further individual measurements are made, the situation is df.sub.2=3+6+3. In order to be able to uniquely determine d.sub.LR, six parameters from {L.sub.1, L.sub.2, d.sub.L, d.sub.CL} have to undergo assignment by assumptions or literature values. The other two result in addition to d.sub.LR from the calculation. In a preferred embodiment, the parameters of the lens back surface, the mean curvature of the lens front surface, and the two length parameters d.sub.L and d.sub.CL undergo assignment a priori (as predetermined standard values).

[0311] In a case that is particularly important for the invention, the anterior chamber depth d.sub.CL, i.e. the distance between the cornea and the lens front surface, is also known e.g. from pachymetric or OCT measurements. The measured parameters thus include {C, d.sub.CL, S.sub.M}. This situation corresponds to the case df.sub.2=4+5+3. The problem is therefore still underdetermined mathematically, so five parameters from {L.sub.1, L.sub.2, d.sub.L} have to be defomed a priori through assumptions or literature values. In a preferred embodiment, these are the parameters of the lens back surface, the mean curvature of the lens front surface, and the lens thickness.

[0312] For the accuracy of the individual adaptation alone, it is advantageous to be able to assign individual measurements to as many parameters as possible. In a preferred embodiment, the lens curvature is additionally provided in a normal section on the basis of an individual measurement. This then results in a situation according to df.sub.2=5+4+3, and it is sufficient to define four parameters from {L.sub.yy, α.sub.L1, L.sub.2, d.sub.L} a priori. Here, too, in a preferred embodiment, the parameters of the lens back surface and the lens thickness are involved. The exact calculation will be described below.

[0313] In particular, as an alternative to the normal section of the lens front surface and particularly preferably in addition to the anterior chamber depth, the lens thickness can also be made available from an individual measurement. This eliminates the need to assign model data or estimated parameters to this parameter (df.sub.2=5+4+3).

[0314] Otherwise, what has already been said above applies. This embodiment is particularly advantageous if a pachymeter is used, the measuring depth of which allows the lens back surface to be recognized, but not a sufficiently reliable determination of the lens curvatures.

[0315] In addition to the anterior chamber depth and a normal section of the lens front surface, one (e.g. measurement in two normal sections) or two further parameters (measurement of both main sections and the axial position) of the lens front surface can be obtained by an individual measurement. This additional information can be exploited in two ways in particular: [0316] Abandonment of a priori assumptions: One or two of the assumptions otherwise made a priori can be abandoned and determined by calculation. In this case the situations df.sub.2=6+3+3 and df.sub.2=7+2+3 arise. In the first case, the mean curvature of the back surface (assuming an astigmatism-free back surface) and in the second case the surface astigmatism (including cylinder axis) with given mean curvature can be determined. Alternatively, the lens thickness can also be determined from the measurements in both cases. [0317] However, such a procedure generally requires a certain degree of caution, since noisy measurement data can easily lead to the released parameters “running away”. As a result, the model can become significantly worse instead of better overall. One possibility to prevent this is to specify anatomically sensible limit values for these parameters and to limit the variation of the parameters to this range. Of course, these limits can also be specified as a function of the measurement values. [0318] Reduction of the measurement uncertainty: If, on the other hand, the same a priori assumptions are made (preferably {L.sub.2, d.sub.L}), the situations df.sub.2=6+4+3 and df.sub.2=7+4+3 exist, the system is therefore mathematically overdetermined. [0319] Instead of a simple analytical determination of D.sub.LR according to the following explanations, D.sub.LR (and possibly the missing parameter from L.sub.1) is determined (“fit”) such that the distance between L.sub.1 resulting from the equations and the measured L.sub.1 (or L.sub.1 supplemented by the missing parameter) becomes minimal. By this procedure—obviously—a reduction in the measurement uncertainty can be achieved.

[0320] In a further preferred implementation, the anterior chamber depth, two or three parameters of the lens front surface, and the lens thickness are measured individually. The other variables are calculated in the same way, whereby the a priori assumption of the lens thickness can be replaced by the corresponding measurement.

[0321] In a further preferred implementation, individual measurements of the anterior chamber depth, at least one parameter of the lens front surface, the lens thickness, and at least one parameter of the lens back surface are provided. This is an addition to the cases mentioned above. The respective additionally measured parameters can be carried out analogously to the step-by-step expansions of the above sections. These cases are particularly advantageous if the above-mentioned pachymetry units, which measure in one plane, two planes or over the entire surface, are correspondingly expanded in the measuring depth and are so precise that the curvature data can be identified with sufficient accuracy.

[0322] If the formula (1b) already mentioned above is solved for D.sub.LR in order to calculate the lens back surface for a given eye length (where D.sub.LR=n.sub.LR/d.sub.LR is the inverse vitreous body length d.sub.LR multiplied by the refractive index n.sub.LR), namely

[00006]$L_2=D_{\mathrm{LR}}-\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right){\left(1-{\tau }_L\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_1\right)\right)}^{-1}$

one obtains

[00007]$D_{\mathrm{LR}}=L_2+\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_2\right)\times {\left(1-{\tau }_L\left(\frac{S_M+C}{1-{\tau }_{\mathrm{CL}}\left(S_M+C\right)}+L_2\right)\right)}^{-1}$

for calculating D.sub.LR. Since D.sub.LR is a scalar, all quantities for the calculation must also be taken as scalar. Preferably, S.sub.M, C, L.sub.1 and L.sub.2 are each the spherical equivalents of the vision disorder, the cornea, the lens front surface, and the lens back surface, respectively. Once the vitreous body length d.sub.LR has been calculated (and thus the eye length as d.sub.A=d.sub.C+d.sub.CL+d.sub.L+d.sub.LR), one of the surfaces can be modified again with regard to the cylinder and the HOA, preferably the lens front surface L1, in order to adapt it consistently.

[0323] For the calculation, the values of the so-called Bennett & Rabbetts eye for the refractive powers of the lens surfaces can be used, which can be taken from Table 12.1 of the book “Bennett & Rabbets' Clinical Visual Optics”, third edition, by Ronald B. Rabbetts, Butterworth-Heinemann, 1998, ISBN-10: 0750618175, for example. The calculation described above leads to results that are very compatible with the population statistics, which state that short-sighted vision disorders tend to lead to large eye lengths and vice versa (see e.g. C. W. Oyster: “The Human Eye”, 1998). The calculation described is, however, even more precise, since a direct use of the correlation from the population statistics can lead to unphysical values for the eye lens, which is avoided by the method according to the invention. Precise knowledge of the lens parameters is all the more important for a method in which these are assumed to be given. For example, if a customer had a vision disorder of −10 dpt before his cataract surgery, he must have an eye length between 28 mm and 30 mm. After surgery, however, due to his emmetropia, an eye length of 24 mm would be concluded, which does not match the actual eye length.

[0324] Preferred embodiments of the invention will be explained below.

Examples Using Bayesian Statistics

[0325] The aim of the method using Bayesian statistics is to use, if possible, all available information sources about an eye or a pair of eyes in a consistent manner in order to achieve an optimal correction of the eye or the eyes with an ophthalmic lens (e.g. a spectacle lens) in the light of this information.

[0326] As a rule, this information is incomplete and/or imprecise, which so far has often led to the fact that only simplified eye models are used to calculate ophthalmic lenses. Such a simplified eye model is e.g. an eye that is characterized solely by its refraction, since this can be determined with a certain accuracy (e.g. with an error of ±0.75 dpt in the spherical equivalent). However, if one wishes to use more complex eye models to calculate ophthalmic lenses, it makes sense to include information about the length of the eye, as well as the position and curvature of the refractive surfaces of the cornea and eye lens in the calculation, but this should only be taken into account as much as is possible within the scope of its accuracy.

[0327] In Bayesian statistics (see e.g. D. S. Sivia: “Data Analysis—A Bayesian Tutorial”, Oxford University Press, 2006, ISBN-13: 978-0198568322 or ET Jaynes: “Probability Theory”, Cambridge University Press, 2003, ISBN-13: 978-0521592710), information is always described in the form of probability distributions (in the case of continuous parameters it is probability densities).

[0328] In this sense, a probability or probability density can be assigned to an individual eye model with a given set of parameters. Individual eye models that are consistent with the available information (e.g. objective wavefront measurement and biometrics of the eye) have a higher probability or probability density, because e.g. the propagation and refraction of a wavefront that a point light source would generate on the retina after exiting the eye reproduces the measured data well within the scope of the measurement accuracy of the objective wavefront measurement, and likewise the parameters of the individual eye match with the available information about the biometry of the eye within the scope of the distributions known e.g. from the literature. Individual eye models that are not consistent with the available information are correspondingly assigned to low probabilities or probability densities. The probability or probability density of an individual eye model can be written as

prob(ϑ.sub.i|d.sub.i,I)

where ϑ.sub.i denotes the parameters of the individual eye model i, and d.sub.i are the measured data (it can e.g. include the current refraction or the refraction measured prior to eye surgery, the measured shape and/or refractive properties of the cornea, the measured eye length or other variables measured on the individual eye). With I, the current state of knowledge upon evaluation of the data, i.e. the existing background information (e.g. about the measurement process of the refraction, the distribution of the parameters of the individual eye model or other related variables in the population) is summarized. The vertical line ‘|’ means that the distribution of the variables to the left of ‘|’ is meant for given (i.e. fixed) variables to the right of ‘|’.

[0329] The information obtained in the measurement process, in which the data d.sub.i is measured, can also be understood as the probability distribution of the data d.sub.i with given parameters ϑ.sub.i of the individual eye model i:

prob(d.sub.i|ϑ.sub.i,I)

[0330] The accuracy of the measurement process is reflected in the width of the distribution:

[0331] an exact measurement has a narrower distribution than an imprecise measurement, which has a wide or wider distribution of the data d.sub.i.

[0332] Now, if one wishes to calculate the distribution of the parameters of an individual eye with given data and background information, the following proportionality can be used:

prob(ϑ.sub.i|d.sub.i,I)∝prob(ϑ.sub.i|I)prob(d.sub.i|ϑ.sub.i,I)

[0333] The term prob(ϑ.sub.i|I) describes the background knowledge about the parameters of the individual eye model. This can be information from literature, for example, but also information from data from past measurements. This can be data from the same person for whom the ophthalmic lens is to be manufactured, as well as data from measurements made for a large number of other people.

[0334] The probability here serves as a measure of consistency. Parameter values of the individual eye model that are consistent with the measurements can be found in particular where both prob(ϑ.sub.i|I) and prob(d.sub.i|d.sub.i,I) are high.

[0335] The probability or probability density prob(ϑ.sub.i|d.sub.i,I) can also be suitably normalized in order to write the proportionality as an equation.

[0336] The term prob(ϑ.sub.i|d.sub.i,I) can also include parameters of the eye lens. For example, some of the parameters ϑ.sub.i can include the refractive power of the eye lens, its position and/or orientation in the eye, or other variables such as the refractive index and curvatures or shape of the surfaces.

[0337] The eye lens can be a natural lens. In this case, literature data on the parameters of natural eye lenses can be used (e.g. distributions of the curvatures of the front and/or back surface, refractive index, etc.).

[0338] If the eye lens is an intraocular lens, the distributions of the parameters of natural eye lenses must not be used. Instead, the parameters of the intraocular lens should be used, provided they are individually known. Otherwise, distributions of these parameters can be used from literature studies of eyes that underwent surgery. If such information is not available, a flat distribution within reasonable limits can be selected. For parameters that are positively definite and define length scales (e.g. radii of curvature or distances), it is also possible to select distributions that are flat in the logarithm of these parameters.

[0339] Formally, the cases “natural eye lens” or “intraocular lens as eye lens” are to be described by different states of background knowledge I (i.e. I=I.sub.NLbzw.Math.I=I.sub.IOL).

[0340] The probability or probability density prob prob(ϑ.sub.i|d.sub.i,I) can have one or more factors. Here, ach factor represents the information about one or more parameters of the individual eye model. For example, the distribution of different independent parameters ϑ.sub.i.sup.1 and ϑ.sub.i.sup.2 from different literature sources can be represented as a product

prob(ϑ.sub.i|d.sub.i,I)=prob(ϑ.sub.i.sup.1,ϑ.sub.i.sup.2|d.sub.i,I)=prob(ϑ.sub.i.sup.1|d.sub.i,I)prob(ϑ.sub.i.sup.2|d.sub.i,I).

[0341] By prob(ϑ.sub.i|I), parameters of the eye model can inadvertently be falsified. For example, if the “true” refraction is understood as a parameter of the individual eye model, the most likely value of the “true” refraction can deviate from the measured refraction. If this is not desired, a distribution that is constant in the corresponding parameter (e.g. spherical equivalent of refraction) within carefully selected limits (e.g. between −30 dpt and +20 dpt for the spherical equivalent M, ±5 dpt for the astigmatic components Jo and J.sub.45) should be chosen.

[0342] If parameters of the individual eye model or other variables related to the parameters or measurement data is known exactly or with a high degree of accuracy, their distribution can be approximated as a Dirac delta distribution. The equations in these parameters or variables can be integrated on both sides, which may simplify subsequent calculations.

Description of the Methods

[0343] Two possible methods for calculating an ophthalmic lens will be presented below (Bayes A and Bayes B). In the Bayes A method, the available information is used to set up a (single) individual eye model, with the help of which an ophthalmic lens optimal for this eye model is calculated. The eye model can e.g. be given by or assigned the set of parameters ϑ.sub.i.sup.max, which maximizes the probability or probability density prob(ϑ.sub.i|d.sub.i,I). Other sets of parameters can also be selected, e.g. the expected value ϑ.sub.i or the median ϑ.sub.i.sup.med of the parameters ϑ.sub.i with regard to the distribution prob(ϑ.sub.i|d.sub.i,I).

[0344] The Bayes B method is more advantageous—but computationally more demanding—compared to Bayes A, since a subset of individual eye models with different sets of parameters can lead to ophthalmic lenses that have very similar (even identical) properties (e.g. refractive power in a reference point of the ophthalmic lens, or the distribution of the refractive deficit across a given area of the ophthalmic lens, or similar criteria for determining the quality of an ophthalmic lens). Overall, an ophthalmic lens that was not calculated with the most likely individual eye model can therefore represent an optimal correction for a subset of individual eye models which overall have a higher probability than the most likely individual eye model. It is therefore advantageous to search for the ophthalmic lens that optimally corrects the distribution of eye models, instead of just determining the most likely individual eye model and manufacturing an ophthalmic lens for it.

[0345] In both methods, an ophthalmic lens (e.g. a spectacle lens) consistent with the information available can be calculated.

Bayes A Method

[0346] In particular, one or more of the following steps can be carried out:— [0347] Providing an initial distribution of parameters of an eye model (ideally as a multivariate probability distribution of all parameters of the eye model, possibly also marginal distributions; the probability distribution corresponds to the information about the distribution of the parameters of the eye model in the population of the persons); [0348] Providing already known (in the best case measured) data on properties of an individual eye (ideally with probability distribution or measurement errors; the probability distribution corresponds to the inaccuracy of the measurements) The already known data can include: already known current subjective and/or objective refraction, already known previous subjective and/or objective refraction (e.g. prior to surgery), power and/or shape and/or position (most important is the axial position) of certain refractive surfaces of the eye, size and/or shape and/or position of the entrance pupil, refractive index of the refractive media, refractive index profile in the refractive media, opacity; possibly determination of these variables depending on the accommodation of the eye on a fixation object (target) at a given close distance; [0349] Determining the parameters of an individual eye model based on the initial distribution of the parameters of the eye model and the already known or measured data on the individual eye using probability calculation. Ideally, the probability distribution or, for example, the set of parameters characterizing a maximum of the probability distribution is determined. [0350] In particular, calculation methods such as Markov Chain Monte Carlo, Variational Inference, Maximum Likelihood, Maximum Posterior, or Particle Filter can be used; [0351] The aim here is to select the parameters of the individual eye model that are consistent both with the initial distribution of eye models provided and with the data that is already known. The product of the probability or probability density of the data with given parameters of the eye model with the probability or probability density of the parameters of the eye model is used as the consistency measure. [0352] Calculating/optimizing/selecting an ophthalmic lens in which at least one parameter of the individual eye model is used.

[0353] The initial distribution of the parameters of eye models provided in the first step can be in a parameterized form, e.g. (possibly multivariate) normal distribution, other distribution of the exponential family, Cauchy distribution, Dirichlet process, etc., or as a set of samples, i.e. one or more (possibly multidimensional) data sets. If the initial distribution of the parameters of eye models is parameterized, the parameters of this distribution are called “hyperparameters”.

[0354] The third step (i.e. determining the parameters of an individual eye model) can include determining a multivariate probability distribution that includes both the parameters of the individual eye model and the hyperparameters of the initial distribution of the parameters of the eye model. In order to calculate the distribution of the parameters of the individual eye model from this, the distribution must be marginalized, i.e. it is integrated via the hyperparameters. The integrals can be solved with common numerical methods (e.g. using Markov Chain Monte Carlo or Hybrid Monte Carlo) and/or analytical methods. The probability or the probability density of the parameters of the eye model can in this case be calculated using the following equation:

prob(ϑ.sub.i|d.sub.i,I)∝prob(d.sub.i|ϑ.sub.i,I)∫dλprob(ϑ.sub.i,λ|I)=prob(d.sub.i|ϑ.sub.i,I)∫dλprob(ϑ.sub.i|λ,I)prob(λ|I)

[0355] Here prob(ϑ.sub.i|λ,I) denotes the probability or probability density to find the parameters of the individual eye model ϑ.sub.i in the population characterized by the hyperparameters λ. The integrals are to be carried out over the entire definition ranges of all hyperparameters λ.

Bayes B Method

[0356] As an alternative or in addition to the Bayes A method, one or more of the following steps can be carried out: [0357] Providing the distribution of at least one parameter of an individual eye model; [0358] Calculating the probability distribution of the parameters of virtual ophthalmic lenses or calculating an ensemble of ophthalmic lenses by optimizing/calculating/selecting virtual ophthalmic lenses using at least one parameter of the individual eye model; [0359] Manufacturing an ophthalmic lens, with the aim that the manufactured parameters of the ophthalmic lens achieve the parameters of the virtual ophthalmic lens with the highest probability.

[0360] In the first step, the distribution calculated analogously to steps 1 to 3 of the Bayes A method can be provided. In the second step, the most likely parameters L.sub.i of the ophthalmic lens are determined, i.e. on the basis of the probability distribution or probability density

[00008]$\begin{array}{c}\mathrm{prob}\left(L_i.Math.d_i,I\right)=\int d{\vartheta }_i\mathrm{prob}\left({\vartheta }_i,L_i.Math.d_i,I\right)=\\ \int d{\vartheta }_{i}prob\left(L_i.Math.{\vartheta }_i,I\right)\mathrm{prob}\left({\vartheta }_i.Math.d_i,I\right)\\ =\int d{\vartheta }_i\delta \left(L_i-L\left({\vartheta }_i\right)\right)pr\mathrm{ob}\left({\vartheta }_i.Math.d_i,I\right)\end{array}$

the parameters of the ophthalmic lens L.sub.i.sup.max are determined, which maximize prob(L.sub.i|d.sub.1,I). Here, L.sub.i initially denotes the parameters of any ophthalmic lens, and in the case of L.sub.i=L(ϑ.sub.i) the parameters of the ophthalmic lens created when an ophthalmic lens is optimized with the aid of an individual eye model with the parameters ϑ.sub.i. The Dirac delta distribution is denoted by δ(.).

[0361] The parameters of the ophthalmic lens can be e.g. vertex depth, refractive power at a reference point of the ophthalmic lens, refractive power distribution over an area of the ophthalmic lens, refractive errors at a reference point of the ophthalmic lens, or the distribution of the refractive errors over an area of the ophthalmic lens.

[0362] It is important that the function L(ϑ.sub.1) can be non-linear, and therefore the maximum of the probability density prob(ϑ.sub.i|d.sub.i,I) (with respect to ϑ.sub.i) with L(ϑ.sub.i) is not necessarily mapped to the maximum of the probability density prob(L.sub.i|d.sub.i,I).

[0363] If the function L(ϑ.sub.i) can be inverted piece by piece, the equation described above can also be solved with the help of partial integration. Other methods are also possible, e.g. numerical methods such as Particle Filter, Markov Chain Monte Carlo, or methods of parametric inference, with which a distribution of the parameters of the ophthalmic lens L.sub.i can be calculated.

[0364] Both in the Bayes A method and in the Bayes B method, independent of both the number and type of variables known by measurement (i.e. the data d.sub.i and the form of the likelihood prob(d.sub.1|ϑ.sub.i,I)) as well as the number and type of parameters of the eye model ϑ.sub.i, there always results a consistent eye model (Bayes A and B method) and possibly a choice of the parameters of the ophthalmic lens (Bayes B method) that matches the ensemble of possible consistent eye models.

Examples Based on Probability Considerations to Solve Inconsistencies

Background to the Maximum Likelihood Approach

Basic Procedure

[0365] The initial situation is that N parameters x.sub.i, 1≤i≤N of a model are to undergo assignment and that the following information is available: [0366] Mean values μ.sub.i, standard deviations σ.sub.i, and correlation coefficients ρ.sup.ij (with 1≤i, j≤N) of these N parameters in the population; [0367] Either there are no measurement values (k=0), or there are measurement values x.sub.i.sup.mess, 1≤i≤k for k of these parameters (where 1≤k≤N). The probability distribution for the measurement value x.sub.i.sup.mess of each parameter x.sub.i is described by a random variable X.sub.i. A reliability measure is preferably available for each measurement value, e.g. a standard deviation σ.sub.i.sup.mess of the random variable X.sub.i, 1≤i≤k [0368] For q=N−k of these parameters there are no measurement values. [0369] Overall, only K of the N parameters are independent since the model requires consistency conditions that can be expressed by Q=N−K constraints.

[0370] Examples can be: [0371] Example without HOA [0372] Parameters (N=15): cornea (SZA), lens front surface (SZA), lens back surface (SZA), vision disorder (SZA), eye length, lens thickness, anterior chamber depth; [0373] Measurement data (k=13): cornea (SZA), lens front surface (SZA), lens back surface (SZA), vision disorder (SZA), anterior chamber depth; [0374] Constraints (Q=3): vision disorder (SZA)=theoretical vision disorder (SZA) (calculated from the eye model assigned); [0375] Example with HOA (up to radial order n=6) [0376] Parameters (N=103): cornea (SZA+HOA), lens front surface (SZA+HOA), lens back surface (SZA+HOA), vision disorder (SZA+HOA), eye length, lens thickness, anterior chamber depth; [0377] Measurement data (k=101): cornea (SZA+HOA), lens front surface (SZA+HOA), lens back surface (SZA+HOA), vision disorder (SZA+HOA), anterior chamber depth; [0378] Constraints (Q=25): vision disorder (SZA+HOA)=theoretical vision disorder (SZA+HOA) (calculated from the eye model assigned).

[0379] The basic problem to be solved is that in the case of measurement values that deviate from the population mean, a decision must be made as to whether the measurement must be discarded (e.g. if it is implausible) or must be adopted. If all measurement values are plausible in themselves, but violate one of the consistency conditions, then they must not all be adopted. Instead, a balance between the various measurement values must then be sought: those that have a very high measurement reliability should at least almost be retained, while uncertain measurement values are more likely to be adapted. Preferably, the best possible values for all N parameters are identified from the known information.

[0380] The inventive idea is based in particular on the assumption that the parameters have certain (unknown but initially fixed) values. Under this assumption, in the light of the above-mentioned information (statistical variables from the population, reliability measures of the measurements), the conditional probability density

P.sup.bed(X.sub.1, . . . ,X.sub.N|x.sub.1, . . . ,x.sub.N)  (1)

is established for the outcome of the measurements, where X.sub.1, . . . , X.sub.N are the random variables that vary for fixed given true values x.sub.1, . . . , x.sub.N. Subsequently, the probability for the observed measurement values P.sup.par is then quantified by evaluating the function P.sup.bed for the k measurement values and marginalizing it for the remaining q=N−k (non-measured) parameters:

P.sup.par(x.sub.1, . . . x.sub.N):=∫P.sup.bed(X.sub.1=x.sub.1.sup.mess, . . . ,X.sub.k=x.sub.k.sup.mess,X.sub.k+1, . . . ,X.sub.N|x.sub.1, . . . ,x.sub.N)dX.sub.k+1. . . dX.sub.N  (2)

[0381] This probability density is understood as a function P.sup.par(x.sub.1, . . . , X.sub.N) of the assumed N parameters x.sub.1, . . . , x.sub.N. Those N parameter values for which this function assumes a maximum are then preferably considered to be the best possible values (maximum likelihood approach):

[00009]$\begin{array}{cc}\frac{\partial P^{par}\left(x_1,.Math.,x_N\right)}{\partial x_i}=0,1\le i\le N& \left(3\right)\end{array}$

[0382] As an alternative to the marginalization in Eq. (2), the N parameter values can also be defined by setting the last parameters equal to the mean values of the population,

x.sub.i=μ.sub.i, k+1≤i≤N  (4)

while the first k parameters x.sub.1, . . . , x.sub.k are determined such that their expected values are equal to the measurement values:

X.sub.i|x.sub.1, . . . ,x.sub.k,x.sub.k+1=μ.sub.k+1, . . . ,x.sub.N=μ.sub.N=x.sub.i.sup.mess  (5)

[0383] As a further alternative, instead of the maximum formation according to equation (3) or the expected value formation according to equation (5), the medians can be used as a criterion as well.

[0384] As a further alternative, the maximum formation according to equation (3) and the expected value formation according to equation (5) as well as the median determination can also be combined as desired in order to determine the N parameter values.

Background to the Maximum Posterior Approach

[0385] The prior knowledge about the population is described by the distribution P.sup.pop(x.sub.1, . . . , x.sub.N), which can correspond to the prior of the Bayesian description. The total probability density, which describes both the distribution of measurement values and of model parameters, is thus given by the distribution function

P.sup.ges(X.sub.1, . . . ,X.sub.k,x.sub.1, . . . ,x.sub.N)=P.sup.mess(X.sub.1, . . . ,X.sub.k|x.sub.1, . . . ,x.sub.N)×P.sup.pop(x.sub.1, . . . ,x.sub.N)  (6)

which can correspond to the posterior of the Bayesian description except for one constant. This is why this approach is also referred to as maximum posterior.

[0386] Preferably, P.sup.pop is described by the multivariate normal distribution

[00010]$\begin{array}{cc}{P}^{\mathrm{pop}}\left(x\right)=\sqrt{\frac{\det \left(C^{-1}\right)}{{\left(2\pi \right)}^N}}\mathrm{Exp}\left(-\frac{1}{2}{\left(x-\mu \right)}^T{C}^{-1}\left(x-\mu \right)\right)& \left(7\right)\end{array}$

where μ is the vector of the mean values and C is the covariance matrix:

[00011]$\begin{array}{cc}\mu =\left(\begin{array}{c}\begin{array}{c}\begin{array}{c}{\mu }_1\\ {\mu }_2\end{array}\\ .Math.\end{array}\\ {\mu }_n\end{array}\right),C=\left(\begin{array}{cccc}{\sigma }_1^2& {\rho }_{12}{\sigma }_1{\sigma }_2& .Math.& {\rho }_{1n}{\sigma }_1{\sigma }_n\\ {\rho }_{12}{\sigma }_1{\sigma }_2& {\sigma }_2^2& & {\rho }_{2n}{\sigma }_2{\sigma }_n\\ .Math.& & \ddots & .Math.\\ {\rho }_{1n}{\sigma }_1{\sigma }_n& {\rho }_{2n}{\sigma }_2{\sigma }_n& .Math.& {\sigma }_n^2\end{array}\right)& \left(7a\right)\end{array}$

[0387] The measurement is described by the distribution P.sup.mess(X.sub.1, . . . , X.sub.k|x.sub.1, . . . , x.sub.N). Preferably, the measurements are independent

P.sup.mess(X.sub.1, . . . ,X.sub.k|x.sub.1, . . . ,x.sub.N)=P.sub.1.sup.mess(X.sub.1|x.sub.1) . . . P.sub.k.sup.mess(X.sub.k|x.sub.k)  (8)

[0388] The entire distribution function P.sup.ges (except for the posterior prefactor) is then given by

P.sup.ges(X.sub.1, . . . ,X.sub.k;x.sub.1, . . . ,x.sub.N)=P.sub.1.sup.mess(X.sub.1|x.sub.k) . . . P.sub.k.sup.mess(X.sub.k|x.sub.k)×P.sup.pop(x.sub.1, . . . ,x.sub.N)  (8a)

[0389] Particularly preferably, each of the measurements is normally distributed with expected value x.sub.i and standard deviation σ.sub.i.sup.mess

[00012]$\begin{array}{cc}{P}_i^{mess}\left(X_i.Math.x_i\right)=\frac{1}{\sqrt{2\pi }{\sigma }_i^{mess}}\mathrm{Exp}\left(-\frac{1}{2{\left({\sigma }_i^{mess}\right)}^2}{\left(X_i-x_i\right)}^2\right)& \left(8b\right)\end{array}$

[0390] The entire distribution function P.sup.ges is then given by inserting the normal distribution from Eq. (8b) into Eq. (8a).

[0391] It is the inventive idea to maximize P.sup.ges as a function of the parameters x.sub.1, . . . x.sub.N. In order to apply the maximum posterior criterion, it is preferred to form the derivatives of the logarithm

[00013]$\begin{array}{cc}\frac{\partial }{\partial x_i}\log P^{ges}\left(X_1,.Math.,X_k,x_1,.Math.,x_N\right)=\frac{\partial }{\partial x_i}\left[\log P^{mess}\left(X_1,.Math.,X_k.Math.x_1,.Math.,x_N\right)+\log P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)\right]=\frac{\partial P^{mess}\left(X_1,.Math.,X_k.Math.x_1,.Math.,x_N\right)/\partial x_i}{P^{mess}\left(X_1,.Math.,X_k.Math.x_1,.Math.,x_N\right)}+\frac{\partial P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)}{P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)},1\le i\le N& \left(9\right)\end{array}$$\begin{array}{cc}\frac{\partial }{\partial x_i}\log P^{ges}\left(X_1,.Math.,X_k,x_1,.Math.,x_N\right)=\frac{\partial }{\partial x_i}\left[\log P_1^{\mathrm{mess}}\left(X_1.Math.x_1\right)+.Math.+\log P_k^{\mathrm{mess}}\left(X_k.Math.x_k\right)+\log P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)\right]=\frac{\partial P_i^{\mathrm{mess}}\left(X_i.Math.x_i\right)/\partial x_i}{P_i^{\mathrm{mess}}\left(X_i.Math.x_i\right)}+\frac{\partial P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)}{P^{\mathrm{pop}}\left(x_1,.Math.,x_N\right)},1\le i\le N& \left(10\right)\end{array}$

[0392] If the distributions are multivariate normally distributed, as is particularly preferred, equation (9) or equation (10) represents a linear system of equations with N equations and N variables that can be solved for x.sub.1, . . . , x.sub.N.

a) No Constraints

[0393] If there are no constraints and equation (9) can be solved, then unique solutions for x.sub.1, . . . , x.sub.N result. If the distributions are multivariate normally distributed, as is particularly preferred, and if the measurement uncertainties are significantly smaller than the ranges of variation of the population, σ.sub.i.sup.mess<<σ.sub.i, 1≤i≤k, then the solutions result

[00014]$\begin{array}{cc}{x}_i\approx x_i^{mess},1\le i\le k& \left(11\right)\end{array}$$x_i\approx {\mu }_i+\Delta x_i,k+1\le i\le N$

i.e. for all parameters for which measurement values are available, one essentially believes the measurement values, and for the remaining values one obtains the mean values μ.sub.i of the population plus shifts Δx.sub.i due to the correlations with the measurement values. One embodiment of the invention then consists in adopting the measurement values for 1≤i≤k directly and neglecting their slight shift due to the underlying population.

b) Constraints

[0394] If there are constraints between the parameters, then every member of the population also satisfies these constraints. Constraints can be described by

f.sub.j(x.sub.1, . . . x.sub.N)=0, 1≤j≤Q⇔f(x.sub.1, . . . ,x.sub.N)=0  (12)

i.e. by Q functions f.sub.j of the parameters x.sub.1, . . . , x.sub.N, which can be combined in a vector f and which, by requirement, are to be equal to zero. The functions f.sub.j are preferably linear or linear approximations to the given constraints.

[0395] In the preferred case of multivariate distributions, this has the consequence that the columns of the covariance matrix are linearly dependent, that is to say that the covariance matrix has a rank r<N and can therefore no longer be inverted. A distribution density P.sup.pop(x.sub.1, . . . , x.sub.N) can then no longer be specified.

[0396] One possibility in practice is to regularize the covariance matrix C by shifting one or more of the correlations ρ.sub.ij or standard deviations σ.sub.i contained in it by ε and then determining x.sub.1, . . . , x.sub.N. The solutions thus obtained then automatically satisfy the constraints for ε.fwdarw.0.

[0397] In the context of the invention, it has been found that this method has disadvantages though. On the one hand, one has to know the distribution in the population, and on the other hand, its covariance matrix is either singular or poorly conditioned. If one researches the correlations ρ.sub.ij and standard deviations σ.sub.i, then small inaccuracies in the information or incomplete information are sufficient for the covariance matrix to be regular, but then possibly generate numerically unstable solutions for the parameters sought.

[0398] In the context of the invention, however, it has been found that this problem can be circumvented by either working on the basis of the distribution (maximum likelihood approach)

P.sup.mess(x.sub.1, . . . ,x.sub.N):=P.sup.mess(X.sub.1=x.sub.i.sup.mess, . . . ,X.sub.k=x.sub.k.sup.mess|x.sub.1, . . . ,x.sub.N)  (13a)

or on the basis of the distribution (maximum posterior approach)

P.sup.ges(x.sub.1, . . . ,x.sub.N):=P.sup.ges(X.sub.1=x.sub.1.sup.mess, . . . ,X.sub.k=x.sub.k.sup.mess;x.sub.1, . . . ,x.sub.N)  (13b)

[0399] The substitution method can preferably be used for this purpose.

Maximum Likelihood Method with Constraints and Substitution

[0400] The first K parameters x.sup.u:=(x.sub.1, . . . , x.sub.K).sup.T are assumed to be independent and equation (12) is solved for the remaining Q=N−K dependent parameters x.sup.a:=(x.sub.K+1, . . . , x.sub.N).sup.T, which can then be understood as a function x.sup.a(x.sup.u) of the independent parameters x.sup.u and can be substituted in f. Then the constraints as a function of x.sup.u are:

f(x.sup.u,x.sup.a(x.sup.u))=0  (14).

[0401] In the context of the invention, it is not necessary to explicitly know the function x.sup.a(x.sup.u). In the context of the invention, one only needs its Jacobi matrix ∂x.sup.a/∂x.sup.u:=∂x.sub.i.sup.a/∂x.sub.j.sup.u, 1≤i≤Q, 1≤j≤K, which according to the theorem of the implicit function is given by

[00015]$\begin{array}{cc}\frac{\partial x^a}{\partial x^u}=-{\left(\frac{\partial f}{\partial x^a}\right)}^{-1}\frac{\partial f}{\partial x^u}& \left(15\right)\end{array}$

where ∂f/∂x.sup.a is the quadratic Jacobi matrix of f with regard to x.sup.a and ∂f/∂x.sup.u is the generally rectangular Jacobi matrix of f with regard to x.sup.u. The probability density P.sup.mess (x.sup.u,x.sup.a(x.sup.u)) thus has to be maximized as a function of x.sup.u, i.e.

[00016]$\begin{array}{cc}\begin{array}{c}\frac{\partial }{\partial x^u}{P}^{mess}\left(x^u,x^a\left(x^u\right)\right)=\frac{\partial P^{mess}}{\partial x^u}+\frac{\partial P^{mess}}{\partial x^a}\frac{\partial x^a}{\partial x^u}\\ =\frac{\partial P^{mess}}{\partial x^u}-\frac{\partial P^{mess}}{\partial x^a}{\left(\frac{\partial f}{\partial x^a}\right)}^{-1}\frac{\partial f}{\partial x^u}\\ =0\end{array}.& \left(16\right)\end{array}$

[0402] The system of equations (16) is K equations that can be solved for the parameters x.sup.u independent for K. The remaining parameters x.sup.a are obtained by inserting them into the context x.sup.a(x.sup.u).

Maximum Likelihood Method with Constraints and Lagrange Parameters

[0403] Alternatively, within the scope of the invention, the entire set of parameters can be considered independent if, instead of the function P.sup.mess(x.sub.1, . . . , x.sub.N), the Lagrange function is maximized

P.sup.mess,Lagrange(x.sub.1, . . . ,x.sub.N,λ)=P.sup.mess(x.sub.1, . . . ,x.sub.N)+λf(x.sub.1, . . . ,x.sub.N)  (17)

where λ=(λ.sub.1, . . . , λ.sub.Q) is a Q-dimensional vector of Lagrange multipliers. It is then to be maximized by setting the derivatives N+Q to zero

[00017]$\begin{array}{cc}\frac{\partial }{\partial x_i}{P}^{\mathrm{mess},\mathrm{Lagrange}}\left(x_1,.Math.,x_N,\lambda \right)=0,1\le i\le N& \left(18\right)\end{array}$$\frac{\partial }{\partial {\lambda }_j}{P}^{\mathrm{mess},\mathrm{Lagrange}}\left(x_1,.Math.,x_N,\lambda \right)=0,1\le j\le Q.$

[0404] Solving equation (18) for the N+Q unknowns (x.sub.1, . . . , x.sub.N) and (λ.sub.1, . . . , λ.sub.Q) leads to the solutions for the parameters.

[0405] Instead of treating the constraints with substitution or long-range parameters, one can alternatively (for example in the case of locally vanishing gradients of the function to be maximized) use a damped Hamilton formalism with a friction term.

[0406] Analogously, the method of Eqs. (16) to (18) can be applied to the function P.sup.ges(x.sub.1, . . . , x.sub.N) instead of P.sup.mess(x.sub.1, . . . , x.sub.N) and then represents a maximum-posterior method with constraints.

Embodiment with Specific Exemplary Numerical Values

[0407] For the sake of simplicity, an eye that is rotationally symmetrical about the optical axis and therefore has neither a cylindrical prescription, nor a cylindrical cornea, nor cylindrical lens surfaces is considered as a starting situation. Exemplary values and parameters prior to the IOL surgery are in detail:

S=−7.0 dpt; vision disorder(measured)

C=41.2 dpt; refractive power of cornea (measured)

L.sub.1=7.82 dpt; refractive power of lens front surface (literature)

L.sub.2=13.28 dpt; refractive power of lens back surface (literature)

d.sub.CL=3.6 mm; anterior chamber depth (measured)

d.sub.L=3.7 mm; lens thickness (literature)

n.sub.CL=1.336; refractive index anterior chamber (literature)

n.sub.L=1.422; refractive index lens (literature)

n.sub.LR=1.336; refractive index vitreous body (literature)  (19).

[0408] After IOL surgery, for example the following values or parameters are transmitted:

S.sub.IOL.sup.mess=0.0 dpt; vision disorder (measured)

L.sub.2,IOL.sup.mess=3.2 dpt; refractive power of lens back surface (manufacturer information)  (20).

[0409] All other parameters after IOL surgery are assumed to be unchanged for the sake of simplicity.

[0410] With the aid or equation

[00018]$\begin{array}{cc}{D}_{LR}=L_2+\left(L_1+\frac{S+C}{1-{\tau }_{\mathrm{CL}}\left(S+C\right)}\right){\left(1-{\tau }_L\left(L_1+\frac{S+C}{1-{\tau }_{\mathrm{CL}}\left(S+C\right)}\right)\right)}^{-1}& \left(21\right)\end{array}$

one can calculate the reduced inverse vitreous length (D.sub.LR=n.sub.LR/d.sub.LR, where d.sub.LR the vitreous length is; further τ.sub.CL=d.sub.CL/n.sub.CL and τ.sub.L=d.sub.CLL/n.sub.L), and thus the eye length d.sub.A=d.sub.CL+d.sub.L+d.sub.LR. Vitreous body length and eye length are so directly related that in the following the vitreous body length can be considered instead of the eye length.

[0411] If one applies equation (21) to the situations before and after surgery, one formally obtains prior to IOL surgery

D.sub.LR=64.69 dpt  (22a)

and formally after IOL surgery

D.sub.LR,IOL.sup.mess=65.65 dpt  (22b).

[0412] However, since the vitreous body length cannot have changed as a result of surgery, there is an inconsistency here that can be solved within the scope of the present invention.

[0413] In order to choose the simplest possible example, the initial situation can be regarded as the case that there are no variations and no correlations in the basic population, and that only the vision disorder measured afterward as well as the IOL itself are subject to uncertainties:

σ.sub.S,IOL.sup.mess=0.25 dpt; vision disorder (Std. deviation of measurement method)

σ.sub.L2,IOL.sup.mess=0.4 dpt; refractive power of lens back surface (manufacturer tolerance)  (23).

[0414] Now, within the scope of the invention, the true values of S.sub.IOL,L.sub.2,IOL can be identified which, as expected, will both deviate from equation (20).

[0415] In the exemplary case, P.sup.pop=1 and the probability density for the distribution of S.sub.IOL,L.sub.2,IOL to be initially assumed based on the measurements is

[00019]$\begin{array}{cc}{P}^{mess}\left(S_{\mathrm{IOL}}^{mess},L_{2,\mathrm{IOL}}^{mess}.Math.S_{\mathrm{IOL}},L_{2,\mathrm{IOL}}\right)=P_1^{mess}\left(S_{\mathrm{IOL}}^{mess}.Math.S_{\mathrm{IOL}}\right).Math.P_k^{mess}\left(L_{2,\mathrm{IOL}}^{mess}.Math.L_{2,\mathrm{IOL}}\right)=\frac{1}{2{\mathrm{\pi \sigma }}_{S,\mathrm{IOL}}^{mess}{\sigma }_{L2,\mathrm{IOL}}^{mess}}\times \mathrm{Exp}\left(-\frac{1}{2{\left({\sigma }_{s,\mathrm{IOL}}^{mess}\right)}^2}{\left(S_{\mathrm{IOL}}-S_{\mathrm{IOL}}^{mess}\right)}^2\right)\times \mathrm{Exp}\left(-\frac{1}{2{\left({\sigma }_{L2,\mathrm{IOL}}^{mess}\right)}^2}{\left(L_{2,\mathrm{IOL}}-L_{2,\mathrm{IOL}}^{mess}\right)}^2\right)& \left(24\right)\end{array}$

[0416] Now, however, the constraint applies that S.sub.IOL,L.sub.2,IOL after insertion in equation (21) have to yield the same value for D.sub.LR after surgery as before surgery. Hence the equation for the constraint is

[00020]$\begin{array}{cc}{D}_{LR}=L_{2,\mathrm{IOL}}+\left(L_1+\frac{S_{\mathrm{IOL}}+C}{1-{\tau }_{\mathrm{CL}}\left(S_{\mathrm{IOL}}+C\right)}\right){\left(1-{\tau }_L\left(L_1+\frac{S_{\mathrm{IOL}}+C}{1-{\tau }_{\mathrm{CL}}\left(S_{\mathrm{IOL}}+C\right)}\right)\right)}^{-1}& \left(25\right)\end{array}$

which when solved for L.sub.2,IOL yields as a function of S.sub.IOL:

[00021]$\begin{array}{cc}{L}_{2,\mathrm{IOL}}\left(S_{\mathrm{IOL}}\right)=D_{LR}-\left(L_1+\frac{S_{\mathrm{IOL}}+C}{1-{\tau }_{CL}\left(S_{\mathrm{IOL}}+C\right)}\right){\left(1-{\tau }_L\left(L_1+\frac{S_{\mathrm{IOL}}+C}{1-{\tau }_{CL}\left(S_{\mathrm{IOL}}+C\right)}\right)\right)}^{-1}.& \left(26\right)\end{array}$

[0417] The constraint means that one may only move on the cutting surface 30 shown in FIG. 2.

[0418] If one substitutes L.sub.2,IOL(S.sub.IOL) in equation (24) and maximizes for S.sub.IOL, i.e. if one solves

[00022]$\begin{array}{cc}\frac{d}{{\mathrm{dS}}_{\mathrm{IOL}}}{P}^{\mathrm{mess}}\left(S_{\mathrm{IOL}}^{\mathrm{mess}},L_{2,\mathrm{IOL}}^{\mathrm{mess}}.Math.S_{\mathrm{IOL}},L_{2,\mathrm{IOL}}\left(S_{\mathrm{IOL}}\right)\right)=0& \left(27\right)\end{array}$

for S.sub.IOL, one obtains

S.sub.IOL=−0.12 dpt

L.sub.2,IOL(S.sub.IOL)=3.01 dpt  (28).

[0419] Both variables S.sub.IOL,L.sub.2,IOL are therefore in the negative direction compared to the measurement values, but not to the same extent. Rather, the method seeks a balance in the light of the different standard deviations and the asymmetrical position of the constraint relative to the Gaussian bell.

[0420] Inconsistencies in the eye model can occur not only for a calculated eye length (or a calculated lens-retina distance), but also e.g. when measuring the eye length. Such inconsistencies can be solved analogously to the example of a calculated eye length described above. Of course, more complex examples in which the eye length itself is also not fixed or where possibly correlations occur, can also be given.

REFERENCE NUMERAL LIST

[0421] 10 main ray [0422] 12 eye [0423] 14 first surface of the spectacle lens (front surface) [0424] 16 second surface of the spectacle lens (back surface) [0425] 18 corneal front surface [0426] 20 eye lens [0427] 30 cutting surface