OR
`
`3,95�,lll
`
`United St11
`
`
`Fisher et al.
`
`[ 11] 3,953,111
`
`
`
`[45J Apr. 27, 1976
`
`(57]
`
`ABSTRACT
`
`[54] NON-LINEAR LENS
`
`[75] Inventors:
`Ralph W. Fisher, St. Charles;
`
`George Licis,
`Manchester; Wayne
`A non-linear lens possesses distortion characteristics
`
`
`
`Bridgeton,
`all of Mo.
`W. Schurter,
`
`
`which are such that objects along the optical axis of
`
`the lens occupy disproportionately large areas of the
`
`
`
`St.[73]Assignee: McDonnell Douglas Corporation,
`
`image cast by the lens, whereas objects near the pe­
`
`Louis, Mo.
`
`riphery of the field of view occupy a disproportion­
`
`ately small area of the image. The distortion charac­
`
`
`teristics approximate the formula H=sin113 0 where H
`
`
`is height measured from the optical axis and 0 is the
`
`
`angle measured from the optical axis. The image cast
`350/181
`[52] U.S. Cl . ................................. 350/189;
`
`
`
`by the lens falls on the vidicon of a television camera
`829D 13/18; G02B 13/08
`[51] Int. Cl.2
`
`
`where it is scanned and transmitted to a projector.
`350/189, 192,198,175,
`[58]Field of Search ...........
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`
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`Since the lens enlarges objects in the vicinity of the
`350/181
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`
`
`optical axis, those objects are transmitted with much
`
`
`
`greater detail than objects in the peripheral region of
`References Cited
`
`
`the view. The transmitted image is reproduced at a
`UNITED ST A TES PATENTS
`
`
`projector and the reproduced image is rectified
`
`
`
`
`through another lens having identical distortion char­
`
`
`
`3,037,426 6/1962 Hughues ............................. 350/192
`
`
`acteristics. This lens casts the rectified image on a
`FOREIGN PATENTS OR APPLICATIONS
`
`
`
`spherical screen. The final image which appears on
`
`
`
`
`1,105,632 4/1961 Germany ......................... 350/175 R
`
`
`the screen possesses a high degree of acuity in the re­
`OTHER PUBLICATIONS
`
`
`
`gion of the optical axis and substantially less acuity in
`
`
`
`
`peripheral regions. The resolution throughout the en­
`
`
`Rigler; A. K. and Vogt; T. P., "Spline Functions: an
`
`
`tire field of the reproduced image corresponds quite
`
`
`
`Alternative Representation of Aspheric Surfaces,"
`
`
`closely to the resolution characteristics of the human
`Vol. 10, No. 7, pp. 1648-1651, July,
`
`Applied Optics,
`eye.
`1971.
`
`[22]Filed:Nov. 4, 1974
`[21]Appl. No.: 520,487
`
`•••••••••••••••.••.•
`
`[ 56]
`
`K. Corbin
`Primary Examiner-John
`Clark
`
`Assistant Examiner-Conrad
`
`
`
`Lieder & Attorney, Agent, or Firm-Graveley,
`Woodruff
`
`
`
`10 Claims, 7 Drawing Figures
`
`
`
`C
`
`I
`
`/
`
`FIG.6
`
`APPLE 1009
`
`1
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`

`

`U.S. Patent
`
`April 27,1976
`
`Sheet 1 of 3.
`
`3,953,111
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`30Hz FRAME RATE
`
`
`
`
`
`ANGULAR
`RESOLUTION
`MINUTES
`ARC
`
`40
`
`120
`80
`FIELD OF VIEW — DEG
`FIG. |
`
`160
`
`200
`
`i O
`
`ACUITY
`
`05
`
`i
`
`0
`
`8
`
`|bh°o
`40
`0
`DEGREES FROM FOVEA
`
`FIG.2
`
`-|'
`
`-2!
`
`-10!
`
`80
`
`MIN OF ARC
`
`H= sm '4& oO
`
`NON-LINEAR LENS
`
`FISHEYE LENS
`H=K ©
`
`
`CONVENTIONAL
`CAMERA LENS
`
`H= Ktan ©
`
`
`1.0
`
`08
`
`t
`~ 06
`
`04
`02
`
`0
`
`f5
`
`Ww
`+
`
`
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`NORMALIZEDIMAGE
`
`1000
`
`100
`
`3
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`BANDWIDTHINMEGAHERTZ
`
`2
`
`

`

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`
`U.S. Patent April27, 1976=Sheet2of3 33,953,111
`
`ERROR SIGNALS
`
` NON-LINEAR
`
`PROJECTION
`SCREEN
`
`FIG. 5
`
`5
`
`LEN
`
`0.9
`
`;
`
`:
`
`NON —LINEAR
`LENS
`
`ii
`i _—<$——-n
`Li)
`ea ANGLE
`“9.9
`2.6
`5.1
`8.3
`I2.2
`16.8
`23.5
`34.3
`524
`90.0
`
`IMAGELENGTH
`O1
`0.2
`0.3
`0.4
`0.5
`0.6
`0.7
`0.8
`09
`lo
`
`4.0
`
`0.8
`
`0.7
`0.6
`0.5
`
`04
`
`0.3
`
`0.2
`
`0.)
`
`MAGE
`PLANE
`
`—0ol
`a2
`03
`04
`05
`0.6
`
`0.7
`
`0.8
`
`OBJECT
`SPACE
`
`0.9
`
`FIG.4
`
`3
`
`

`

`U.S. Patent
`
`April 27,1976
`
`Sheet 30f 3.
`
`3,953,111
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`
`
`SQ YY,
`q
`
`
`SSSA
`
`<Rew
`
`
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`
`
`
`
`
`
`
`
`
`
`
`4
`
`

`

`3,953,111
`
`1
`
`NON-LINEAR LENS
`
`The Governmenthasrights in this invention pursuant
`to Contract Number NO0014-73-C-0154 awarded by
`the Department of the Navy.
`BACKGROUNDOF THE INVENTION
`
`The present invention relates in general to lenses and
`moreparticularly to a lens having non-linear distortion
`characteristics.
`The typical remote viewing system utilizes a televi-
`sion camera at the remote location, some type of pro-
`jector at the observer location, and a television trans-
`mitting system linking the two. These viewing systems
`fall far short of duplicating the visual characteristis of
`the human eye in that
`they have extremely limited
`fields of view or else poor resolution in a large field of
`view.
`
`In particular, for any fixed angular resolution (mea-
`sured in minutes of arc) and frame rate (usually 30 Hz
`or frames/sec.) a definite relationship exists between
`field of view and bandwidth for transmitting that field
`of view. For example, commercial television, which
`utilizes a 525 line raster traced 30 times per second,
`operates on a bandwidth of 3.93 MHz. To match the
`resolution of the human eye, which is one minute of arc
`along its foveal or optical axis, the field of view for
`commercial television must be restricted to less than
`10° (see FIG. 1). On the other hand,if the field of view
`is increased to about 180°, whichis the field of view for
`the human eye, the bandwidth must be increased to
`1000 Mhz to maintain one minute of arc resolution
`over the entire field. This demands a raster of 10,000
`lines and is far in excess of the capabilities of current
`television systems.
`,
`Indeed, the most advancedtelevision currently avail-
`able utilizes an 875 line system and requires a band-
`width of 10.9 Mhz. This provides a field of view of
`about 20° with one minute arc resolution throughout
`the entire field, which is far less than the 180° field of
`view possessed by the humaneye.
`From the foregoing,it is clear that present television
`viewing systems present a dilemma.If the field of view
`is sufficient to encompassall possible locationsof inter-
`est, resolution is so low that detection or clear observa-
`tion is impossible. On the other hand,if the resolution
`is adequate to insure that the objects will be seen
`clearly, the field of view is quite limited and many
`objects of interest are located outside of the field of
`view.
`In a sense the human eye provides a solution for the
`foregoing dilemma. The human eye possesses high
`optical acuity along and in the vicinity ofits foveal or
`optical axis, but the acuity diminishes outwardly there-
`from. In other words, the eye distinguishes fine detail
`directly in front of it, but not to the sides. This charac-
`teristic is not derived from the shape of the eye lens,
`but instead results from the fact that most of the optical
`fibers for the eye are concentratedin the vicinity of the
`optical axis. Hence, only along the optical axis does the
`eye possess one minute of arc resolution. The resolu-
`tion becomesprogressively less toward the periphery of
`the field of vision (see FIG. 2). Nevertheless, the reso-
`lution in the peripheral areais sufficient to detect the
`presence of many objects in that area as well as much
`movementin that area. Of course, when the eye detects
`anything ofinterest in the peripheral areas, the head or
`
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`eye is immediately moved to bring the foveal axis to the
`thing of interest and thereby provide a clearer image of
`it.
`
`SUMMARYOF THE INVENTION
`
`One ofthe principal objects of the present invention
`is to provide a lens having non-linear distortion charac-
`teristics which are such that objects located along and
`nearthe optical axis of the lens occupy a disproportion-
`ately large area of the image produced by the lens.
`Another object is to provide a lens of the type stated
`which closely approximates the resolution characteris-
`tics of the human eye over a wide field of view. A fur-
`ther object is to provide a lens of the type stated which
`is ideally suited for use in remote viewing systems in
`that it provides a wide field of view with maximum
`acuity along the optical axis. These and other objects
`and advantages will become apparent hereinafter.
`The present invention is embodied in a lens which
`distorts a field of view such that objects in the vicinity
`of the optical axis occupy a disproportionately large
`area of the image cast by the lens and objects in the
`peripheral region of the field of view occupy a dispro-
`portionately small area of the image. The invention also
`consists in the parts and in the arrangements and com-
`binations of parts hereinafter described and claimed.
`DESCRIPTION OF THE DRAWINGS
`
`In the accompanying drawings which form part of the
`specification and wherein like numerals and letters
`refer to like parts wherever they occur:
`FIG. 1 is a graph showingthe relationship between
`field of view, angular resolution, and bandwidth for
`transmitting a picture of a remote location by televi-
`sion;
`.
`FIG. 2 is a graph showing relative acuity of the
`human eye throughoutthe field of view for the eye;
`FIG. 3 is a graph showing the distortion characteris-
`tics of the lens of the present invention in terms of
`normalized image height andfield of view and compar-
`ing such distortion characteristics with the distortion
`characteristics of a fisheye lens and a conventional
`camera lens;
`FIG. 4 is a graphic representation of the non-linear
`distortion characteristics and showing how equalincre-
`ments on the image plane correspond to unequal incre-
`ments in the field of view;
`FIG. 5 is a schematic perspective view of the remote
`viewing system in which the non-linear lens may be
`utilized;
`FIG. 6 is a sectional view of the non-linear lens; and
`FIG.7 is an enlarged sectional view of the second and
`third lens groupings for the non-linearlens.
`DETAILED DESCRIPTION
`
`The lens of the present invention (FIGS. 6 and 7)
`provides non-linear image distortion characteristics
`over an extremely wide field of view which approaches
`160°. This is in contrast to so-called fisheye lenses
`which provide linear distortion characteristics.
`In
`paticular, with a linear or fisheye lens the image height
`is directly proportional to the field angle (FIG. 3). The
`relationship is defined by the formula H=K @ where H
`is the image height from the optical axis, K is a con-
`stant, and @ is the angle measured from the optical axis.
`Thus, with a linear lens an object occupying twice the
`angle as another object, when measured from the opti-
`cal axis, will cast an image twice as high as the other
`
`5
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`5
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`

`

`3,953,111
`
`4
`object. On the other hand, with a non-linearlens of the
`The projector projects the transmitted image through
`presentinvention the image heightis equalto a variable
`its non-linear lens which casts the image uponaspheri-
`function of the field angle (FIG. 3). The relationship is
`cal screen surrounding the projector. The observer
`approximated by the formula H=sin"9. Thus, objects
`views the screen from theposition of the projector.
`centered along the optical axis of the non-linear lens L
`Positioned on the projector is an oculometer which
`cast a muchlarger image than objects located near the
`views the observer's eye through a transparent beam-
`peripheryofthe field of view with the size diminishing
`splitter, and in effect tracks the observer's eye, produc-
`as the angle from the optical axis increases. The result
`ing error signals whenever the foveal axis of the eye
`of the distortion is that objects along the optical axis
`deviates from the optical axis of the projector lens. In
`Occupy a disproportionately large share of the image
`other words, error signals are produced whenthe fo-
`cast by the lens when compared with other objects
`veal axis of the eye and the optical axis of the projector
`Closerto the periphery of the field of view for the lens.
`lens intersect the screen at different locations. These
`In effect, the center of the non-linearlensis a telephoto
`signals are converted into elevation and azimuth com-
`lens, while the periphery of the lens amounts to a wide
`mands which are transmitted to the servo system for
`angle lens with the annular region between the center
`the camera through the transmission system. The com-
`and periphery varying from telephoto to wide angle.
`mands cause the camerato changeelevation and azi-
`Naturally, the image produced is quite distorted. The
`muth, and the movementis such that the corresponding
`typical cameralens is represented by the formula H=K
`movement ofthe projector reducesthe error and brings
`tan 6 (FIG. 3) and is non-linear, but in a sense opposite
`the foveal axis of the camera back toward coincidence
`from thatof the lens of the present invention.
`with the optical axis of the lens, at least at the screen.
`The non-linear transfer characteristics of the lens
`Thus, the oculometer controls the position of the cam-
`maybeillustrated by breaking the image into equal
`€ra and the camera controls the position of the projec-
`angular
`increments (FIG. 4) and comparing each
`tor, so in effect the camera and projector are both
`image increment with the corresponding incrementit
`slaved to the observer’s eye through the oculometer.
`represents in the actual field of view. Clearly, equal
`The oculometer and servo mechanisms for the camera
`angular increments on the imagesideofthe lens repre-
`and projector should all respond fast enough to bring
`sent unequalincrements on the object side of the lens.
`the optical axis of the projector lens into coincidence
`Morespecifically, near the optical axis relatively small
`with, or at least within 2 percentof, the foveal axis for
`arcs on the object side are converted to large arcs on
`the eye within 0.2 seconds. This is about as rapidly as
`the imageside, thus enlarging the image. At about 25°
`the eye can fixate and perceive when changing from
`from the optical axis arcs on the Object side equal the
`one objectof interest to another, so the lag in the pro-
`arcs on the image side andthis portion ofthe lens may
`jectoris barely discernible,if atall. A suitable oculom-
`be considered linear. Objects from about 25° to 80°
`eter is marketed by Honeywell Inc., Radiation Center,
`(lens periphery) occupy arcs much larger than they
`Boston, Mass.
`cast on the image side with the variance in arcs becom-
`Referring again to the television camera at the re-
`ing greater as the object approaches the periphery of
`mote location,
`the camera lens casts the distorted
`the field. Hence, objects within 25° of the optical axis
`image ofall objects in the field of view on the vidicon
`for the lens are magnified with the magnification being
`of the camera,and this vidicon is scanned in the usual
`substantial along the lens axis, whereas objects in the
`manner, that is with a beam which traces a raster pat-
`annular region located beyond 25° are reducedin size,
`tern at uniform velocity. The conventional commercial
`with the reduction becoming progressively greater as
`television system of 525 lines per scan and 30 scans per
`the maximumfield angle for the lensis approached.
`second may be employed. This requires a bandwidth of
`To appreciate the lens requires an understanding of
`3.93 MHz (FIG. 1). The beam in effect picks the image
`the remote viewing system in whichitis utilized. That
`off of the camera vidicon- Since the objects along the
`viewing system basically comprises (FIG. 5) a televi-
`optical axis are magnified and occupy a disproportion-
`sion camera at the remote location, a projector at the
`ately large area of the vidicon, they are picked off the
`observer location, and a transmission system linking
`vidicon in great detail. On the other hand, objects in
`the camera and projector in both directions. Both the
`the peripheral region ofthefield of view are reduced in
`camera and projector are fitted with non-linear lenses
`size and occupyrelatively little area on the vidicon.
`having identical distortion characteristics. However,
`Hence, they are picked off of the vidicon with substan.
`the projector lens is mounted just the reverse of the
`tially less detail. The picture is transmitted accordingly.
`camera lens so thatit rectifies the distortion created by
`The magnification along and near the optical axis is
`the camera lens. The camera is supported on a gim-
`great enough to enable the beam to extract one minute
`balled mountandis therefore capable of swinging both
`ofarc detail, which is all an eye with 20—20vision can
`vertical and horizontal angles with respect to fixed
`perceive along its foveal axis. The beam extracts
`coordinates at the remote location. A suitable servo
`greater angles ofarc detail away from the optical axis
`mechanism bridges the gimballed mountto control the
`and hencepoorerresolutionis available in this area. In
`position of the camera. Theprojectoris likewise sup-
`this regard,it will be recalled that to extract one minute
`ported on a gimballed mount which permits it to swing
`arc detail overa full 180° requires a 10,000 line vidicon
`both vertical and horizontal angles with respectto fixed
`or in other words a bandwidth of 1000 MHz which is
`coordinates at the observer location. Another servo
`far in excess of presenttelevision capabilities.
`mechanism bridges the gimballed mountof the projec-
`The distorted image cast upon the vidicon of the
`tor, and this servo is slaved to the camera through the
`camera is reproduced bythelight valve ofthe projector
`transmission system such that a changein elevation or
`and this image likewise has at least one minute of arc
`azimuth of the camera with respecttoits fixed coordi-
`resolution along the optical axis with the resolution
`nates results in a corresponding change in elevation
`diminishing toward the periphery of the image so that
`and azimuth of the projector with respect to its fixed
`only muchlarger angles of arc are discernible beyond
`coordinates,
`the optical axis. The image so produced is rectified by
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`ment c. The inside element c also has a spherical sur-
`the projector lens which casts it upon the spherical
`face R, which faces the tapered interior of the housing
`screen. The resulting screen image constitutes a faithful
`2 and is presented toward the second lens grouping B.
`reproduction of the scene which lies within the field of
`Along the optical axis Z for the lens, the element a has
`view for the camera lens. The projector lens in no way
`a thickness t,, the element b a thickness f,, and the
`affects the resolution of the imageit transmits and as a
`elementc a thickness¢3. Index matching oil couples the
`result the image appearing on the screen showsdetail
`matching surfaces R, of the lens elements a and 5 and
`as close as one minute ofarc at the optical axis for the
`the matching surfaces Rof the lens elements 5 and c.
`lens and the area immediately surroundingit, but in the
`The outside andinside lens elements a and ¢ are formed
`remaining area such detail is notavailable. In other
`from type SK16 glass, whereas the intermediate lens
`wards, the resolution in the other areas is somewhat
`element is formed from type F2 glass. The index of
`less. Hence, the projected image is very sharp and clear
`refraction for SK16 glass is 1.62041 and for F2 glassis
`on the screenat the optical axis, that is directly in front
`1.62004. The Abbe numberfor SK16 is 60.27 and for
`of the projector, and then turns somewhat fuzzy or
`F2 is 36.25.
`blurred in the surrounding area particularly at the max-
`Turning now to the second lens grouping B, it con-
`imum angle of 80° from the optical axis.
`The variance in clarity or in resolution of the final
`sists of four lens elements, namely, a convex-concave
`lens element d, a double convex lens elemente, a dou-
`image cast upon the screen closely resembles the opti-
`ble concave lens element f, and a double convex lens
`cal characteristics of the eye (FIG. 2). In this connec-
`elementg, all arranged in that order from thefirst lens
`tion, it will be recalled that most of the optical sensing
`grouping A. The innermost lens element d has spherical
`elements for the human eye are concentrated along the
`foveal axis.
`surfaces R, and R, andathickness ¢;, along the optical
`As previously mentioned, the oculometer tracks the
`axis Z. The next lens element e has spherical surfaces
`R, and R, and a thickness ¢, along the optical axis Z.
`eye position and causes the camera to change position
`Next,
`is the double concave lens element f having
`in response to eye movement while the projector un-
`dergoes corresponding movementas a result of being
`spherical surfaces Rg and Ry, and a thicknessf,) along
`the optical axis Z. The surface Rio of the element f
`slaved to the camera. Consequently, the foveal or opti-
`cal axis of the eye is always directed at the center of the
`correspondsto andis against a matching surface Ry) on
`projected image, that is the portion along the optical
`the lens element g which has another spherical surface
`axis for the projector lens. This is the portion having
`Rj, presented toward the third lens grouping C. The
`the one minute of arc resolution. Since the resolution
`surfaces R, and R,of the lens elements d ande, respec-
`of the eye falls off with the angle from the foveal axis,
`tively, are separated by an air gap ts measured along the
`little is lost by having the resolution of the projected
`axis Z of the lens, while the surfaces Rg and Rg of the
`image diminish with the angle from the optical axis.
`lens elements e andf, respectively, are separated by an
`The resolution in the surroundingarea ofthe picture is
`air gap tg measured alongtheoptical axis Z. The match-
`still good enough to permit the eye, as a result of the
`ing surfaces Rig of the lens element f and g are ce-
`built-in peripheral vision, to detect movement and ob-
`mented. The lens elements d and f are formed from
`jects of interest, and if whatever is detected appears
`type F2 glass, while the lens elements ¢ and g are
`interesting enough, the viewer will turn his eye toward
`formed from type SK16 glass.
`The third lens grouping C containsa single lens ele-
`it. This, of course, causes the camera and projector to
`changeposition so that the formerly blurred area of the
`ment 4 having a non-spherical and non-planar surface
`image to which the eye is turned lies along the optical
`Riz which is presented toward the second lens group B
`axis of the camera and projector lenses and is projected
`and a spherical surface Rys exposed outwardly. The
`lens element / has a thickness 1) along the optical axis
`with high resolution.
`The non-linear lens (FIGS. 6 and 7) has three lens
`Z and is made from the type SK16 glass.
`sets or groupings A, B and C.Its aperture ratio is 5.6
`The first and second lens groupings A and B are
`and it forms a 0.358 F diameter image where F is the
`separated by an air gap ¢, which is measured from the
`focal length along the optical axis Z. The first lens
`surface R, to the surface R; along the axis Z of the lens.
`grouping A is considerably larger than the other group-
`The second andthird lens groupings B and C are sepa-
`ing B and C andis contained in the large end ofa ta-
`rated byan air gap ¢,, which is the distance between the
`pered lens housing. The other lens groupings B and C
`surfaces R,, and Ry measured alongthe opticalaxis Z.
`are contained within a subhousing whichfits into the
`The distorted image formed by the non-linear lens
`small end of the tapered main housing. The first lens
`exists in an image plane p located beyondthethird lens
`groupingis a triplet and provides the mapping function,
`grouping C. The vidicon of the television camera
`thatis the unique distortion whichis essentially defined
`should be at this plane p.
`by the formula H=sin"? @. The second grouping B,
`The surfaces Re, Ry, Rs, Re, Rz, Re, Ro» Rios Ru and
`which has four elements, contains the aperture stop
`R,3 are all spherical and have their centers of curvature
`and forms an imageof the sceneas distorted bythefirst
`alongthe optical axis Z of the lens. The surfaces R, Rs,
`grouping A. The third grouping C is a single element
`and Ry, while being curved, are not spherical. The
`which functions as a field flatener, that is it makesthe
`radii of curvatute for the surfaces R, through Rg fol-
`image cast by the second grouping B planar.
`low:
`a
`Thefirst lens grouping A (FIG. 6) consists of three
`lens elementsa, b, and c with no air gaps between adja-
`cent elements. The outside lens element a has a non-
`spherical surface R; exposed outwardly and a spherical
`surface R, presented inwardly against a matching sur-
`face R, on the intermediate element b. The opposite
`surface R; of the intermediate elementb is non-spheri-
`cal and abuts a matching surface Rg on the inside ele-
`
`1.37F
`<1LSF
`0.729F
`< 1,092F
`< 0.2376F
`< 0.273F
`< 0.3940F
`<0.335F
`<0.2271F
`
`$5
`
`60
`
`65
`
`1.4F
`
`‘1.091F
`0.2373F
`0.272F
`0.3936F
`0.334F
`0.2268F
`
`R,
`< R:
`Rs
`<R,
`<-R;
`<=—Ry
`<R,
`<-Ri
`<~Ry
`
`(at optical axis only)
`
`(at optical axis only)
`
`7
`
`

`

`0.280F
`0.S71F
`
`0.4168F
`
`< Re
`<-R,,
`—Ri:
`<-R,3
`
`7
`-continued
`< 0.285F
`< 0.572F
`> O.314F
`< 0.4173F
`
`(at optical axis only)
`
`3,953,111
`
`8
`
`Surface
`Surface
`Surface
`Riz
`Ry
`Ri
`0.0000000
`0.0000000
`0.0000000
`X(0)
`~—0.0029324
`0.1795877
`0.2108668
`X(1)
`X(2)
`0.8554116
`0.7202623
`—0.0041 130
`X(3)
`1.7838662
`1.4361922
`—0.0004761
`whereFis the focal length ofthe lens along its optical
`X(4)
`1.9775600
`1.6027882
`0.0065340
`X(5)
`—
`-
`0.0100000
`axis Z. Note, that since the surfaces R,, R; and Ry,are
`X(6)
`~
`_
`0.0100000
`not spherical, the radii of curvature listed above for
`p(0)
`0.0000000
`0.0000000
`0.0000000
`p(t)
`1.0780200
`0.7222734
`0.0825000
`those surfaces represents only radii along the optical
`p(2)
`2.1560400
`1.4444546
`0.1650000
`axis Z of thelens.
`e(3)
`3.2340801
`2.1668202
`0.2475000
`The thicknesses of the various lens elements mea-
`p(4)
`3.636080!
`2.4760010
`0.3300000
`p(5)
`_
`-
`0.4010000
`sured along the optical axis Z follow:
`p(6)
`_
`_
`0.5000000
`0.199F <1,<0.202F
`M(0)
`0.3644990
`0.6859612
`—1.5916510
`M(1)
`0.3596961
`0.6935749
`0.5983017
`0.399F<1,.<0.402F
`M(2)
`0.4358774
`0.6937277
`0.7436587
`0.399F<2,;<0.402F
`M(3)
`—0.6375478
`—1.4543979
`0.6856855
`M(4)
`~2.1376087
`—0.6921574
`—0.5200549
`0.047F<2,<0.049F
`M(5)
`-
`~
`0.0000000
`0.186F<¢,<0.189F
`M(6)
`_
`_
`0.0000000
`0.019F<2,<0.022F
`0.06F<?,9<0.07F
`0.08F<t,.<0.09F
`The thicknesses of the air gaps measuredalong the
`optical axis Z follow:
`3.248F<t,<3.251F
`0.0004F<1,<0.0014F
`0.266F<t,,<0.267F
`0.037F<2,<0.041F
`As previously noted, the surfaces R,, R3, and Ry; are
`neither spherical nor planar. Furthermore, not one of
`them fits any single known mathematical formula. They
`are defined in terms of splines, that is each surface is
`broken up into increments or intervals which are de-
`fined separately. The surfaces R,, Rg and R,3 are con-
`sidered spline surfaces and are defined by the following
`cubic spline equation:
`
`0
`
`20
`
`25
`
`35
`
`(pce)?
`S(p)=Mi-n
`M,
`6 hy
`+ («-
`)
`Miyh?
`
`6
`
`-
`
`(emp ie. y
`6 i
`hy
`(pip)
`
`+ («-
`
`Mae
`
`6 Verran
`
`where
`fii == The value of the spline surface height at the
`start of the ith interval.
`Pp; = The value of the spline surface height at the end
`of the ith interval.
`X11 = The value of the spline surface sag at the start
`of the ith interval.
`X;= The value of the spline surface sag at the end of
`the ith interval.
`hy = 0; — Pia = The length of the ith interval.
`M,.; = The value of the slope derivative at the start of
`the ith interval.
`M; = The value of the slope derivative at the end of
`the ith interval.
`.
`p= Thespline surface height (independentvariable)
`S(p) = Thespline surface sag as a function of height
`(dependentvariable)
`,
`The slope of a spline surface elementat a particular
`height (p) is given by
`
`Slope = d S(p)/dp
`
`The values of spline surface sag (X), spline surface
`height (p), and slope derivative (M) for various spline
`intervals 0, 1, 2, etc. follow:
`
`40
`
`45
`
`50
`
`.
`
`55
`
`60
`
`65
`
`Whatis claimedis:
`1. A non-linear lens comprising first lens means for
`distorting a scene in the field of view for the lens such
`that objects in the vicinity of the optical axis are given
`substantially greater prominence than objects in the
`peripheral region of the field of view, and second lens
`means for forming a real image of the scene as distorted
`by the first lens means, whereby objects in the vicinity
`of the optical axis will occupy a disproportionately
`large area of the real image and objects in peripheral
`regions of the scene will occupy a disproportionately
`small area of the real image.
`2. A lens according to claim 1 wherein the field of
`view is at least approximately 160°.
`3. A non-linear lens according to claim 1 and further
`characterized by third lens means for causing the real
`image formed by the second lens meansto lie in a
`plane.
`4. A non-linear lens according to claim 1 wherein the
`distortion in the real image approximates the formula
`H=sin 36
`
`whereH is the distance in the image measured from the
`optical axis and @ is the angle measured from the opti-
`cal axis.
`5. A non-linear lens according to claim 1 wherein the
`first lens means comprisesa plurality of individual lens
`elements and the second lens meansincludesa plurality
`of different lens elements.
`6. A non-linear lens comprising a first lens grouping
`for distorting a scene in the field of view for the lens
`such that objects in the vicinity of the optical axis are
`given greater prominencethan objects in the peripheral
`region of the field of view, the first lens grouping in-
`cluding first, second, and third lens elements, thefirst
`lens element having surfaces R, and Rg, the second lens
`element having surfaces R, and Rs, and the third lens
`element having surfaces Rs; and Ry, the surface R, of
`the first lens element matching the surface R, of the
`second lens element and being substantially in contact
`therewith, the surface R, of the second lens element
`matching the surface R, of the third lens element and
`being substantially in contact therewith,the surfaces R,
`and R; being curved at the optical axis and being non-
`spherical beyond the optical axis, the surfaces R, and
`R, being spherical substantially throughout, the radii of
`
`8
`
`

`

`9
`the surfaces along the optical axis being substantially as
`follows:
`
`10
`wherein the value for p at splineintervals 0,1, 2,etc. is
`
`3,953,111
`
`1.4F
`
`L.O9LF
`
`:
`<R,
`Rs:
`<R,
`
`1.3
`45
`0.729
`1.092
`
`<
`
`<
`
`F
`F
`F
`F
`
`p(0)
`p(t)
`(2)
`p(3)
`(4)
`
`0.0000000
`|.0780200
`2.1560400
`3.2340801
`3.6360801
`
`0.0000000
`0.7222734
`1.4444546
`2.1668202
`2.4760010;
`
`whereFis the focal length of the lens along the optical
`and wherein the value for M at spline intervals 0, 1, 2,
`10
`etc. is
`axis of the lens; and a second lens grouping for forming
`a real image ofthe sceneas distorted by thefirst group-
`ing, whereby objects in the vicinity of the optical axis
`will occupy a disproportionately large area of the real
`image and objects in peripheral regions of the scene
`will occupy a disproportionately small area of the real
`image.
`7. A lens according to claim 6 wherein the first, sec-
`ond, and third lens elements have thicknesses t,, t, and
`ts, respectively, along the optical axis of the lens and
`the thicknesses are as follows:
`0.199F<1,<0.202F
`0.399F<1,<0.402F
`0.399F<t,<0.402F.
`8. A non-linear lens according to claim 6 wherein the
`non-spherical surfaces R, and R3 are defined by the
`cubic spline equation
`
`20
`
`(pi-e)*
`(p-
`p11)?
`SPM oytM,
`+ (x-
`wa : )
`hs, (x 6 Vier
`M,.h?_\
`(orp)
`
`
`Mie
`
`where
`i-l = The value of the spline surface height at the
`start of the ith interval.
`i= The value of the spline surface height at the end
`of the ith interval.
`X;., = The value of the spline surface sag at the start
`of the ith interval,
`X; = The value of the spline surface sag at the end of
`the ith interval.
`hy = p; — pi, = The length of the ith interval.
`M.., = The value of the slope derivative at the start of
`the ith interval.
`M, = The value of the slope derivative at the end of
`the ith interval.
`p= Thespline surface height (independentvariable)
`S(p) = Thespline surface sag as a function of height
`(dependentvariable); and wherein the value for X
`at spline intervals 0, 1, 2, etc. is
`
`X(0)
`X(1)
`X(2)
`X(3)
`X(4)
`
`Surface
`R;
`0.0000000
`0.2108668
`0.8554116
`1.7838662
`1.9775600
`
`Surface
`Rs
`0.0000000
`0.1795877
`0.7202623
`1.4361922
`1.6027882;
`
`30
`
`35
`
`40
`
`45
`
`50
`
`35
`
`60
`
`65
`
`M(0)
`M(t)
`M(2)
`M(3)
`M(4)
`
`0.3644990
`0.359696 1
`0.4358774
`—0.6375478
`—2.1376087
`
`0.6859612
`0.6935749
`0.6937277
`~1.4543979
`-0.6921574.
`
`9. A non-linear lens according to claim 6 wherein the
`second lens grouping comprises a convex-concavefirst
`lens element, a double convex second lens element, a
`double concave third lens element, and a double con-
`vex fourth lens element, arranged in that order from
`the first lens grouping, the first lens element having a
`thickness t; along the optical axis and spherical sur-
`faces Rs and Rg, the second lens element having a
`thickness f; and spherical surfaces R; and Rg, the third
`lens element having a thickness t9 and spherical sur-
`faces Rg and Ryo, and the fourth lens element having a
`thickness ¢;)9 and spherical surfaces Ryo and R,,, the
`surface Ryo of the third lens element matching the sur-
`face Ry of the fourth lens element and being substan-
`tially in contact therewith, the surfaces Rg and R; being
`separated by a distance tg along the optical axis and the
`surfaces Rg and Ry being separated by a distance ft,
`along the optical axis; wherein the radii of curvature for
`the surfaces are:
`1.