# US 8,786,521 B2 · Representation system

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- Number: US 8,786,521 B2
- Document ID: US8786521B2
- Status: Granted
- Kind: B2
- Application number: 12/665,078
- Filed: 2008-06-25
- Published: 2010-07-22
- Granted: 2014-07-22
- Earliest claimed: 2007-06-25
- Assignees: Giesecke & Devrient GmbH
- Applicants: Wittich Kaule; Michael Rahm; Wolfgang Rauscher
- Inventors: Wittich Kaule; Michael Rahm; Wolfgang Rauscher
- Web page: https://patentgrep.com/patent/US8786521B2

## CPC

- B42D 25/342: Moiré effects
- B42D 25/00: Information-bearing cards or sheet-like structures characterised by identification or security features; Manufacture thereof (printing processes to produce identification or security features B41M3/14)
- B42D 25/21: for multiple purposes
- B42D 25/29: Securities; Bank notes
- G02B 27/60: Systems using moiré fringes
- B42D 25/324: Reliefs
- B42D 2033/24
- B42D 2035/20
- Y10T 29/49: Method of mechanical manufacture
- Y10T 29/49826: Assembling or joining
- B42D: BOOKS; BOOK COVERS; LOOSE LEAVES; PRINTED MATTER CHARACTERISED BY IDENTIFICATION OR SECURITY FEATURES; PRINTED MATTER OF SPECIAL FORMAT OR STYLE NOT OTHERWISE PROVIDED FOR; DEVICES FOR USE THEREWITH AND NOT OTHERWISE PROVIDED FOR; MOVABLE-STRIP WRITING OR READING APPARATUS
- G02B: OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- Y10T: TECHNICAL SUBJECTS COVERED BY FORMER US CLASSIFICATION

## Abstract

The present invention relates to a depiction arrangement for security papers, value documents, electronic display devices or other data carriers, having a raster image arrangement for depicting a planar target image that is given by an image function f(x,y), having a motif image that is subdivided into a plurality of cells (24), in each of which are arranged imaged regions of the target image a viewing grid (22) composed of a plurality of viewing elements for reconstructing the target image when the motif image is viewed with the aid of the viewing grid (22), the motif image exhibiting, with its subdivision into a plurality of cells, an image function m(x,y).

## Claims

### Claim 1

A security element for security papers, value documents, or other non-transitory data carriers, the security element comprising:
(A) a motif layer including a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of a planar target image defined by an image function f(x,y),
(B) a viewing grid composed of a plurality of viewing elements for reconstructing the target image when the motif image is viewed with the aid of the viewing grid,
the motif image of the motif layer having an image function m(x,y) that is given by
m
⁡
(
x
,
y
)
=
f
⁡
(
(
x
y
)
+
(
A
⁡
(
x
,
y
)
-
I
)
·
(
(
(
(
x
y
)
+
w
d
⁡
(
x
,
y
)
)
⁢
mod
⁢
⁢
W
)
-
w
d
⁡
(
x
,
y
)
-
w
c
⁡
(
x
,
y
)
)
)
·
g
⁡
(
x
,
y
)
⁢
⁢
⁢
where
⁢
⁢
w
d
⁡
(
x
,
y
)
=
W
·
(
d
1
⁡
(
x
,
y
)
d
2
⁡
(
x
,
y
)
)
⁢
⁢
and
⁢
⁢
w
c
⁡
(
x
,
y
)
=
W
·
(
c
1
⁡
(
x
,
y
)
c
2
⁡
(
x
,
y
)
)
,
such that the planar target image is reconstructed when the motif image of the motif layer is viewed through the viewing grid;
wherein
a unit cell of the viewing grid being described by lattice cell vectors
w
1
=
(
w
11
w
21
)
⁢
⁢
and
⁢
⁢
w
2
=
(
w
12
w
22
)
and combined in the matrix
W
=
(
w
11
w
12
w
21
w
22
)
,
the matrix
A
⁡
(
x
,
y
)
=
(
a
11
⁡
(
x
,
y
)
a
12
⁡
(
x
,
y
)
a
21
⁡
(
x
,
y
)
a
22
⁡
(
x
,
y
)
)
describes a desired magnification and movement behavior of the reconstructed planar target image,
the vector (c1(x,y), c2(x,y)), where 0≦c1(x,y),c2(x,y)<1, indicates position of the center of the viewing elements relative to the cells of the motif image of the motif layer,
the vector (d1(x,y), d2(x,y)), where 0≦d1(x,y), d2(x, y)<1, represents a location-dependent displacement of cell boundaries of the plurality of cells in the motif image of the motif layer,
g(x,y) is a mask function for adjusting visibility of the reconstructed planar target image, and
wherein the reconstructed planar target image either exhibits no periodicity, or exhibits a periodicity that is described by a 2×2 matrix P, wherein the periodicity unit cell is not equal to (A−I) W.

### Claim 2 (depends on 1)

The security element according to claim 1, characterized in that the plurality of viewing elements are arranged periodically or locally periodically,
wherein if arranged locally periodically, the local period parameters either
changing, in relation to the periodicity length, only slowly, or
an abrupt change in the periodicity length occurring if it was previously kept constant or nearly constant over a segment that is large compared with the periodicity length.

### Claim 3 (depends on 1)

The security element according to claim 1, characterized in that a periodicity length or a local periodicity length of the viewing elements is between 3 μm and 50 μm, preferably between 5 μm and 30 μm, particularly preferably between about 10 μm and about 20 μm.

### Claim 4 (depends on 1)

The security element according to claim 1, characterized in that the viewing elements are formed by non-cylindrical microlenses or concave microreflectors, especially by microlenses or concave microreflectors having a circular or polygonally delimited base area.

### Claim 5 (depends on 1)

The security element according claim 1, characterized in that the viewing elements are formed by elongated cylindrical lenses or concave cylindrical reflectors whose dimension in the longitudinal direction measures more than 250 μm, preferably more than 300 μm, particularly preferably more than 500 μm and especially more than 1 mm.

### Claim 6 (depends on 1)

The security element according to claim 1, characterized in that the viewing elements are formed by circular apertures, slit apertures, circular or slit apertures provided with reflectors, aspherical lenses, Fresnel lenses, GRIN (Gradient Refractive Index) lenses, zone plates, holographic lenses, concave reflectors, Fresnel reflectors, zone reflectors or other elements having a focusing or also masking effect.

### Claim 7 (depends on 1)

The security element according to claim 1, characterized in that a support of the image function
f
⁡
(
(
A
-
I
)
·
(
x
y
)
)
is greater than the unit cell of the viewing grid W.

### Claim 8 (depends on 1)

The security element according to claim 1, characterized in that the viewing grid and the motif layer are firmly joined together to form the security element having a stacked, spaced-apart viewing grid and motif layer.

### Claim 9 (depends on 8)

The security element according to claim 8, characterized in that the motif layer and the viewing grid are arranged at opposing surfaces of an optical spacing layer.

### Claim 10 (depends on 8)

The security element according to claim 8, characterized in that the security element is a security thread, a tear strip, a security band, a security strip, a patch or a label for application to a security paper, value document or the like.

### Claim 11 (depends on 10)

The security element according to claim 10, characterized in that the magnification and movement matrix A is given by
A
=
(
a
11
a
12
a
21
a
22
)
=
(
v
1
·
cos
⁢
⁢
ϕ
1
v
2
·
cos
⁢
⁢
ϕ
2
v
1
·
sin
⁢
⁢
ϕ
1
v
2
·
sin
⁢
⁢
ϕ
2
)
such that the reconstructed planar target image, upon tilting the security element horizontally about a vertical axis, moves with a speed v1 in the direction φ1 in relation to the x-axis, and upon tilting the security element vertically about a horizontal axis, moves with the speed v2 in the direction φ2 in relation to the x-axis.

### Claim 12 (depends on 11)

The security element according to claim 11, characterized in that φ1 is not equal to 0° and not equal to 180° and/or φ2 is not equal to 90° and not equal to −90° and/or v1 is not equal to v2.

### Claim 13 (depends on 11)

The security element according to claim 11, characterized in that, upon tilting the security element, the reconstructed planar target image always moves in the same direction, irrespective of the tilt direction.

### Claim 14 (depends on 8)

The security element according to claim 8, characterized in that the total thickness of the security element is below 50 μm, preferably below 30 μm and particularly preferably below 20 μm.

### Claim 15 (depends on 8)

The security element according to claim 8, characterized in that, in the magnification and movement matrix A, a11=z1/e and a22=z2/e, where e denotes the effective distance of the viewing grid from the motif image of the motif layer such that, upon viewing with an eye separation being in the x-direction, the reconstructed planar target image is visible at a depth or height z1, and when the security element, rotated by 90°, is viewed with the eye separation being in the y-direction, is visible at a depth or height z2.

### Claim 16 (depends on 15)

The security element according to claim 15, characterized in that, in the magnification and movement matrix A,
a11=z1/e, a21=(z1/e)tan φ1
a22=z2/e, a12=(z2/e)cot φ2
such that,
upon viewing with the eye separation being in the x-direction, the reconstructed planar target image is visible at a depth or height z1 and, upon tilting the security element horizontally or upon a horizontal change in the viewing direction, moves in the direction φ1 in relation to the x-axis, and
when the security element, rotated by 90°, is viewed with the eye separation being in the y-direction, the reconstructed planar target image is visible at a depth or height z2 and, upon tilting the security element vertically or upon a vertical change in the viewing direction, moves in the direction φ2 in relation to the x-axis.

### Claim 17 (depends on 15)

The security element according to claim 15, characterized in that one or more of the magnifications z1, z2 and movement directions φ1, φ2 are location dependent.

### Claim 18 (depends on 8)

The security element according to claim 8, characterized in that the viewing grid is a slot grid, cylindrical lens grid or cylindrical concave reflector grid whose unit cell is given by
W
=
(
cos
⁢
⁢
γ
-
sin
⁢
⁢
γ
sin
⁢
⁢
γ
cos
⁢
⁢
γ
)
·
(
d
0
0
∞
)
where d is the slot or cylinder axis distance and the direction of the slot or cylinder axis is specified by an angle γ, and in that the magnification and movement matrix A is given by
A
=
(
cos
⁢
⁢
γ
-
sin
⁢
⁢
γ
sin
⁢
⁢
γ
cos
⁢
⁢
γ
)
·
(
a
11
0
a
21
1
)
·
(
cos
⁢
⁢
γ
sin
⁢
⁢
γ
-
sin
⁢
⁢
γ
cos
⁢
⁢
γ
)
where
⁢
⁢
(
a
11
0
a
21
1
)
=
(
v
1
·
cos
⁢
⁢
ϕ
1
0
v
1
·
sin
⁢
⁢
ϕ
1
1
)
=
(
z
1
e
0
z
1
e
·
tan
⁢
⁢
ϕ
1
1
)
,
where φ1 is a prechosen movement direction.

### Claim 19 (depends on 1)

The security element according to claim 1, characterized in that the viewing grid and the motif layer are arranged at different positions of a non-transitory data carrier such that the viewing grid and the motif layer are stackable for self-authentication and form the security element in the stacked state.

### Claim 20 (depends on 19)

The security element according to claim 19, characterized in that the viewing grid and the motif layer are stackable by bending, creasing, buckling or folding the non-transitory data carrier.

### Claim 21 (depends on 1)

The security element according to claim 1, characterized in that the magnification and movement matrix A is location independent.

### Claim 22 (depends on 1)

The security element according to claim 1, characterized in that the cell boundaries in the motif image of the motif layer are not displaced location dependently and in that the vector (d1(x,y), d2(x,y)) occurring in the image function m(x,y) is constant.

### Claim 23 (depends on 1)

The security element according to claim 1, characterized in that the cell boundaries in the motif image of the motif layer are displaced location dependently, preferably in that the motif image of the motif layer exhibits two or more subregions having a different, in each case constant, cell grid.

### Claim 24 (depends on 1)

The security element according to claim 1, characterized in that the mask function g is identical to 1.

### Claim 25 (depends on 1)

The security element according to claim 1, characterized in that the mask function g is zero in subregions, especially in edge regions of the cells of the motif image of the motif layer, and in this way describes an angle limit when the reconstructed planar target image is viewed, or an image field limit in which the reconstructed planar target image does not become visible.

### Claim 26 (depends on 1)

The security element according to claim 1, characterized in that the relative position of the center of the viewing elements is location independent within the cells of the motif image of the motif layer, and the vector (c1(x,y), c2(x,y)) is constant.

### Claim 27 (depends on 1)

The security element according to claim 1, characterized in that the relative position of the center of the viewing elements is location dependent within the cells of the motif image of the motif layer.

### Claim 28 (depends on 1)

The security element according to claim 1, characterized in that the motif image is present in a printing layer or an embossed layer.

### Claim 29 (depends on 1)

The security element according to claim 1, characterized in that the image contents of individual cells of the motif image of the motif layer are interchanged according to the definition of the image function m(x,y).

### Claim 30 (depends on 1)

A method for manufacturing a security element according to claim 1, in which is calculated, from the planar target image that is given by the image function f(x,y), the motif image of the motif layer that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the planar target image,
the motif image having an image function m(x,y) that is calculated by
m
⁡
(
x
,
y
)
=
f
⁡
(
(
x
y
)
+
(
A
⁡
(
x
,
y
)
-
I
)
⁢
(
(
(
(
x
y
)
+
w
d
⁡
(
x
,
y
)
)
⁢
mod
⁢
W
)
-
w
d
⁡
(
x
,
y
)
-
w
c
⁡
(
x
,
y
)
)
)
·
g
⁡
(
x
,
y
)
⁢
where
⁢
⁢
w
d
⁡
(
x
,
y
)
=
W
·
(
d
1
⁡
(
x
,
y
)
d
2
⁡
(
x
,
y
)
)
⁢
⁢
and
⁢
⁢
w
c
⁡
(
x
,
y
)
=
W
·
(
c
1
⁡
(
x
,
y
)
c
2
⁡
(
x
,
y
)
)
⁢
such that the planar target image is reconstructed upon viewing the motif image with the aid of the viewing grid,
wherein
the unit cell of the viewing grid is described by lattice cell vectors
w
1
=
(
w
11
w
21
)
⁢
⁢
and
⁢
⁢
w
2
=
(
w
12
w
22
)
and combined in the matrix
W
=
(
w
11
w
12
w
21
w
22
)
,
the matrix
A
⁡
(
x
,
y
)
=
(
a
11
⁡
(
x
,
y
)
a
12
⁡
(
x
,
y
)
a
21
⁡
(
x
,
y
)
a
22
⁡
(
x
,
y
)
)
describes a desired magnification and movement behavior of the reconstructed planar target image upon tilting the security element,
the vector (c1(x,y), c2(x,y)), where 0≦c1(x,y), c2(x,y)<1, indicates the relative position of the center of the viewing elements within the cells of the motif image of the motif layer,
the vector (d1(x,y), d2(x,y)), where 0≦d1(x,y), d 2(x,y)<1, represents a location-dependent displacement of the cell boundaries in the motif image of the motif layer,
g(x,y) is a mask function for adjusting the visibility of the reconstructed planar target image, and
wherein the reconstructed planar target image either exhibits no periodicity, or exhibits a periodicity that is described by a 2×2 matrix P, wherein the periodicity unit cell is not equal to (A−I) W.

### Claim 31 (depends on 30)

The method according to claim 30, characterized in that the viewing grid and the motif layer are firmly joined together to form the security element having a stacked, spaced-apart viewing grid and motif layer.

### Claim 32 (depends on 30)

The method according to claim 30, characterized in that the viewing grid and the motif layer are arranged at different positions of a non-transitory data carrier such that the viewing grid and the motif layer are stackable for self-authentication and form the security element in the stacked state.

### Claim 33 (depends on 1)

A security paper for manufacturing security or value documents, such as banknotes, checks, identification cards, certificates or the like, having the security element according to claim 1.

### Claim 34 (depends on 1)

A non-transitory data carrier, especially a branded article, value document, decorative article or the like, having the security element according claim 1.

### Claim 35 (depends on 34)

The non-transitory data carrier according to claim 34, characterized in that the viewing grid and/or the motif layer of the security element is arranged in a window region of the non-transitory data carrier.

### Claim 36

A security element for security papers, value documents, or other non-transitory data carriers, the security element comprising:
(A) a motif layer having a motif image that is subdivided into a plurality of cells, in each of which are arranged imaged regions of a plurality of planar target images that are given by image functions fi(x, y), i=1, 2, . . . n, where n≧1,
(B) a viewing grid composed of a plurality of viewing elements for reconstructing the planar target images when the motif layer is viewed with the aid of the viewing grid,
the motif image of the motif layer having, an image function m(x, y) that is given by m(x,y)=F(h1, h2, . . . hn), having the describing functions
h
i
⁡
(
x
,
y
)
=
f
i
⁡
(
(
x
y
)
+
(
A
i
⁡
(
x
,
y
)
-
I
)
·
(
(
(
(
x
y
)
+
w
di
⁡
(
x
,
y
)
)
⁢
mod
⁢
W
)
-
w
di
⁡
(
x
,
y
)
-
w
ci
⁡
(
x
,
y
)
)
)
·
g
i
⁡
(
x
,
y
)
⁢
where
⁢
⁢
w
di
⁡
(
x
,
y
)
=
W
·
(
d
i
⁢
⁢
1
⁡
(
x
,
y
)
d
i
⁢
⁢
2
⁡
(
x
,
y
)
)
⁢
⁢
and
⁢
⁢
w
ci
⁡
(
x
,
y
)
=
W
·
(
c
i
⁢
⁢
1
⁡
(
x
,
y
)
c
i
⁢
⁢
2
⁡
(
x
,
y
)
)
,
such that the plurality of planar target images are reconstructed when the motif image is viewed through the viewing grid;
wherein:
F(h1, h2, . . . hn) is a master function that indicates an operation on the n describing functions hi(x, y),
a unit cell of the viewing grid being described by lattice cell vectors
w
1
=
(
w
11
w
21
)
⁢
⁢
and
⁢
⁢
w
2
=
(
w
12
w
22
)
and combined in the matrix
W
=
(
w
11
w
12
w
21
w
22
)
,
the matrices
A
i
⁡
(
x
,
y
)
=
(
a
i
⁢
⁢
11
⁡
(
x
,
y
)
a
i
⁢
⁢
12
⁡
(
x
,
y
)
a
i
⁢
⁢
21
⁡
(
x
,
y
)
a
i
⁢
⁢
22
⁡
(
x
,
y
)
)
each describe a desired magnification and movement behavior for the reconstructed planar target image fi,
the vectors (ci1(x, y), ci2(x, y)), where 0≦ci1(x, y), ci2(x, y)<1, indicate in each case, for the target image fi, the relative position of the center of the viewing elements within the cells i of the motif layer,
the vectors (di1(x, y), di2(x, y)), where 0≦di1(x, y), di2(x, y)<1, each represent a location-dependent displacement of the cell boundaries in the motif image of the motif layer,
gi(x, y) are mask functions for adjusting the visibility of the reconstructed planar target image fi, and
wherein at least one of the reconstructed planar target images either exhibits no periodicity, or exhibits a periodicity that is described by a 2×2 matrix P, wherein the periodicity unit cell is not equal to (A−I) W.

### Claim 37 (depends on 36)

The security element according to claim 36, characterized in that at least one of the describing functions hi(x, y) is designed according to an image function m(x, y) that is given by
m
⁡
(
x
,
y
)
=
f
⁡
(
(
x
y
)
+
(
A
⁡
(
x
,
y
)
-
I
)
⁢
(
(
(
(
x
y
)
+
w
d
⁡
(
x
,
y
)
)
⁢
mod
⁢
W
)
-
w
d
⁡
(
x
,
y
)
-
w
c
⁡
(
x
,
y
)
)
)
·
g
⁡
(
x
,
y
)
⁢
where
⁢
⁢
w
d
⁡
(
x
,
y
)
=
W
·
(
d
1
⁡
(
x
,
y
)
d
2
⁡
(
x
,
y
)
)
⁢
⁢
and
⁢
⁢
w
c
⁡
(
x
,
y
)
=
W
·
(
c
1
⁡
(
x
,
y
)
c
2
⁡
(
x
,
y
)
)
.
⁢

### Claim 38 (depends on 36)

The security element according to claim 36, characterized in that the security element depicts an alternating image, a motion image or a morph image.

### Claim 39 (depends on 36)

The security element according to claim 36, characterized in that the mask functions gi define a strip-like or checkerboard-like alternation of the visibility of the reconstructed planar target images fi.

### Claim 40 (depends on 36)

The security element according to claim 36, characterized in that the master function F constitutes the sum function.

### Claim 41 (depends on 36)

The security element according to claim 36, characterized in that two or more reconstructed planar target images fi are visible simultaneously.

### Claim 42 (depends on 41)

The security element according to claim 41, characterized in that the master function F constitutes the sum function, the maximum function, an OR function, an XOR function or another logic function.

### Claim 43 (depends on 41)

The security element according to claim 41, characterized in that n reconstructed planar target images fj(x,y), j=1, . . . n are specified that each lie at an apparent depth zj, where zj>zj-1, and in that, as the master function F, a modified minimum function is chosen through which, in the event that, at a position (x,y), the describing functions hj are not equal to zero for multiple j, the smallest j is selected.

### Claim 44 (depends on 36)

The security element according to claim 36, characterized in that the viewing grid and the motif layer are firmly joined together to form the security element having a stacked, spaced-apart viewing grid and motif layer.

### Claim 45 (depends on 44)

The security element according to claim 44, characterized in that the motif layer and the viewing grid are arranged at opposing surfaces of an optical spacing layer.

### Claim 46 (depends on 44)

The security element according to claim 44, characterized in that the security element is a security thread, a tear strip, a security band, a security strip, a patch or a label for application to a security paper, value document or the like.

### Claim 47 (depends on 44)

The security element according to claim 44, characterized in that the total thickness of the security element is below 50 μm, preferably below 30 μm and particularly preferably below 20 μm.

### Claim 48 (depends on 36)

The security element according to claim 36, characterized in that the viewing grid and the motif layer are arranged at different positions of a non-transitory data carrier such that the viewing grid and the motif layer are stackable for self-authentication and form the security element in the stacked state.

### Claim 49 (depends on 48)

The security element according to claim 48, characterized in that the viewing grid and the motif layer are stackable by bending, creasing, buckling or folding the non-transitory data carrier.

### Claim 50 (depends on 36)

A method for manufacturing a security element according to claim 36, in which is calculated, from the plurality of planar target images that are given by image functions fi(x,y), i=1, 2, . . . n, where n≧1, a motif image of the motif layer that is subdivided into a plurality of cells, in each of which are arranged imaged regions of the planar target images,
the motif image having an image function m(x,y) that is calculated by
m
⁡
(
x
,
y
)
=
F
⁡
(
h
1
,
h
2
,
…
⁢
⁢
h
n
)
⁢
having
⁢
⁢
the
⁢
⁢
describing
⁢
⁢
functions
⁢
h
i
⁡
(
x
,
y
)
⁢
f
i
(
(
x
y
)
+
(
A
i
⁡
(
x
,
y
)
-
I
)
·
(
(
(
(
x
y
)
+
w
di
⁡
(
x
,
y
)
)
⁢
mod
⁢
W
)
-
w
di
⁡
(
x
,
y
)
-
w
ci
⁡
(
x
,
y
)
)
·
g
i
⁡
(
x
,
y
)
⁢
⁢
⁢
where
⁢
⁢
w
di
⁡
(
x
,
y
)
=
W
·
(
d
i
⁢
⁢
1
⁡
(
x
,
y
)
d
i
⁢
⁢
2
⁡
(
x
,
y
)
)
⁢
⁢
and
⁢
⁢
⁢
w
ci
⁡
(
x
,
y
)
=
W
·
(
c
i
⁢
⁢
1
⁡
(
x
,
y
)
c
i
⁢
⁢
2
⁡
(
x
,
y
)
)
,
⁢
such that the reconstructed planar target images are reconstructed upon viewing the motif image with the aid of the viewing grid,
wherein
F(h1, h2, . . . hn) is a master function that indicates an operation on the n describing functions hi(x, y),
the unit cell of the viewing grid is described by lattice cell vectors
w
1
=
(
w
11
w
21
)
⁢
⁢
and
⁢
⁢
w
2
=
(
w
12
w
22
)
⁢
and combined in the matrix W=
W
=
(
w
11
w
12
w
21
w
22
)
,
the matrices
A
i
⁡
(
x
,
y
)
=
(
a
i
⁢
⁢
11
⁡
(
x
,
y
)
a
i
⁢
⁢
12
⁡
(
x
,
y
)
a
i
⁢
⁢
21
⁡
(
x
,
y
)
a
i
⁢
⁢
22
⁡
(
x
,
y
)
)
each describe a desired magnification and movement behavior for the reconstructed planar target images fi,
the vectors (ci1(x,y), ci2(x,y)), where 0≦ci1(x,y), ci2(x, y)<1, indicate in each case, for the reconstructed planar target image fi, the relative position of the center of the viewing elements within the cells i of the motif image of the motif layer,
the vectors (di1(x,y), di2(x,y)), where 0≦di1(x,y), di2(x,y)<1, each represent a location-dependent displacement of the cell boundaries in the motif image of the motif layer,
gi(x, y) are mask functions for adjusting the visibility of the reconstructed planar target images fi, and
wherein at least one of the reconstructed planar target images either exhibits no periodicity, or exhibits a periodicity that is described by a 2×2 matrix P, wherein the periodicity unit cell is not equal to (A−I) W.