Symmetries of 2-D Images: cases without periodic translations Lewis D Griffin Computer Science University College London London WC1E 6BT, UK The different ways in which images, defined as scalar functions of the Euclidean plane, can be symmetrical is considered. The symmetries found are relative to the class of image isometries, each of which is a spatial and an intensity isometry combined. All symmetry types, apart from those with periodic translations, are derived. Fifteen such types are found, including one that has not previously been reported. The novel type occurs when an image has a continuous line of centres of symmetry each like the one found in the Taiji (Yin-Yang) symbol.
1. Introduction In [1] a class of transformations, called image isometries, applicable to scalar functions on Euclidean domains was introduced. Each image isometry is an ordered pair of a domain isometry and a range isometry. In the same paper, the possible automorphism groups of 1-D images, relative to the class of image isometries, were determined. In this paper, we extend the analysis to 2-D images; but limit the scope by not considering cases where the projection of the automorphism groups onto its domain components results in a periodic group. The paper is organized as follows. In sections 2 and 3 linear and planar isometries and their groups will be reviewed. This is standard material that can be found in many works. We recommend [2]. In section 4 image-isometries, as introduced in [1] and [3] will be defined. In section 5, the automorphism groups (excluding periodic cases) of 2-D images, relative to the class of image isometries, will be determined and presented. We will use the following notations: • E for the Euclidean line, E 2 for the Euclidean plane, L for the lines in E 2 . • Points of the Euclidean plane will be written in bold; p ∈ E 2 will always be such a point. • Lines of the Euclidean plane will be written in bold with a bar above; l ∈ L will always be such a line. • The existence of a fiducial point 0 ∈ E 2 is assumed. • The existence of a fiducial line 0 ∈ 0 ∈ L through the origin is assumed. • The composition of a transformation B followed by a transformation A is written AB. • n will always be a natural number ( n ∈ ); unrestricted unless stated otherwise. • Unspecified variables can be assumed to be real numbers.
We will encounter the following groups, which we define now for convenience: • Z n is the cyclic group of n elements. Defined as the unique groups of that size
•
generated by a single element. Z1 = {e} is the trivial group. e will be used throughout for the identity transformation, context determining of which class. Z ≅ , + the infinite cyclic group. ≅ denotes isomorphism.
•
Dihn is the nth dihedral group. It has 2n elements and is the automorphism group of
the cyclic graph with n vertices. N.B. Dih1 ≅ Z 2 • • •
S 1 - the circle group, isomorphic to the group of 2D rotations about a point. O( 2 ) - the orthogonal group of 2-by-2 orthogonal matrices.
, + - the additive group of the reals.
2. Line Isometries Line isometries are distance-preserving maps from the Euclidean line ( E ) to itself. Two types only are possible — translations Tλ : x x + λ and reflections M λ : x 2λ − x — and there
is a 1-D family of each. The identity isometry ( e : x x ) is a special case translation T0 = e . Linear isometries compose according to: Tλ Tµ = Tλ + µ , M λ M µ = T2( λ − µ ) , Tλ M µ = M µ + λ 2 , and
M λ Tµ = M λ − µ 2 . Translations, other than the identity, have no fixed points; each reflection ( M λ ) has a unique single fixed point ( x = λ ). I will write M := M 0 , so M ( x ) = − x . The following will be useful later:
Remark 1 (from [4]). Points of agreement of distinct linear isometries A distinct pair of line isometries either transform no point the same, if the isometries are of the same type, or a single point the same, if they are of different type. There are six types of group of line isometries. We list these and note some alternative naming conventions and isomorphisms with common abstract groups: • the trivial group {e} ≅ Z1 ,
•
a simple reflection D1 ≅ {e, M } ≅ Z 2 ,
•
the translation group E + (1) = T (1) = {Tλ | λ ∈ } ,
•
a periodic translation group F1 ≅ {Tz | z ∈ } ,
•
a periodic translating and reflecting group F4 ≅ {T2 z , M z | z ∈ } ≅ Z Z 2 , and
•
the full Euclidean group E (1) = {T2 λ , M λ | λ ∈ } .
We will write Γ(1) for the set of possible groups of linear isometries.
3. Planar Isometries Planar isometries are distance-preserving maps from the Euclidean plane ( E 2 ) to itself. They are provably of four types only: translations, rotations, reflections and glide reflections [2]. The identity can be regarded as a special case translation, or as a special case rotation. We will write: T
•
Tx , y for translation by the vector ( x
y) .
•
Rp ,θ for rotation by an angle θ about the centre p .
•
M ( l ) for reflection in the line l .
We do not use any special notation for glide reflections, expressing them when needed as compositions of the other types. For convenience we will use the following abbreviations: • Tx := Tx ,0 •
Rθ := R0,θ
•
M := M ( 0 )
•
M θ := Rθ M R−θ
•
N = Mπ 2
λ
• M := Tλ M T− λ The four types of isometry differ in their patterns of fixed points. The identity fixes everything, glide reflections and non-identical translations fix nothing, rotations fix a single point, and reflections fix a line of points. Groups of planar isometries may be distinguished by the fixed points common to all their elements. If the group contains any translations or glide reflections there will be no common fixed points. Groups that consist of reflections and rotations only, known as Rosette Groups, can and do have fixed points. It can be shown that any rotations in a Rosette group must be about a common centre, and the mirror lines of any reflections must intersect such a centre. There are two types of Rosette group. The Cyclic groups
{
}
Cn ≅ Rm 2 π | 0 ≤ m < n , n ≥ 2 n
of rotations only, and the Dihedral groups which contain reflections and an equal number of rotations (when counting the identity as a rotation):
{
}
Dn ≅ Rm 2π , M m π | 0 ≤ m < n n
n
Since the Rosette Groups exhaust the possibilities for planar isometry groups that do not contain translations or glide reflections elements, by considering the addition of translational elements we can determine the remaining planar isometry groups. In particular, we consider only the addition of infinitesimal translations, as we are not concerned with the periodic translation groups in this paper. Table 1 shows the results of this analysis, and makes use of some new notation. For instance, adding an infinitesimal translation to the two-element Cyclic group C2 , and closing the group with respect to composition, results in a group isomorphic to
E +(1) × C2 = {(Tλ , e ) , ( Tλ , Rπ ) | λ ∈ } which gives us a more convenient way to write it. Other new notations are: T ( 2 ) = {Tx , y | x, y ∈ } for all the translations; Cn ( 2 ) ≅ T ( 2 ) × Cn for all the translations, plus
{Tx , Tx Rπ T− x | x ∈ } , but this group is isomorphic to
a cyclic group centred at every point; Dn ( 2 ) ≅ T ( 2 ) × Dn for all the translations, plus a dihedral group centred at every point; C∞ for the continuous cyclic group, isomorphic to S 1 ;
D∞ for the infinite dihedral group, isomorphic to O( 2 ) ; E + ( 2 ) the group of all translations and rotations; and E ( 2 ) the Euclidean group of all planar isometries.
Rosette Group
number of independent, infinitesimal, translational generators 0 1 2
{e}
{e}
E +(1)
T ( 2)
C2
C2
E +(1) × C2
C2 ( 2 )
C3 , C4 , …
C3( 2 ) , C4( 2 ) , …
C∞
C∞
E +( 2 )
D1
D1
E +(1) × D1 , E (1)
D1( 2 )
D2
D2
E (1) × D1
D2( 2 )
C3 , C4 , …
D3 , D4 , … D∞
D3 , D4 , …
D3( 2 ) , D4( 2 ) , …
D∞
E( 2)
Table 1 – Combining Rosette Groups with infinitesimal translational generators, produces the non-periodic groups of planar isometries.
In Table 2, we re-organize the groups set out in Table 1 and also include (for didactic completeness) the periodic isometry groups. In this table, the Fi are the Frieze Groups, and the Wi are the wallpaper groups. We will use Γ( 2 ) for the full set of types of planar isometry group (as in table 2), and Γ∗( 2 ) for the non-periodic subset shown in Table 1.
Rosette
Translational
Trivial Cyclic
Dihedral
Continuous
Continuous & Periodic
Periodic
E + (1) , E (1) E + (1) × D1 , E (1) × D1
{e}
C2 , C3 , …
D1 , D2 , …
C∞
D∞
F1 , … , F7
1-D
W1 , … , W17
2-D
E + (1) × C2
E +(1) × E +(1) C2( 2 ) , C3( 2 ) , … D1( 2 ) , D2( 2 ) , … E +( 2 ) , E ( 2 )
F1× E + (1) , F1× E (1) F4 × E + (1) , F4 × E (1)
Table 2 – All possible groups of planar isometries. Only the unshaded cells are dealt with in this paper; their combined contents is the same as the contents of Table 1.
In figure 1, the group/subgroup lattice for the non-periodic groups of planar isometries is shown. This lattice indicates when a group of planar isometries appears as a subgroup of a larger group of planar isometries.
E( 2)
D∞
C∞
C4 + 2b
C3+ 2a
D4+ 2 h ( 2 )
C4 + 2 f ( 2 )
D3+ 2 g ( 2 )
E (1) × D1
C3+ 2 e( 2 )
D4+ 2d
D3+ 2c
E +( 2 )
E (1)
D2
D1
D1( 2 )
T (1) × D1
C2
D2( 2 )
C2( 2 )
T ( 2)
T (1) × C2
T (1)
{e}
Figure 1 – The group/subgroup lattice for the non-periodic groups of planar isometries. Higher-up groups contain lower-down groups, so for example D2 contains D1 as a subgroup. Containment is transitive, so for example, T (1) is a subgroup of E (1) × D1 . All groups contain themselves as a trivial subgroup. The presence of translational elements in groups is indicated by shading: dark grey boxes contain T ( 2 ) ; light grey contain T (1) but not T ( 2 ) ; unshaded boxes contain neither. Round cornered boxes indicate countably infinite families of groups, for example the box D4+ 2d corresponds to the groups D4 , D6 , D8 ,…. Dotted links are between the round-cornered boxes only; and exist if and only if the lower index divides the upper e.g. when ( 3 + 2a ) | ( 3 + 2c ) . Group/subgroup relationships can also exist between elements of one family of groups, when one index divides another.
In table 3, the factor groups of elements of Γ∗ ( 2 ) are laid out. These will be used in section 5. Factor groups are isomorphic to the cosets of a normal subgroup. So to determine these one finds subgroups, eliminates non-normal ones, and then constructs and analyzes the cosets of the remainder. To carry this out we make use of the group/subgroup diagram shown in figure 1 and the matrix of pairs of Γ∗ ( 2 ) in Table 3. The trivial factor groups of any group are the trivial group and the group itself.
candidate normal subgroup {e} {e}
Z1
C3+ 2a
Z3+ 2a
D1
Z2
C2
Z2
T (1)
»
C4+ 2b
D1
C2
T (1)
D2
E (1)
T (1) × D1
T (2)
T (1) × C2
C∞
D4+ 2δ
C3+ 2ε ( 2 )
E (1) × D1 D1 ( 2 )
C2 ( 2 )
D∞
C4 + 2 λ ( 2 )
D3+ 2 µ ( 2 )
D2 ( 2 )
E + ( 2)
D4 + 2θ ( 2 )
E (2)
Z1 Z1 Z1
Z 4+ 2b
Z 4+ 2b Dih 3+ 2 c
Z 2 +b
Z 2 +b
3+ 2α
2+ β
×
Z1 ⇔ c = γ
D2
Dih2
Z2
E (1)
E (1)
×
Z2
×Z
»
Z2
3+ 2α
2
T (2)
»2
T (1) × C2
E (1)
S
D3+ 2γ
3+ 2α
Dih3+ 2c
C∞
C4 + 2 β
Z 3+ 2 a
D3+ 2c
T (1) × D1
Group
C3+ 2α
1
Z2
× S
Dih4 + 2 d
E (1)
×
D1 ( 2 )
Z2 ( 2 )
×
C2 ( 2 )
Z2 ( 2 )
Z 3+ 2 e ( 2 )
E (1) × D1
3+ 2α
×
D∞
O (2)
O (2)
Z 4+ 2 f ( 2 )
× ×
D3+ 2 g ( 2 )
Dih3+ 2 g ( 2 )
D2 ( 2 )
Dih2 ( 2 )
E + ( 2)
E + ( 2)
Z1 1
Dih 2+b 2+ β
Z1 Z1 ⇔ d = δ ,
Z 2 ⇔ d = 1 + 2γ Z 2 ⇔ d = 0
Z 2 ⇔ d = 2 (1 + δ )
×
×
× × ×
×
×
× ×
D4 + 2 h ( 2 )
Dih4 + 2 h ( 2 )
×
×
×
E (2)
E (2)
×
×
×
Z2
Z2
»
×
E (1)
O (2)
×
× ×
Z2
Z1
×
×
× ×
× ×
Z1 Z2
Z 4+ 2 f
E (1) × Z 2
×
Z1
Z2 O (2)
× ×
Z1 ⇔ e = ε
Z3+ 2e
Dih2 E (1)
×
C4 + 2 f ( 2 )
Z1
S
E (1) × Z 2
C3+ 2 e ( 2 )
Z1
×
1
Dih2=d
Dih4+ 2d
Z1
»
×
D4+ 2d
Z1
E (1)
× ×
×
×
×
S
1
Dih3+ 2 g
×
×
×
×
×
Dih4+ 2h
×
×
×
×
×
×
O (2)
×
× ×
×
S
Z2
1
×
Dih4 + 2 h
×
O (2)
3+ 2ε
2 +λ
×
3+ 2ε
× ×
Z 2+ f
Z 2+ f
3+ 2ε
Dih3+ 2 g
Dih2
Z1 Z 4+ 2 f
Z1 ⇔ g = µ Z2 S
Z1
1
×
×
Dih2+ h
×
×
O (2)
S
1
Z1 Z1 ⇔ h = θ ,
Z2 ⇔ h = 1 + 2µ Z2 ⇔ h = 0
×
O (2)
×
×
Z 2 ⇔ h = 2 (1 + θ )
Z2
Table 3 – Determination of the factor groups of elements of Γ∗ ( 2 ) . Cells relate to the factorization of the row group by the column group. Variables a–h and α–θ are natural numbers. Factorization is possible if and only if the column group appears as a normal subgroup of the row group. Shaded cells indicate that factorization is impossible because the column group does not appear even as a subgroup, let alone normal subgroup. The × indicates where a subgroup exists but not a normal one. In the other cells the factor groups that arise are listed. For some cells, the factor group exists only when an explicitly stated condition holds (e.g. Z 2 ⇔ d = 0 ). Other cells, where the subscript of the listed factor group involves a fraction, have associated in the implicit condition that the factor group exists if and only if the fraction evaluates to an integer.
Z1
Z1
4. Image Isometries We define images to be scalar functions of Euclidean domains. In this paper the domains will always be 2-D. I will always denote an image i.e. I : E 2 → . An image isometry ( r = s, i ) is an ordered pair of a (spatial) planar isometry (s) and an (intensity) linear isometry (i). It is applied to an image as follows ( r I )( x ) = i( I ( s x ) ) , so composition of image isometries is component-wise i.e. s, i t , j = s t , i j . When the intensity component of an image isometry is the identity, we will make use of the shorthand of writing s instead of s, e , and similarly for a group of image isometries where the intensity component of all the elements is the identity e.g. T ( 2 ) will be written as shorthand for
{ t, e
| t ∈ T ( 2 )} .
An image is symmetrical, with respect to some image isometry g, if and only if it is unchanged by application of the isometry i.e. g I = I , such an isometry is said to be an automorphism of the image I. Some types of image isometry are never image automorphisms – it depends on the type of the spatial and intensity components. Table 4 shows the possible types of image isometry, and an indication (Y or N) of whether or not they can be automorphisms. Notes of the form R2 and R3 indicate which remark, following the table, is relevant to that type of combination. Intensity Isometry Type Identity Translation Reflection Identity Translation Planar Isometry Rotation Type Reflection Glide Reflection
Y Y Y Y Y
N(R2) Y N(R2) N(R2) Y
Y(R3) Y Y Y Y
Table 4 – The types of image isometry, and whether they can be image automorphism
Remark 2: Types of image isometry that are never automorphisms e, Tλ ≠ 0 , Rθ ≠ 0 , Tλ ≠ 0 , M , Tλ ≠ 0 ∉ Aut[ I ] because if such an element was an automorphism, it would imply that I ( 0, 0 ) = I ( 0, 0 ) + λ , which would contradict the condition that λ ≠ 0 .
Remark 3: An image isometry sufficient to show the image to be constant e, M c ∈ Aut[ I ] ⇔
I =c
because the l.h.s. implies I ( x ) = 2c − I ( x ) ; and the inference from right to left is trivial.
5. Determining possible Image Symmetries The set of automorphisms of an image I , form its automorphism group Aut[ I ] . We follow the method presented in [4] for determining the possible types of automorphism groups, but here for 2-D images rather than 1-D. We will treat two automorphism groups as being of equivalent form if they are conjugate (and hence isomorphic). So the form of the automorphism group of an image is unchanged by application of an image isometry to the image. We start by characterizing the automorphism group of a constant image ( I ≡ 0 ). We define: J const := { g , e , g , M | g ∈ E ( 2 )} ≅ E ( 2 ) × D1 J const is the first of the possible automorphism groups of images that we define in this paper.
All of these groups will similarly be denoted in the form J subscript . We now state and prove:
Lemma 1: Automorphism group of a constant image The following are true or false together: C1) I = c C2) Aut [ I ] ≅ J const C3) ∃ s, i , s, j ∈ Aut[ I ] , i ≠ j Proof. C1 ⇒ C 2 because the transformation e, T − c does not change the form of Aut [ I ] C 2 ⇒ C 3 because e, e , e, M ∈ I const C 3 ⇒ C1 follows from R1.
Ñ
L1 shows that, for J const at least, the spatial parts of the group form a group ( E ( 2 ) ) of planar isometries, and the intensity parts form a group ( D1 ) of linear isometries. R4 notes that this is true in general.
Remark 4: Projections of groups image isometries are groups If G is a group of image isometries, then its projection onto its spatial components ( π s [G ] := { p | ∃ p, i ∈ G} ) and its projection on its intensity components ( π i [G ] := {i | ∃ p, i ∈ G} ) are groups of planar and line isometries respectively. T1 establishes that, apart from one exception, the automorphism groups we wish to establish are one-to-one with epimorphisms from a group of planar isometries onto a group of linear isometries. Epimorphisms are surjective mappings that preserve the group structure i.e. ϕ ( a b) = ϕ( a ) ϕ (b ) .
Theorem 1 – Limited forms of automorphism groups Aut[ I ]
must be isomorphic to either
{ s, ϕ ( s )
J const , or a group of the form
}
| s ∈ S , ϕ ( s ) ∈ I where S is a group of planar isometries, and I a group of
line isometries. Proof. Similar theorem stated and proved in [4].
Ñ
T1 shows that to discover the possible automorphism groups of images it is sufficient, having taken note of the special case of E ( 2 ) × D1 , to consider pairs of an elements of Γ∗ ( 2 ) and an element of Γ(1) , and to determine what if any epimorphisms exist from the first of the pair to the second. Each epimorphism found generates a candidate automorphism group. These candidates must be checked for whether there is some image with that automorphism group. Groups can be ruled out at this stage if there are no images with all their symmetries, or if any image with the symmetries would have even more and hence a larger automorphism group. To employ the second of these reasons, R3 which concerns the automorphism group of a constant image is useful, as is the following T2 which concerns the automorphism group of a linear (i.e. slope) image. We first specify the automorphism group of a slope:
{
}
J slope = Tx , y , Tx , M x , M x , N , e | x, y ∈
and then state and prove:
Lemma 2 – Sufficient symmetries to be a slope Aut[ I ] ≅ J slope ⇔ I ( x, y ) = a + bx + cy, ¬ ( b = c = 0 ) Proof. ⇐ : by considering each type of image-isometry in turn. ⇒ : Use a change of coordinates to achieve Aut[ I ] = J slope , and then note that
{T
x, y
}
, Tx | x, y ∈ ⊂ Aut [ I ] ⇒ I ( x, y ) = I ( 0, 0 ) + x .
Ñ
Table 5 shows the results of carrying out the plan of considering each possible pair of an element of Γ∗ ( 2 ) and an element of Γ(1) . The symbols (e.g. J 3 , J11 ) identify for which pairs which automorphism groups are possible: some groups come in different orders which is indicated by a second subscript e.g. J 6,3 . The table caption gives further details on how to read this table. Following the table are the list of lemmas used as reasons for candidate automorphism groups not being possible.
Spatial Projection
Intensity Projection
{e}
D1
E + (1)
E (1)
F1
F4
{e}
J0
×
×
×
×
×
C3+ 2 a
J1,3 , J1,5 ,…
×
×
×
×
D1
J 2,1
× J 6,1
×
×
×
×
C2
J1,2
J 7,2
×
×
×
×
T (1)
N(L3)
J10
×
×
×
C4 + 2 b
J1,4 , J1,6 , …
× J 7,4 , J 7,6 ,…
×
×
×
×
D3+ 2 c
J 2,3 , J 2,5 ,…
J 6,3 , J 6,5 , …
×
×
×
×
D2
J 2,2
J 6,2 , J 8,2
×
×
×
×
E (1)
J3
N(L6)
×
N(L10)
×
×
T (1) × D1
N(L3)
N(L3)
J11
×
×
×
T ( 2)
N(L4)
×
N(L9)
×
×
×
T (1) × C2
N(L3)
N(L3)
×
J12
×
×
C∞
N(L5)
×
×
×
×
D4+ 2d
J 2,4 , J 2,6 ,…
× J 6,4 , J 6,6 ,…
×
×
×
×
C3+ 2 e ( 2 )
N(L4)
×
×
×
×
×
E (1) × D1
J4
J 9 , N(L7)
×
N(L10)
×
×
D1 ( 2 )
N(L4)
N(L4)
N(L9)
N(L10)
×
×
C2 ( 2 )
N(L4)
N(L4)
×
N(L11)
×
×
D∞
J5
N(L8)
×
×
×
×
C4 + 2 f ( 2 )
N(L4)
N(L4)
×
×
×
×
D3+ 2g ( 2 )
N(L4)
N(L4)
×
×
×
×
D2 ( 2 )
N(L4)
N(L4)
×
J slope
×
×
E ( 2)
N(L4)
N(L4)
×
×
×
×
D4+ 2h( 2 )
N(L4)
N(L4)
×
×
×
×
E( 2)
N(L4)
J const
×
×
×
×
+
J 8,4 , J 8,6 ,…
Table 5 - Determination of the possible automorphisms groups of images. In each cell is considered whether and how the row group and the column group can be combined, and whether the result is an automorphism group (indicated by a J subscript ). Cells where no automorphism group is possible are either light-shaded with a N(Tn), indicating that Theorem n gives the reason why not, or are dark-shaded with a × , indicating that no quotient group (of the spatial projection) is isomorphic to the desired intensity projection.
Lemma 3 - T (1) ⊆ Aut[ I ] ⇒ E (1) ⊆ Aut[ I ] Proof. Assume the antecedent and hence that T2 x ∈ Aut [ I ] , from which it follows that
I ( x, y ) = I ( − x, y ) which is equivalent to M ∈ Aut[ I ] . Then, by closure of the group, E (1) ⊆ Aut[ I ] .
Ñ
Lemma 4 - T ( 2 ) ⊆ Aut[ I ] ⇒ Aut[ I ] ≅ J const Proof. Assume the antecedent and hence that Tu − a , v −b ∈ Aut[ I ] , from which it follows
that I ( a, b ) = I ( u, v ) . Since this is true for a, b, u, v it follows that I ≡ const . The consequent then follows by L1. Ñ
Lemma 5 - C∞ ⊆ Aut[ I ] ⇒ D∞ ⊆ Aut[ I ] Proof. Assume the antecedent and hence that R2θ ∈ Aut[ I ] , from which it follows that
I ( r cos θ , r sin θ ) = I ( r cos( −θ ) , r sin ( −θ ) ) = I ( r cosθ , − r sinθ ) , which is equivalent to M π ∈ Aut[ I ] . Then by closure of the group D∞ ⊆ Aut [ I ] 2
Ñ
Lemma 6 - E (1) ⊗ D1 ⊆ Aut[ I ] ⇒ Aut[ I ] ≅ J const Assume the antecedent; then, from M x , M ∈ Aut[ I ] it follows that I ( x, y ) = 0 , and hence I ≡ 0 . The consequent then follows by L1.
Ñ
There are two ways to combine E (1) × D1 with D1 . The automorphism group J 9 arises from the factorization ( E (1) × D1 ) / E (1) ≅ Z 2 , an alternative group K 9 arises from the alternative factorization ( E (1) × D1 ) / (T (1) × D1 ) ≅ Z 2 . An explicit expression for this new group is:
{
}
K 9( l ) = Tλ , e , M λ , M , NTλ , e , NM λ , M | λ ∈ . However, L7 shows that in fact K 9 is never the automorphism group of an image.
Lemma 7 - K9 ⊆ Aut[ I ] ⇒ Aut[ I ] ≅ J const Proof. Assume the antecedent; then, from M λ , M ∈ K 9 it follows that I ( λ , y ) = 0 ,
and hence I ≡ 0 . The consequent then follows by L1.
Ñ
Lemma 8 - D∞ ⊗ D1 ⊆ Aut [ I ] ⇒ Aut[ I ] ≅ J const Proof. Assume the antecedent; then from
M θ , M ∈ Aut[ I ] it follows that
I ( r cos θ , r sin θ ) = 0 , and hence I ≡ 0 . The consequent then follows by L1.
Ñ
Lemma 9 - { Tx, y , Tx | x, y ∈ } ⊆ Aut [ I ] ⇒ Aut[ I ] ≅ J slope Proof. Assume the antecedent; it follows that I ( x, y ) = x + I ( 0 ) . The consequent then follows from L2. Ñ
Lemma 10: { Tx , Tx | x ∈ } ∪ { M , M } ⊆ Aut[ I ] ⇒ Aut[ I ] ≅ J slope Proof. (i) M , M ∈ Aut[ I ] ⇒ ∀y ∈ I ( 0, y ) = 0
(ii)
{ T ,T x
x
| x ∈ } ⊆ Aut[ I ] ⇒ ∀x, y ∈ I ( x, y ) = I ( 0, y ) + x
( i ) ∧ ( ii ) ⇒ I ( x, y ) = x I ( x, y ) = x ∧ T 7 ⇒ Aut [ I ] ≅ J slope
Ñ
Lemma 11 - C2( 2 ) ⊗ E (1) ⊆ Aut[ I ] ⇒ Aut[ I ] = J slope
{
} {
}
Proof. C2( 2 ) ⊗ E (1) ≅ Tx , y , Tx | x, y ∈ ∪ Tx , y Rπ T− x ,− y , M x | x ∈
{ T ,T x
xy
}
| x, y ∈ ⊆ Aut[ I ] ⇒ ∀x, y ∈ I ( x, y ) = I ( 0, 0 ) + x
I ( x, y ) = I ( 0, 0 ) + x ∧ T 7 ⇒
Aut [ I ] ≅ J slope
Ñ
6. The possible Image Symmetries In this section I report the complete set of fifteen types of (non-periodic) image automorphism groups. Two types, J const and J slope were already defined in section 5. The remaining types were derived using the method described in the previous section. The results are shown in three ways: in table 5, which allows groups to be looked up in terms of their spatial and intensity projections; in figure 2, which shows the group/subgroup lattice of the different types; and in the following explicit descriptions of the groups. J 0 := { e, e
} is the trivial group of 1-D image isometries. This is the automorphism group of
any image without symmetries. There is only one of these groups. J1,n ( p ) ≅ { s, e | s ∈ Cn } , n ≥ 2 are the image-isometry versions of the cyclic groups of
planar isometries. There is a 2-D family of groups for each order, indexed by the position of the centre of rotation. J 2,n ( p, α ) ≅ { s, e | s ∈ Dn } , α ∈ [ 0, 1) are the image-isometry versions of the dihedral groups
of planar isometries. There is a 3-D family of groups for each order, indexed by the position of the centre of rotation and by a real variable that specifies the absolute angle of the system of lines of reflection. J 3 (θ ) ≅ { g , e | g ∈ E (1)} , θ ∈ [ 0, π ) is the image-isometry version of the planar isometry
group E(1). It is the automorphism group of an image which does not vary in some direction,
and has no other symmetries. There is a 1-D family of such groups, indexed by the direction of no variation. J 4 ( l ) ≅ { T− λ g Tλ , e | λ ∈ , g ∈ D2 } is the image-isometry version of the planar isometry
group E (1) × D1 . It is the automorphism group of an image which does not vary in some direction, and also has a single line of reflection parallel to the direction of constancy, but no other symmetries. There is a 2-D family of such groups, indexed by the line of reflection. J 5 ( p ) ≅ { s, e | s ∈ D∞ } is the image-isometry version of the infinite dihedral planar
isometry group. It is the automorphism group of an image which is continuously rotationally symmetric about some point. There is a 2-D family of points indexed by the centre of rotation.
J 6,n ( p, α , λ ) ≅
{R
m 2π n
}
, e , M mπ n , M | 0 ≤ m < n , α ∈ [ 0,1) are like dihedral groups of
planar isometries, except that each planar spatial reflection is combined with an intensity reflection (the same one for each); whereas the spatial rotations are of ordinary type. They are the automorphism groups of images with a system of lines of negating-reflection, spaced at regular angles through a common point – such images also inevitably have a matching number of ordinary rotational symmetries. There is a 4-D family of such groups, indexed by the centre of rotation, a real variable specifying the absolute angle of the system of lines of anti-reflection, and a median intensity value, about which intensities are reflected in each of the negating-reflections.
J 7,2 n ( p, λ ) ≅
{R
2 mπ n
}
, e , R( 2 m +1)π n , M | 0 ≤ m < n are like even-order cyclic groups of planar
isometries, except that the rotations in the group alternate between ordinary ones and negating-rotations (i.e. rotations combined with an intensity reflection). The lowest order of these groups J 7,2 is the automorphism group of the Taiji symbol of Taoist philosophy [5]. For each order, there is a 3-D family of groups, indexed by the centre of rotation, and a median intensity value. J 8,2 n ( p, α , λ ) ≅
{R
2 m πn
}
, e , R( 2 m +1) π , M , M m π , e , M ( 2 m +1) π , M | 0 ≤ m < n n
2n
n
are
like
even-order dihedral groups of planar isometries except that both the reflections and the rotations alternate between ordinary ones and negating ones. The lowest order of these groups J 8,2 is the automorphism group of an image with a line of ordinary reflection, and perpendicular to this a line of negating-reflection. Necessarily, such an image is also unchanged by a 180± negating reflection. For each order of this type of group there is a 4-D family, indexed by the centre of rotation, a real variable fixing the alignment of the mirrors, and a median intensity value about which negation happens.
{
}
J 9 ( l ) = Tλ , e , M λ , e , NTλ , M , NM λ , M | λ ∈
is the automorphism group of an
image which does not change in some direction, and also has a single line of negating-reflection parallel to the direction of constancy, but no other symmetries. There is a 3-D family of such groups, indexed by the line of reflection, and the median intensity value. J10 (θ , k ) ≅ { Tλ , Tλ | λ ∈ } , θ ∈ [ 0, 2π ) , k ≠ 0 is the automorphism group of an image that it invariant to any translation, in some fixed direction, so long as that translation is paired
with the addition (or subtraction) of an intensity increment of size proportional to the signed translation distance. The contour plot of such an image consists of identical, equally-spaced, non-straight curves. There is a 2-D family of such groups, indexed by the direction of the translation, and the constant of proportionality between the spatial and intensity shifts. J11 ( l , k ) ≅ { Tλ , Tλ , NTλ , Tλ | λ ∈ } , k ≠ 0 . Images with this automorphism group have the shift and increment symmetry of a J10 -symmetric image, but also have a single line of reflection, parallel to the direction of invariant translation. There is a 3-D family of such groups, indexed by the line of reflection and the constant of proportionality between the spatial and intensity shifts.
{
} {
}
J12 ( l , k ) ≅ Tx , y , Tx | x, y ∈ ∪ Tx Rπ T− x , M x | x ∈ , k ≠ 0
Images
with
this
automorphism group have the shift and increment symmetry of a J10 -symmetric image, but also have a line of centres of negating-rotation, each like the J 7,2 symmetry. The line of centres is parallel to the direction of invariant translation. There is a 3-D family of such groups, indexed by the line of reflection and the constant of proportionality between the spatial and intensity shifts. Section 7 has some additional commentary on this symmetry.
Jconst
…
J5
Out[ 89]=
Jslope
J8,4, J8,6, …
…
J2,4, J2,6, …
…
J6,2
Out[ 94]=
J6,3, J6,5, …
J1,4, J1,6, …
…
Out[ 90] =
…
…
… J6,4, J6,6, …
J6,1
J9
J4
J7,4, J7,6, …
Out[84]=
…
J2,2
Out[ 85]=
J8,2
Out[ 95]=
J11
J12
J2,3, J2,5, …
J1,2
J2,1
Out[ 87] =
J3
Out[ 91]=
J7,2
J10
J1,3, J1,5, …
J0
Figure 2 – Shows the different types of non-periodic image automorphism group. The links show group/subgroup relationships, with smaller automorphism groups being lower in the figure. The colour of non-grey panels indicate different orders of the same type of automorphism group. Dashed links show that the group/subgroup relationships only hold between certain orders of group at either end of the link. Group/subgroup relationship can also hold within a node when the orders permit it; for example J1,3 is a subgroup of J1,9 . The contour plots within the panels show example images with the corresponding automorphism group. Overlaid on these plots are shown lines of reflection (red), lines of negating-reflection (green), centres of rotation (red), centres of negating-rotation (yellow). The plot for group J12 has a dotted blue line which is explained in figure 3.
7. Concluding Remarks Most of the automorphism groups listed in the previous section are already known. Referring back to table 5, those in the column headed {e} have been studied under the simplest notion of planar symmetry where the transformations are planar isometries only [6]; those in the column headed D1 have been studied under the notion of ‘coloured symmetry’ when a planar isometry is combined with a permutation of a finite label set [7-9]. Of the four other types ( J10 , J11 , J12 , J slope ) the least obvious is J12 , the example of which in in figure 2 is shown larger in figure 3. As can be seen from figure 3, images with the J12 symmetry remain the same after translations in some direction when combined with an intensity increment of magnitude
proportional to the translation distance. J12 -symmetric images are also unchanged by 180± rotations about certain points, when the rotation is combined with a reflection of the intensity about a median value that depends on the centre of rotation. This type of negating-rotation is familiar from the Taiji (or Taijitu or Yin-Yang) symbol of Taoist philosophy [5]. For the J12 symmetry though, there is not just one such centre of negating-rotation, but a continuous line of them.
J12 Figure 3 – Shows the example image from figure 2 with J12 automorphism group, magnified for easier inspection.
8. Acknowledgements Research undertaken as part of the EPSRC-funded project ‘Basic Image Features’.
9. References 1. 2. 3. 4. 5. 6. 7. 8. 9.
Griffin, L.D., Symmetries of 1-D Images. Submitted, 2008. Yale, P.B., Geometry and Symmetry. 1968: Dover Press. Koenderink, J.J. and A.J. van Doorn, Image processing done right, in Computer Vison - Eccv 2002, Pt 1. 2002. p. 158-172. Griffin, L.D., Automorphism groups of 1-D images. Submitted, 2007. Browne, C., Taiji variations: yin and yang in multiple dimensions. Computers & Graphics, 2007. 31(1): p. 142-146. Vol A. Space-Group Symmetry. International Tables for Crystallography, ed. T. Hahn. 2006, Chester: International Union of Crystallography. Schattschneider, D., MC Escher. Visions of Symmetry. 1990: Plenum Press. Loeb, A.A., Color and Symmetry. 1978: Robert E. Krieger. Shubnikov, A.V. and V.A. Kopstik, Symmetry in Science and Art. 1974: Plenum Press.
