I Introduction
In image target detection, one often desires to detect all geometric symmetry transformed versions of targets, for two reasons. One reason is that for any given target pattern, all symmetry transformed versions also are valid target patterns, to at least a good approximation. The other reason is that so few separate target patterns are available for detector training, relative to the number of detector coefficients, that the detector has the capacity to learn the peculiarities of each individual pattern, rather than being forced to learn the universal properties of the target pattern class. Symmetry transformation of available target patterns creates additional patterns, which even if not valid target patterns, at least share many of the universal properties of genuine target patterns. Training the detector on both actual and transformed target patterns discourages overlearning of individual pattern peculiarities, and encourages learning of universal pattern properties.
There are four distinct classes of geometric point transformations^{1}^{1}1 Point transformations are geometric transformations that leave at least one point invariant.: rotation; reflection; shear; and dilation. Rotation is a sensepreserving rigidbody transformation. Reflection is a sensereversing rigidbody transformation. Shear is an anglemodifying, areapreserving transformation. Dilation is an anglepreserving, areamodifying transformation. Usually only rotation operations are considered in image target detection [1]–[4]. Reflection operations also are valid where targets have approximate mirror symmetry, or where the second reason in the previous paragraph is the motivating factor for considering symmetry. Dilation and shear have the fundamental theoretical drawback that no matter how symmetrical the detector support, there is always a flux of image content into or out of the support under these transformations, so that the detector support is not transformed onto itself. Dilations have the additional theoretical drawback for sampled (discrete) images that all strict dilations map nonlattice points onto lattice points, and all strict contractions map lattice points onto nonlattice points; once again, the discretespace detector support is not mapped onto itself. These properties of dilation and shear preclude their inclusion in group theoretical methods of accounting for geometric symmetry^{2}^{2}2An example of the problem is that a dilation followed by its reverse contraction is not an identity transformation, because the original dilation ejected the periphery of the support, and the following contraction leaves a void in the same periphery, even though the interior of the support has been perfectly restored. in pattern recognition, as developed here and elsewhere [5]–[8]. Group theoretical considerations make rotation and reflection the only allowable geometric symmetries in this analysis. Even then, the detector support requires at least the same rotation and mirror symmetry, and orientation^{3}^{3}3A polygon and lattice of the same symmetry have the same orientation if their mirror lines coincide., as the sampling lattice, for the discrete space support to map onto itself, as it must. If one treats images as being spatially continuous and detector supports infinitely large, then dilation and shear are permissible symmetry operations [9].
This article presents the group theory based geometric symmetry analysis of the Quadratic Fisher Discriminant (QFD) operating on image pixels. Encapsulated in the analysis is a method of symmetrising any quadratic detector, not just the QFD. The mathematical formalism generalises in a straightforward manner to polynomial detectors of arbitrary degree (Volterra filters). Symmetrisation of the QFD, or indeed any polynomial filter, is a means of reducing the number of detector coefficients by identifying and discarding redundant coefficients, without introducing approximations. In contrast, most approaches to reducing polynomial filter complexity approximate the desired filter by a simpler version [10].
A synopsis of this article is as follows. Section II introduces notation and establishes the square lattice symmetry group. Section III derives the consequences for the QFD of sampling lattice symmetry. Section IV solves for the symmetrised QFD, using a procedure that is suitable for arbitrary quadratic filters, and that easily extends to higher degree polynomial filters. Section V establishes the hexagonal lattice symmetry group, and notes the detail changes in the preceding analysis if images are sampled on an hexagonal lattice instead of a square lattice. Section VI experimentally demonstrates the symmetrisation of a QFD for target detection in synthetic aperture radar (SAR) images sampled on a square lattice.
Ii Square lattice symmetry
As shown in Figure 1
, the square lattice is generated by two equal length basis vectors subtending angle
. To exploit the symmetry of the sampling lattice, the detector support must be a polygon with the same symmetry as the lattice, except for translational symmetry. Conventionally, a square lattice is taken to use a square detector support of dimension pixels, with total pixel count(1) 
and with lattice basis vectors as perpendicular bisectors of its sides. Although not considered here, another (unusual) possibility is a square detector support centred on a lattice point and oriented so that the lattice basis vectors point at its vertices (i.e. a diamond shape).
The set of symmetry operations that rigidly transform a lattice onto itself while leaving one lattice point fixed, form the point group of the lattice^{4}^{4}4The space group of a lattice comprises point symmetries plus translational symmetries.. The fixed lattice point is the centre of symmetry. Point symmetries are restricted to rotations and reflections, since dilations and shears are prohibited for the reasons given in Section I. For a square lattice, the point group symmetry operations with centre displaced by one half lattice spacing horizontally and vertically from a lattice point, also map the square lattice onto itself, albeit without leaving any lattice point fixed (Figure 1). If the detector support has odd linear dimension (odd ) then it is centred on a lattice point, whereas if the detector support has even linear dimension (even ) then it is centred on an interstitial site. Although the two centres are not equivalent^{5}^{5}5Nonequivalent points are separated by a nonintegral combination of lattice vectors., the square lattice has identical symmetry about both centres, and that symmetry is the point symmetry.
A symmetry group having elements , that is,
(2) 
represents symmetry of order (i.e. fold symmetry). Square lattice point symmetry is of order
(3) 
Table I lists the 8 elements of the square lattice point group—being the dihedral group ^{6}^{6}6Dihedral group is the group of symmetry transformations of an sided regular polygon, being an fold rotation axis and mirror lines.—in the first column. The second column of Table I describes the geometric operations corresponding to the point symmetries. Symmetries to satisfy the four mandatory group properties: closure; presence of identity element; presence of all inverse elements; and associativity. The point group is nonAbelian, because in general symmetry operations do not commute (e.g. for the square lattice point group ).
Symmetry:  Description of symmetry  Factorisation  Rank 2 tensor representation: 

operation  

Identity or unit operation.  

Anticlockwise rotation by  
1/4 turn.  

Rotation by 1/2 turn, being  
inversion through centre.  

Clockwise rotation by 1/4  
turn.  

Reflection in the axis.  

Reflection in line 1/8 turn  
anticlockwise from axis.  

Reflection in the axis.  

Reflection in line 1/8 turn  
clockwise from axis.  

Subgroups of represent image ensembles with intermediate geometric symmetry. For square lattice sampling: represents a 4fold rotation axis; and represent 4fold rectangular symmetry; represents a 2fold rotation axis; , , and represent 2fold reflection symmetry (one mirror line); and represents the complete absence of geometric symmetry. In the context of airborne surveillance, imaging in the nadir direction has the full symmetry of group , but sidelooking SAR only has a mirror line along the range direction (axis), corresponding to subgroup .
Introduce the 2component index (2index) to convert the two indices of detector pixels (row and column ) into a single index,
(4) 
thereby allowing pixels in the support to be columnordered into the dimension column vector with components , where the (1) superscript signifies that is inherently a rank 1 tensor (i.e. vector). It is possible to construct QFD theory using 2indices alone, at the expense of using rank 4 tensors. Alternatively, combining pairs of 2indices ( and ) into 4component indices () according to the prescription
(5)  
allows rank 4 tensors to be expressed as higher dimension rank 2 tensors (i.e. square matrices), as will be seen forthwith. As a reminder that 4indices are properly a pair of 2indices, the notation retains a comma between the first and second 2indices. Pairwise products of pixels in the support are collected into the dimension column vector with components
(6) 
where the (2) superscript signifies that is inherently a rank 2 tensor^{7}^{7}7 is the outer product of with itself..
Symmetry operation linearly transforms unprimed vectors into primed vectors as follows:
(7) 
, with components , is the rank 2 tensor representation of symmetry operation . , with components , is the rank 4 tensor representation of symmetry operation . Use of 4index notation allows to be expressed in matrix form, but this is merely a notational convenience. Lattice point symmetries rearrange image pixels without altering pixel values (i.e. they are permutation operations on the pixel values), implying invariance of the Euclidean norm^{8}^{8}8This length property establishes that tensors are defined in a Euclidean vector space, so they are cartesian tensors with no distinction between covariant and contravariant indices.
(8) 
which ensures that all transformation matrices are orthogonal, that is,^{9}^{9}9The transpose of a rank 4 tensor with four 1component indices is defined as the swapping of the first and second index pairs, with the index ordering in individual pairs not changing (i.e. becomes ).
(9) 
have determinants 1 or ^{10}^{10}10It is not the case that all rotations have determinant 1 and all reflections have determinant , because the matrices transform pixel values and not pixel coordinates (the latter being the common usage for transformation matrices).. For a square lattice,
has eigenvalue 1,
and have eigenvalues 1 and , and and have eigenvalues 1, , and . Expanding (7) in terms of tensor components, and substituting (6), connects components of the two representations by(10) 
Equation (10) implies that the rank 4 tensor components have permutation symmetry
(11) 
The rank 2 tensor representation of the square lattice point group is listed in the fourth column of Table I; the rank 4 tensor representation is computed from the rank 2 tensor representation according to (10). Each matrix has a single one entry in every row and column. Computing the rank 2 tensor representation is efficiently done as follows. Derive matrices and by inspection. Elements and generate group , as indicated in the third column of Table I, so the remaining elements of the rank 2 representation of are computed by repeated multiplication of matrices and ^{11}^{11}11
, representing the identity operation, is immediately set to the identity matrix, without being explicitly generated as
..Iii Symmetry invariants
The underlying postulate of this analysis is that training ensemble statistics (clutter and target separately) are invariant with respect to all point symmetry transformations about the centre of detector support. Such statistical symmetry arises if every image in the actual ensemble contributes to a notional ensemble all symmetry transformations of itself (with equal weighting), and the notional ensemble is a more complete portrayal of the totality of images than the actual ensemble. There is no requirement for images in the ensembles to be individually invariant with respect to any symmetry transformations.
The quadratic detector response is given by the scalar product
(12) 
where detector coefficients and pixel terms partition into rank 1 and rank 2 tensors as
(13) 
Pixel terms have mean
(14) 
and covariance
(15) 
where (as usual) superscripts of the form indicate that the quantity is inherently a rank tensor.

Degrees of freedom—  Degrees of freedom—  

permutation symmetry  permutation and  
Detector coefficients:  only:  geometric symmetry:  
linear  quadratic  total  linear  quadratic  total  linear  quadratic  total  
1 
1  1  1  2  1  1  2  1  1  2 
2 
4  4  16  20  4  10  14  1  3  4 
3 
9  9  81  90  9  45  54  3  11  14 
4 
16  16  256  272  16  136  152  3  24  27 
5 
25  25  625  650  25  325  350  6  55  61 
6 
36  36  1296  1332  36  666  702  6  99  105 
7 
49  49  2401  2450  49  1225  1274  10  181  191 
8 
64  64  4096  4160  64  2080  2144  10  288  298 
9 
81  81  6561  6642  81  3321  3402  15  461  476 
10 
100  100  10000  10100  100  5050  5150  15  675  690 
12 
144  144  20736  20880  144  10440  10584  21  1368  1389 
14 
196  196  38416  38612  196  19306  19502  28  2499  2527 
16 
256  256  65536  65792  256  32896  33152  36  4224  4260 
18 
324  324  104976  105300  324  52650  52974  45  6723  6768 
20 
400  400  160000  160400  400  80200  80600  55  10200  10255 
25 
625  625  390625  391250  625  195625  196250  91  24805  24896 

Ensemble statistics are invariant with respect to all symmetry transformations if and only if
(16) 
Using the fact that any group element multiplying separately all group elements (including itself) yields all the group elements in a different order, one may verify that
(17) 
satisfy invariance properties (III) for arbitrary and . Furthermore, and have the correct form to be ensemble statistics. Although has the correct invariance property, it is a crosscovariance between all pairs of symmetry transformed images, whereas a suitable must be an autocovariance of all symmetry transformed images. The normalising factors in (17) ensure that if and already satisfy statistics invariance conditions (III), then resymmetrisation by (17) will leave and unchanged. The QFD analysis proceeds with symmetrised ensemble statistics and in place of their unsymmetrised counterparts and .
The QFD is the quadratic filter whose output maximises the ratio of the squared difference of means of targets and clutter (dividend), to the sum of variances of targets and clutter (divisor). This QFD objective function is
(18) 
which is maximised by the solution to
(19) 
In (18) and (19) is the difference of means, and is the sum of covariances, for clutter and target ensembles:
(20) 
Ensemble statistics invariance properties (III) applied to (19) derive detector coefficient invariance properties
(21) 
under the assumption that is nonsingular and so (19) has a unique solution^{12}^{12}12 actually has essential singularities due to permutation symmetry, but their removal will be addressed soon.. For the same reasons that apply to (17),
(22) 
satisfies invariance property (21) for arbitrary . Detector coefficients invariance (21) applied to (12) yields the detector response invariance property
(23) 
which implies that the detector response is the same for all images that are point symmetry transformations of each other. Unlike this work, most approaches to filter symmetrisation [11]–[16] postulate detector response symmetry, from which detector coefficient symmetry follows.
Iv Degeneracy matrix
Solving (19) directly yields detector coefficients that are properly symmetrised to within numerical accuracy. Then if necessary, explicit symmetrisation of the detector coefficients by (22) will give perfectly symmetrised detector coefficients. However, there is a more elegant and computationally efficient method of computing perfectly symmetrised detector coefficients. This superior method introduces a degeneracy^{13}^{13}13A set of quantities is said to be degenerate if all of the quantities are identically equal, and not simply equal by accident. The concepts of degeneracy and symmetry are synonymous. matrix to account for permutation and geometric symmetry degeneracies.
As a consequence of the group properties satisfied by matrices , matrix
(25) 
satisfies the symmetry invariance properties
(26) 
Matrix is symmetric,
(27) 
due to the orthogonality property (9) of matrices . has nonzero entries everywhere along the main diagonal. If column has nonzero entries in rows (maximum of distinct rows), then columns are identical. The nonzero entries in a given column all have the same value and add to ^{14}^{14}14A square lattice has , and columns of may have 8 1s, or 4 2s, or 2 4s, or 1 8 entry, although only actually has columns with 2 4 entries..
Matrix has a one entry wherever has any nonzero entry:
(28) 
where the notation is an abbreviation for a component index. Matrix is symmetric. Because each matrix contains only a single one entry per row or column, and zeros elsewhere, satisfies the symmetry invariance properties
(29) 
Additional to the geometric symmetry being postulated in this analysis, it is convenient to impose the permutation symmetry
(30) 
on the rank 2 tensor detector coefficients. Permutation symmetry (30) of is chosen to correspond to the permutation symmetry of (6). Matrix has a one entry if has a one entry at the same row and column, or the same row but permuted column 4index:
(31)  
satisfies the geometric symmetry invariance properties
(32) 
and the permutation symmetry invariance properties
(33) 
Additional to (33) is the property that is a symmetric matrix, which may be regarded as another, unavoidable permutation symmetry. The redefinition
(34) 
is adopted for notational consistency. Covariance matrix and its derivatives have multiple essential singularities if permutation symmetry is not explicitly taken into account, as is done here by replacing by .
Define the reduced width matrix as having columns that are all of the distinct columns of matrix , without repetition. Repeated columns in are identified by the property that if column has one entries in rows ^{15}^{15}15Maximum of distinct rows for ; maximum of distinct rows for ., then columns are identical, and there are no other columns that are identical to column . This property derives from the analogous property of that was noted earlier in this section. Matrices and respectively are the rank 2 and rank 4 tensor^{16}^{16}16Since the column index of can not be expressed in terms of 2indices, are strictly not tensors in the context of this analysis. But matrices are derived from rank tensors, hence the terminology. parts of degeneracy matrix :
(35) 
satisfies the geometric symmetry invariance properties
(36) 
and the permutation symmetry invariance properties
(37) 
Symmetry:  Description of symmetry  Factorisation  Rank 2 tensor representation: 

operation  

Identity or unit operation.  

Anticlockwise rotation by  
1/6 turn.  

Anticlockwise rotation by  
1/3 turn.  

Rotation by 1/2 turn, being  
inversion through centre.  

Clockwise rotation by 1/3  
turn.  

Clockwise rotation by 1/6  
turn.  

Reflection in the axis.  

Reflection in line 1/12 turn  
anticlockwise from axis.  

Reflection in line 1/6 turn  
anticlockwise from axis.  

Reflection in the axis.  

Reflection in line 1/6 turn  
clockwise from axis.  

Reflection in line 1/12 turn  
clockwise from axis.  

Let be an unconstrained nondegenerate detector coefficient vector. Then
(38) 
is a degenerate coefficient vector that has both the required geometric symmetry (21) and permutation symmetry (30). Each row of matrix has a single one entry and zeros elsewhere. Multiple one entries in a column of associate degenerate coefficients in the symmetrised detector . The number of columns in , and the number of components in , is the number of degrees of freedom^{17}^{17}17Degrees of freedom of the detector are the independent coefficients remaining after accounting for all symmetries and before maximising the objective function. of the detector. Substituting (38) into (24) reexpresses the objective function as
(39) 
where the nondegenerate mean vector is
(40) 
and the nondegenerate covariance matrix is
(41) 
The second equalities in (40) and (41), which also may be derived from (17) and (36), affirm the redundancy of statistics symmetrisation already noted in association with (24). Equation (41) shows that the degeneracy matrix formalism maximally compresses the covariance matrix without loss of information; an alternative approach due to Lenz [17, 18] partially compresses the covariance matrix into an equal size block diagonal matrix, whose topleft block has precisely the same size as of (41).
Maximisation of objective function (39) is equivalent to solving (cf. (19))
(42) 
whose solution is
(43) 
where is the MoorePenrose generalised inverse (or pseudoinverse) [19] of . A property of solutions of the form (43) obtained by generalised inversion is that they are the minimum norm vector that minimises the Euclidean norm of the residual [20]. The residual vanishes if is nonsingular, in which case the generalised inverse is equivalent to the conventional inverse. QFD generalisation^{18}^{18}18Generalisation of a detector refers to the consistency of detector performance when assessed separately against training and test ensembles. may be improved by regularising [21, 22] before solving (42).
A necessary and sufficient condition for to be nonsingular is that the clutter and target training ensembles between them must contain no fewer linearly independent images than the number of symmetrised detector degrees of freedom. In contrast, symmetrised covariance matrix (and unsymmetrised covariance matrix ) contains many essential singularities, no matter how large the training ensembles. Zero eigenvalues of are at least equal in number to the number of permutation symmetry degeneracies in the detector coefficients; beyond that there may be extra zero eigenvalues due to small training ensemble sizes. The present degeneracy matrix removes the essential singularities in or by operation (41).
Table II shows the reduction in degrees of freedom afforded by permutation and geometric symmetries. Permutation symmetry among the coefficients leaves degrees of freedom. This is a reduction in degrees of freedom by a factor of up to 2. Maximum reduction is achieved for large detector supports, where the proportion of quadratic terms that are pixel selfproducts becomes small. Geometric symmetry further reduces the degrees of freedom by a factor of up to ( for a square lattice, to which Table II applies). Maximum reduction occurs for large detector supports, where proportionately few pixels and pixel pairs map onto either themselves under symmetry transformations, or common pixels or pixel pairs under different symmetry transformations. The reduction in degrees of freedom attributable to geometric symmetry should constitute a significant improvement in detector generalisation, when the unsymmetrised detector^{19}^{19}19That is, a detector that accounts for permutation symmetry, but not geometric symmetry. generalises poorly.
V Hexagonal lattice symmetry

Degrees of freedom—  Degrees of freedom—  

permutation symmetry  permutation and  
Detector coefficients:  only:  geometric symmetry:  
linear  quadratic  total  linear  quadratic  total  linear  quadratic  total  
1 
1  1  1  2  1  1  2  1  1  2 
2 
7  7  49  56  7  28  35  2  6  8 
3 
19  19  361  380  19  190  209  4  26  30 
4 
37  37  1369  1406  37  703  740  6  77  83 
5 
61  61  3721  3782  61  1891  1952  9  189  198 
6 
91  91  8281  8372  91  4186  4277  12  394  406 
7 
127  127  16129  16256  127  8128  8255  16  742  758 
8 
169  169  28561  28730  169  14365  14534  20  1281  1301 
9 
217  217  47089  47306  217  23653  23870  25  2081  2106 
10 
271  271  73441  73712  271  36856  37127  30  3206  3236 
11 
331  331  109561  109892  331  54946  55277  36  4746  4782 
12 
397  397  157609  158006  397  79003  79400  42  6781  6823 
13 
469  469  219961  220430  469  110215  110684  49  9421  9470 
14 
547  547  299209  299756  547  149878  150425  56  12762  12818 
15 
631  631  398161  398792  631  199396  200027  64  16934  16998 

The most symmetric possible plane lattice has a 6fold rotation axis through a lattice point, around which are 6 mirror lines in the lattice plane uniformly spaced by angle ; this lattice is the hexagonal lattice^{20}^{20}20The hexagonal lattice is often called the triangular lattice, since interstices have three lattice points at the vertices of equilateral triangles as nearest neighbours, much as square lattice interstices have four lattice points at the vertices of squares as nearest neighbours. The alternative convention used here is that lattices are named according to their point symmetry and their WignerSeitz primitive cell.. As shown in Figure 2, the hexagonal lattice is generated by two equal length basis vectors subtending angle . With centre of symmetry at a lattice point, hexagonal lattice point symmetry is of order
(44) 
Unlike the square lattice, interstitual sites in the hexagonal lattice have reduced symmetry compared with lattice point centres. The 6fold point symmetry of hexagonal lattice interstices^{21}^{21}21Unlike square lattices, hexagonal lattice interstices are of two nonequivalent types, depending on whether they are centred on upright or inverted triangles of nearest neighbour lattice points. is even lower than the 8fold point symmetry of square lattices. Clearly, one would not sample on an hexagonal lattice, only to use detector supports centred on interstices. Having established that the detector support will be centred only on lattice points, Figure 2 shows a regular hexagonal support of linear dimension , and a suitable pixel indexing scheme. The support linear dimension is the number of concentric hexagons that account for all pixels, where the central lattice point counts as a hexagon of vanishing size; equivalently, the linear dimension is the number of lattice points between adjacent vertices of the support perimeter. A spiral indexing scheme is adopted for the hexagonal lattice, where represents the th lattice point in the th concentric hexagon. An analogous spiral indexing scheme could have been used for the square lattice, the result being a closer correspondence between the point symmetry tensor representations for square and hexagonal lattices than is evident in comparing Tables I and III. The rowcolumn indexing scheme actually used for the square lattice in Table I is conventional, and allows simpler expression of tensor representations, as is evident in comparing Tables I and III.
The square lattice analysis of Section II also holds for the hexagonal lattice, with the following detail amendments. The total number of pixels in the hexagonal support is (c.f. (1))
(45) 
Radial index and circumferential index transform to 2component index according to (c.f. 4)
(46) 
Details of the hexagonal lattice point group—being the dihedral group —are presented in Table III^{22}^{22}22Derivation of the rank 2 tensor representation of the hexagonal lattice point group requires the following identity: , for , where the definition is adopted. (c.f. Table I). Point subgroups of the hexagonal lattice are: representing a 6fold rotation axis; and representing 6fold triangular symmetry; representing a 3fold rotation axis; , , , , and representing 2fold reflection symmetry (one mirror line); and representing complete absence of geometric symmetry. Depending on the degree of symmetry of the continuous image ensembles, it may or may not be beneficial to use hexagonal sampling instead of square sampling. For an hexagonal lattice, has eigenvalue 1, and have eigenvalues 1 and , and have eigenvalues 1 and , and and have eigenvalues , , and . The only amendment to the degeneracy matrix analysis of Section IV is that for the hexagonal lattice columns of have either 12 1s, or 6 2s, or 1 12 entry; likewise for with the inclusion of columns with 3 4 entries. Columns with 4 3s or 2 6s, although plausible on number theoretical grounds, never occur in (c.f. Footnote 14).
Reduction in detector degrees of freedom consequent on hexagonal lattice permutation and geometric symmetries is quantified in Table IV (c.f. Table II). Support linear dimensions are quantified by for both square and hexagonal lattices, but is akin to the support diameter for the square lattice and radius for the hexagonal lattice. Accordingly, square and hexagonal supports with the same contain very different numbers of pixels—almost 3 times more in the hexagonal support for large . A hexagonal support of linear dimension is closer in pixel population to a square support of linear dimension —just over 1/3 more in the square support for large . Permutation symmetry reduces degrees of freedom by a factor of almost 2 for larger supports, for both square and hexagonal lattices^{23}^{23}23The reduction in degrees of freedom due to permutation symmetry is independent of sampling lattice, but depends on the degree of the detector’s polynomial response function. For an th degree polynomial detector, permutation symmetry reduces degrees of freedom by a factor of almost for large supports.. Geometric symmetry reduces degrees of freedom by a factor of almost for large supports, for both square and hexagonal lattices. The hexagonal lattice, with , achieves greater reduction in degrees of freedom than the square lattice, with .
Vi Experimental results
The effect of QFD symmetrisation on target detection in real imagery is investigated. Discrimination is between SAR images of bushland and SAR images of a variety of vehicles immersed in the same bushland, samples of which are shown in Figure 3. Images are sampled on a square lattice. Image ensembles are a mixture of different flights of a spotlight SAR illuminating the ground at different angles of incidence. Clutter images are not just random samples of the background, but are background regions that ‘tricked’ a simple prescreener into declaring a target; hence the difficulty of distinguishing between clutter and targets in Figure 3. In these experiments ensemble sizes are: clutter training—17197 ; clutter test—37909 ; target training—3425 ; target test—3579. Detector support is pixels, so . Unsymmetrised and symmetrised QFDs are computed by the same procedure, but using different degeneracy matrices . The unsymmetrised detector degeneracy matrix accounts for only permutation symmetry, while the symmetrised detector degeneracy matrix accounts for the full 8fold square lattice point symmetry as well as permutation symmetry. It has been noted in Section II that square lattice sampled SAR image ensembles strictly have only 2fold symmetry, so in principle the detector is being excessively symmetrised here. Justification for maximally symmetrising the detector is that SAR image ensembles have approximately 8fold symmetry, and that imposing the full 8fold symmetrisation on the detector allows verification of the full square lattice point group representation tabulated in Table I. However, there is a risk that the excessive symmetrisation will reduce the detector effectiveness compared with its unsymmetrised counterpart.
Experimental ‘detection error tradeoff’ (DET)^{24}^{24}24DET curves present the same information as ‘receiver operating characteristics’ (ROC) curves, but better illustrate low error regions. curves are plotted in Figure 4 in various combinations to assist comparison. Error bars (standard deviation) are plotted in graphs that compare training and test ensemble DET curves (i.e. graphs (a) and (b)). Graph (a) demonstrates that the unsymmetrised QFD generalises poorly, while graph (b) demonstrates that the symmetrised QFD generalises well. The reason for this behaviour is revealed in Table II, which states that the unsymmetrised QFD has 3402 degrees of freedom compared with the 476 of the symmetrised QFD. There is more scope for the unsymmetrised QFD to overtrain on the training ensemble than there is for the symmetrised QFD, as is apparent in graph (c). That the unsymmetrised QFD is going to generalise poorly in this example, is predictable beforehand on the basis of the number of detector degrees of freedom (3402) not being much less than the size of the smaller of the target and clutter training ensembles (3425). Both unsymmetrised and symmetrised detectors should have their generalisation improved by regularisation procedures [21, 22]. Graph (d) demonstrates that—despite the caution expressed at the end of the previous paragraph—the symmetrised QFD is more effective than the unsymmetrised QFD when tested on the test ensembles, and the advantage is most pronounced at the important low falsealarm probabilty extreme of the DET. The correctness of the square lattice point group representation tabulated in Table I has been verified, and the practical value of QFD symmetrisation—even theoretically excessive symmetrisation—has been established.
Acknowledgements
References
 [1] Y.N. Hsu, H. Arsenault and G. April, “Rotationinvariant digital pattern recognition using circular harmonic expansion”, Appl. Opt. 21, 4012–4015 (1982).

[2]
R. Wu and H. Stark, “Rotationinvariant pattern recognition using optimum feature extraction”, Appl. Opt.
24, 179–184 (1985).  [3] C.F. Chiu and C.Y. Wu, “The design of rotationinvariant pattern recognition using the silicon retina”, IEEE J. Sol.State Circ. 32, 526–534 (1997).

[4]
M. Fukumi, S. Omatu and Y. Nishikawa, “Rotationinvariant neural pattern recognition system estimating a rotation angle”, IEEE Trans. Neur. Net.
8, 568–581 (1997).  [5] R. Lenz, “Grouptheoretical model of feature extraction”, J. Opt. Soc. Am. A 6, 827–834 (1989).
 [6] R. Lenz, “Group invariant pattern recognition”, Patt. Recog. 23, 199–217 (1990).
 [7] R. Lenz, Group Theoretical Methods in Image Processing, SpringerVerlag, Berlin (1990).
 [8] Y. Liu, R. Collins and Y. Tsin, “A computational model for periodic pattern perception based on frieze and wallpaper groups”, IEEE Trans. Pattern Anal. Mach. Intell. 26, 354–371 (2004).
 [9] T. Leen, “From data distributions to regularization in invariant learning”, Neural Comp. 7, 974–981 (1995).
 [10] R. Nowak and B. Van Veen, “Tensor product basis approximations for Volterra filters”, IEEE Trans. Signal Process. 44, 36–50 (1996).

[11]
C. Giles and T. Maxwell, “Learning, invariance, and generalization in highorder neural networks”, Appl. Opt.
26, 4972–4978 (1987).  [12] G. Ramponi and G. Sicuranza, “Quadratic digital filters for image processing”, IEEE Trans. Acoust., Speech, Signal Process. 36, 937–939 (1988).
 [13] M. Reid, L. Spirkovska and E. Ochoa, “Simultaneous position, scale, and rotation invariant pattern classification using thirdorder neural networks”, Neural Net. 1, 154–158 (1989).
 [14] G. Ramponi, “Biimpulse response design of isotropic quadratic filters”, Proc. IEEE 78, 665–677 (1990).
 [15] G. Gheen, “Distortion invariant Volterra filters”, Pattern Recog. 27, 569–576 (1994).

[16]
T. Sams and J. Hansen, “Implications of physical symmetries in adaptive image classifiers”, Neural Net.
13, 565–570 (2000).  [17] R. Lenz, “Using representations of the dihedral groups in the design of early vision filters”, ICASSP93, V165–V167 (1993).
 [18] R. Lenz, “Investigation of receptive fields using represenations of the dihedral groups”, J. Vis. Commun. Image Rep. 6, 209–227 (1995).
 [19] R. Penrose, “A generalized inverse for matrices”, Proc. Camb. Phil. Soc. 51, 406–413 (1955).
 [20] R. Penrose, “On best approximate solutions of linear matrix equations”, Proc. Camb. Phil. Soc. 52, 17–19 (1956).
 [21] J. Friedman, “Regularized discriminant analysis”, J. Am. Statist. Assoc. 84, 165–175 (1989).
 [22] T. Hastie, A. Buja and R. Tibshirani, “Penalized discriminant analysis”, Ann. Stat. 23, 73–102 (1995).