ABSTRACT

The decomposition of an image into a linear combination of digitized basis functions is an everyday task in astronomy. A general method is presented for performing such a decomposition optimally into an arbitrary set of digitized basis functions, which may be linearly dependent, non‐orthogonal and incomplete. It is shown that such circumstances may result even from the digitization of continuous basis functions that are orthogonal and complete. In particular, digitized shapelet basis functions are investigated and are shown to suffer from such difficulties. As a result the standard method of performing shapelet analysis produces unnecessarily inaccurate decompositions. The optimal method presented here is shown to yield more accurate decompositions in all cases.

1 INTRODUCTION

The linear decomposition of a digitized astronomical data set into a series of basis functions is a fundamental task in many areas of astrophysics and cosmology. Indeed, in the analysis of one‐dimensional (1D) spectra, two‐dimensional (2D) images or higher‐dimensional data sets, one is often faced with the problem of determining the coefficients in a linear expansion of the data set in some set of (sampled) basis functions. To illustrate our discussion, we will focus here on the important specific example of decomposing a 2D astronomical image, although the general approach that we advocate will be applicable to data sets of arbitrary dimensionality.

One often describes an image in terms of, for example, a set of orthogonal Fourier modes, by calculating the coefficients in the expansion using a discrete Fourier transform. Significant attention has also been given to representing an image as a linear combination of discrete wavelets basis functions, both orthogonal and non‐orthogonal (see, for instance, Hobson, Jones & Lasenby 1999; Sanz et al. 1999a,b; Tenorio et al. 1999). More recently, several authors have investigated the use of Gaussian–Hermite modes (or shapelets) in representing images of galaxies (Refregier 2003; Refregier & Bacon 2003). Although such image decomposition is commonplace, there exist a number of subtleties in the procedure that are not widely appreciated within the astronomical community. These include the effect of digitization, or sampling, on familiar notions from the analytic theory of continuous functions, in particular the concepts of completeness and linear independence of sampled basis functions, and the use of non‐orthogonal bases.

In this paper, we therefore discuss a general mathematical framework for the linear decomposition of an astronomical image into a set of arbitrary sampled functions (or modes). These may, in general, form a ‘basis’ that is non‐orthogonal and either undercomplete, perfectly complete or overcomplete. In the undercomplete case, the modes do not support all the degrees of freedom in an arbitrary image. By perfectly complete we mean that the modes support all the degrees of freedom without linear dependence between them, whereas for an overcomplete mode set there are additional modes relative to the complete case and hence linear dependence in the set. In particular, we show how to quantify those degrees of freedom in an image that can be supported by any particular mode set, and we explicitly consider how to obtain the optimal set of mode coefficients for representing the image. These methods present an opportunity for the optimization of sampling within modal analysis, ensuring that computational operations are reduced to a minimum.

The structure of the paper is as follows. In Section 2, we outline the modal decomposition problem and present a general prescription for obtaining optimal results, which is based on the singular value decomposition of the matrix of digitized mode vectors. In Section 3, we illustrate our general approach by investigating the decomposition of simple 1D function into Gauss–Hermite (shapelet) modes. This investigation is pursued further in Section 4, where we consider the decomposition of images of Hubble Deep Field galaxies into 2D Gauss–Laguerre (polar shapelet) modes. Finally, our conclusions are presented in Section 5.

2 THE MODAL DECOMPOSITION PROBLEM

Consider a 2D digitized image consisting of N pixels, which we denote by the N‐dimensional column vector d. Our goal is to represent the image, as accurately as possible, in terms of a set of M (2D) modes {ek}(k= 1, 2, … , M) defined at the same points at which d is sampled; thus each ek is also a column vector of length N. It is convenient to combine the mode vectors into the single N×M mode matrix defined by 

formula
Our objective is thus to determine a suitable coefficient vector graphic that minimizes the residual 
1
formula
Note that we make no requirement for the mode vectors to be mutually orthogonal, linearly independent or of unit length.

Several cases may arise in this problem. There may exist a unique coefficient vector graphic that minimizes equation (1); this corresponds to the quantity ε2 possessing a single well‐defined multidimensional quadratic minimum in the space of coefficients a. Surfaces of constant ε2 are given by the Hermitian form graphic, where R=EE (which is called the Gram matrix). The eigenvectors of R determine the principal directions of this multidimensional ellipsoid, and the extent along each principal direction is inversely proportional to the corresponding eigenvalue. If there exist approximate degeneracies in the coefficient space, some of the eigenvalues become very small and so the multidimensional ellipsoid is considerably elongated in these directions; this leads to a wide range of coefficients vectors a for which ε takes its minimum value to within the numerical precision. In the limiting case of exact degeneracies, some eigenvalues are identically zero leading to an infinite extent for the ‘ellipsoid’ along the corresponding principal directions. In the subspace spanned by these directions the value of ε takes its minimum value precisely. In either of the last two cases, to the numerical precision, there exist an infinite number of possible coefficient vectors graphic that minimize equation (1). The value of εmin is also of central importance. Clearly, if εmin= 0, then the modal expansion graphic provides an exact representation of the original image, whereas if εmin > 0 it represents only an approximation to d.

Which of the above cases occurs is dependent on the completeness and linear dependence of the mode set ek, and on the particular image d being decomposed. Clearly, if M < N then the modes are ‘undercomplete’, and so cannot represent an arbitrary image, but only an approximation to it. This may also occur when MN, however, as the modes may still not form a complete basis as a result of linear dependence between them. Nevertheless, even with an undercomplete set, the particular image d under analysis may lie in the subspace spanned by the modes, and so may be represented exactly. When MN cases arise where the modes form a complete basis, in which any image can be represented exactly. In this case, if M=N the modes must be linearly independent and form a ‘perfectly complete’ basis. If M > N, some linear dependence must exist between the modes and they form an ‘overcomplete’ basis.

In general, the degree of completeness of the chosen mode set may not be known at the outset. Fortunately, a straightforward technique exists for determining the degree of completeness of the mode set, and simultaneously determining an ‘appropriate’ coefficient vector graphic that minimizes equation (1). This is achieved by performing the singular‐value decomposition (SVD) of the mode matrix E, which we now discuss.

2.1 SVD

The SVD of the N×M mode matrix E (which may, in general, contain complex‐valued entries) may be written (see, for example, Press et al. 1994; Golub & van Loan 1996) 

2
formula
where U is unitary matrix of dimensions N×N, V is a unitary matrix of dimensions M×M, and Σ is a N×M matrix (the same dimensions as E) that is diagonal in the sense that Σiii for ip, where p= min[M, N], and zero otherwise. The coefficients σi are the singular values of the matrix E.

As is well known, the number r (say) of non‐zero singular values is equal to the rank of E, which in turn is the dimensionality of the image subspace that is spanned by the M modes {ek}; clearly it must be the case that rM and rN. The dimensionality of the nullspace of E (the subspace of vectors n in the coefficient space for which En=0) is given by Mr. It is thus a simple matter to determine the completeness of the mode set contained in E as follows: (i) if r < N the modes are undercomplete; (ii) if r=N and M=N the modes are perfectly complete; and (iii) if r=N and M > N the modes are overcomplete. Moreover, it is straightforward to show that the columns of U corresponding to non‐zero singular values constitute an orthonormal basis for the range of E, and the columns of V that do not correspond to a non‐zero singular value form an orthonormal basis for the nullspace. In practical numerical problems, it is often the case that none of the singular values σi are identically zero. Instead, one usually sets to zero those singular values for which |σi|/|σ1| < η, where η is some small factor (for example, 10−5 in single precision arithmetic) and σ1 is the first (and largest) singular value.

Once the SVD (equation 2) has been calculated, it is also straightforward to obtain an ‘appropriate’ coefficient vector graphic that minimizes equation (1). As we outline below, in all cases one should calculate the coefficient vector using 

3
formula
where the M×N matrix denoted by graphic is constructed by taking the transpose of Σ in equation (2) and replacing each non‐zero singular value σi by 1/σi.

It is clear that, with the above construction, graphic is an N×N diagonal matrix with diagonal entries that equal unity for those values of j for which σj≠ 0, and zero otherwise. Using this result it is straightforward to show that equation (3) does indeed minimize the residual (1). Modifying slightly the argument of Press et al. (1994), suppose we were to add to graphic some arbitrary vector a′=a1+a2, where a2 is the part of the vector that lies in the nullspace of E (if one exists) and a1 is the part that lies in the complement to the nullspace. This would result in the addition of the vector d′=Ea1 to graphic. We would then have 

4
formula
where in the last line we have made use of the fact that the length of a vector is left unchanged under the action of the unitary matrix U. Now, the jth component of the vector graphic will only be non‐zero when σj= 0. However, the jth element of the vector Ud′ is non‐zero only if σj≠ 0, as d′ lies in the range of E. Thus, as these two terms only contribute to equation (4) for two disjoint sets of j values, its minimum value, as a′ is varied, occurs when d′=0; this requires a1=0.

If the image d lies in the subspace spanned by the M modes {ek}, then the minimum value of the residual (1) is zero, and the image is represented exactly by graphic. Clearly, to represent an arbitrary image we require r=N (and so MN). If d does not lie in the range of E, then graphic yields only an approximate representation of the image (for this to occur, the modes must be undercomplete; this will always be true if M < N, but may also occur when MN).

In either case, we see from the argument above that, if E does not possess a null space, the coefficient vector (3) is unique in minimizing the residual (1). The condition for this to occur is that r=M, indicating that the M modes are linearly independent (and hence requiring MN). If E does possess a nullspace, however, any vector a2 in this nullspace can be added to equation (3), without changing the value of the residual. Thus there are an infinite number of coefficient vectors that minimize equation (1). The condition for this to occur is that r < M, indicating linear dependence between the modes (which may occur for MN and M > N).

In the case where E possesses a nullspace, from the infinite number of coefficient vectors that minimize the residual (1), the vector graphic in equation (3) is that which contains no contribution from the nullspace. An equivalent statement is that equation (3) is the vector of shortest length that minimizes equation (1). Consider again adding some arbitrary vector a′ to equation (3). The length of the resulting vector is 

formula
where in the last line we again make use of the fact that the length of a vector is left unchanged under the action of a unitary matrix. The jth component of the vector graphic will only be non‐zero when σj≠ 0, whereas the jth element of the vector Va′ is non‐zero only if σj= 0. Thus the minimum length vector has a′=0.

2.2 Dual modes

Although the appropriate coefficient vector may always be obtained straightforwardly from equation (3), it is conceptually appealing to consider the coefficient graphic as the scalar product of the image vector d with some new mode vector graphic that is dual to the original mode ek. In other words, it is often convenient to rewrite equation (3) as 

5
formula
where graphic is the N×M dual mode matrix, which contains the M dual mode vectors as its columns, i.e. 
formula
From equation (3), we see immediately that the dual mode matrix is given in terms of the SVD of the original mode matrix by 
formula
Thus, once a mode set has been defined, the dual mode set can be calculated directly by performing a SVD, without reference to any image. The coefficients for any particular image are then quickly obtained using equation (5).

It should be noted that, in general, 

formula
where we now start to place subscripts on identity matrices to emphasize their dimensionality. It is only in the case where the original mode set is complete that graphic, from which it is a simple matter to verify that graphic This, of course, corresponds to the case in which an arbitrary image d can be represented exactly, because then 
formula
Otherwise only an approximation to the original image is possible.

It is also worth pointing out that, in general, 

formula
In other words, the dual modes do not necessarily obey the standard orthogonality condition graphic with the original mode set. This orthogonality condition is only satisfied if the original mode set is linearly independent (for which r=M and hence MN). In this case, graphic, from which one quickly finds that graphic, which recovers the standard result that the dual modes form the reciprocal basis of the original mode set. This corresponds to the case where equation (5) is unique in minimizing the residual (1). If, in addition, we have M=N and so the basis perfectly complete, E is invertible and it follows that graphic. We reiterate, however, that all possible cases are automatically accommodated using SVD.

2.3 Orthogonal mode sets

So far, we have imposed no restrictions on the orthogonality, normalization or the number of members of our original mode set {ek}. It is worth investigating, however, the simplifications that occur in the case where the modes are mutually orthogonal and each mode is unique (so MN). Hence, we have EE=IM.

In this case, the mode set is automatically linearly independent and so the coefficient vector (equation 3) or (equation 5) is unique in minimizing the residual. Moreover, the dual modes and original modes obey the standard orthogonality condition graphic. In the case of orthogonal modes, it is clear that, in addition, this condition is satisfied if the dual modes are given by 

formula
Thus, for orthogonal modes (even with M < N), we see that each dual is simply proportional to the original mode. In the event that the original modes are normalized to unit length, this proportionality becomes an equality, and the dual modes are identical to the original modes, so that graphic and hence 
6
formula
This last result is, of course, the reason for the common practice, in the case of orthonormal modes, of calculating the kth mode coefficient graphic by simply evaluating the scalar product of the image d with the kth mode ek. Of course, when M=N, a set of orthogonal modes also forms a perfectly complete basis.

2.4 Practical considerations

Throughout our discussion, we have adopted the practical approach of working with finite dimensional vectors, rather than continuous functions. Thus, both the image d and the modes ek are considered simply as N‐dimensional vectors. This allows the direct application of our approach to a wide variety problems in the decomposition of digital astronomical images. The image pixel values di correspond to samples of the underlying continuous distribution d(x) at a particular set of points {xi}. In practice, it is most often the case that sample points are regularly spaced, but this restriction is not necessary. The entire approach can just as easily be applied to the case where the pixels values correspond to samples that are not regularly spaced. In an analogous manner, the elements of each mode vector ek are the sample values from some underlying continuous mode function fk(x) at the same set of points {xi}.

It is clear that the positions of the sample points will have a profound effect on whether the linear‐independence, orthogonality or other properties of the original set of continuous mode functions {fk(x)} are inherited by the mode vectors {ek}. Consider, for example, a set of continuous mode functions {fk(x)} that are orthonormal over some continuous domain graphic, so that 

formula
Even in the case where the sample points {xi} are evenly spaced, the spacing between them and the extent of the domain graphic that is sampled are important in determining whether the corresponding mode vectors {ek} are orthonormal. Indeed, as we shall see below, in some applications it is common for the corresponding mode vectors not to be orthonormal. If this is the case, then it is is no longer true that the dual mode vectors are identical to the original mode vectors, and so it is incorrect to calculate the coefficient vector graphic using equation (6), i.e. by taking the scalar product of the data d with the mode vectors. This will lead to a modal decomposition of the image that is unnecessarily inaccurate. Instead, one must return to using the general result (equation 5), where the mode coefficients are calculated by taking the scalar product of the data with the corresponding dual mode vectors.

Finally, a crucial point is that the converse may also occur. For example, if a set of continuous mode functions are not orthogonal, by sampling them at a particular set of (non‐uniform) points one can arrive at a set of mode vectors that are orthogonal, in which case the duals are trivially obtained.

3 ONE‐DIMENSIONAL ILLUSTRATIONS

Although the main focus of this paper is the modal decomposition of 2D astronomical images, it will be informative first to illustrate the discussion given above by investigating the decomposition of 1D sampled functions (which may be considered as cuts through some image). In particular, we will focus our numerical investigations in this section on decompositions using 1D Gaussian–Hermite (or shapelet) mode functions, as recently advocated by Refregier (2003) for representing images of galaxies.

In this case, the continuous mode functions are given by 

formula
for k= 0, 1, 2, … , ∞. The quantity β is some fixed ‘characteristic scale’ for the set of mode functions and the function φk(u) reads 
formula
where Hk(u) is a Hermite polynomial of order k. From these expressions it is clear that the mode functions are real and that β is equal to the dispersion ‘σ’ of the k= 0 Gaussian mode function. It is also straightforward to show that the mode functions are orthonormal over the domain −∞ < x < ∞, i.e. 
7
formula
We note that, for convenience, the mode functions are centred at x= 0. It is, of course, possible to centre them about some arbitrary point x=xc, but in most cases the coordinate system is chosen so that the centroid of the galaxy image (say) is located at the origin. The first six mode functions are shown in Fig. 1 for β= 12.

1

The Gaussian–Hermite continuous mode functions fk(x;β) for k= 0, 1, … , 5 and β= 12.

1

The Gaussian–Hermite continuous mode functions fk(x;β) for k= 0, 1, … , 5 and β= 12.

Let us now consider the decomposition of a 1D pixelized ‘image’ into shapelets. In any such decomposition, one is free to choose both the characteristic scale β of the modes set and the total number M of modes used (note that M=kmax+ 1). As shown by Refregier (2003), such a set of shapelet modes is suitable for describing features in an image on scales between the two limits 

8
formula
Thus, if the image has features on scales ranging from θmin (e.g. the pixel size or the size of the point spread function) to θmax (e.g. the size of the image or the overall extent of the structure it contains), then a good choice of β and M is 
9
formula

For each of the pixelized test images we consider below, we take the coordinate positions of the N (regularly spaced) sample points to be 

formula
thereby giving an image of spatial extent of N, with a pixel size of unity. In each case, we take N= 81. Taking into account that our test images contain structure on the order of the image size, but do not contain structure on scales less than a few pixels, acceptable choices for the scale parameter and the number of modes to use are β≈ 10 and M≈ 20. For convenience, we fix M= 20, and investigate mode sets for which β= 1–20 in steps of unity. In each case, the elements of the corresponding pixelized mode vectors are easily calculated from (ek)i=fk(xi; β).

In Fig. 2 we plot the elements of the first six mode vectors for β= 12. We see that, for this value of β, the support of the mode vectors for k > 2 exceeds the length of the image. Indeed, as k increases the support of the mode vectors increases according to equation (8), and so this effect becomes extremely pronounced for the high‐k modes. As a result, we would expect the mode set does not satisfy the discretized form of the orthogonality condition (7). We will show below that this is indeed the case.

2

The Gaussian–Hermite mode vectors ek for k= 0, 1, … , 5 and β= 12.

2

The Gaussian–Hermite mode vectors ek for k= 0, 1, … , 5 and β= 12.

It is also of interest to investigate the structure of the mode vectors for different values of β. As an illustration, in Fig. 3 we plot the highest‐k mode vector in the set (kmax= 19) for six different values of the scale parameter β. In particular, we note that the mode vector is severely undersampled for β= 1 and β= 2, in which case the mode set may again fail to be orthogonal. Moreover, as the highest‐k mode in the set has the largest spatial extent, we see that, in fact, for β≥ 6 the support of the mode set exceeds the length of the image, once again suggesting a loss of orthogonality.

3

The kmax= 19 Gaussian–Hermite mode vector e19 for β= 0, 1, 2, 3, 6, 10, 15.

3

The kmax= 19 Gaussian–Hermite mode vector e19 for β= 0, 1, 2, 3, 6, 10, 15.

3.1 Properties of the mode vectors

As we have chosen M < N, the mode vectors clearly cannot form a complete basis for the image space. Nevertheless, it is of interest to investigate formally the linear dependence and orthogonality of the mode vectors ek (k= 0, 1, … , M− 1) for various values of the scale parameter β. To this end, for each value of β under consideration, we construct the corresponding mode matrix E and then determine its rank r by calculating its SVD and counting the number of non‐zero singular values. If the mode vectors are linearly independent, one would expect r=M. In Fig. 4, we plot the value r/M as a function of the parameter β. When the number of mode vectors is M= 20, we see that they form a linearly independent (r/M= 1) set only if the scale parameter lies in the range β= 2–11. For smaller values of β, the undersampling of the underlying continuous mode functions leads to linear dependence in the resulting mode set. We note that this corresponds to the regime relevant to weak shear applications using shapelets. For large β, linear dependence occurs because the support of (some of) the mode vectors exceeds the length of the image. We also note that the range of β for which the mode vectors are linearly independent increases as the number of modes M is increased.

4

The ‘normalized’ rank r/M of the mode matrix E as a function of the scale parameter β for M= 10 (dashed line), M= 20 (solid line) and M= 30 (dot‐dashed line) Gaussian–Hermite mode vectors.

4

The ‘normalized’ rank r/M of the mode matrix E as a function of the scale parameter β for M= 10 (dashed line), M= 20 (solid line) and M= 30 (dot‐dashed line) Gaussian–Hermite mode vectors.

To investigate whether the mode vectors are orthonormal we calculate the Gram matrix R=EE for each value of β under consideration. Clearly, for an orthonormal set of mode vectors would expect R=IM. In Fig. 5 (top), for each value of β, we plot the logarithms of the largest and smallest diagonal elements of R. For mode vectors of unit length, we would expect all the diagonal elements to equal unity, and so the two plotted curves should coincide at log(Rii) = 0. We see that this occurs only if the scale parameter lies in the range β= 3–6. In the bottom panel of Fig. 5, we plot the value of the logarithm of the largest off‐diagonal element of R as a function of β. For an orthogonal set of mode vectors, one would expect all the off‐diagonal elements of R to be zero in the ideal case, or below the machine precision in a numerical implementation. We see that this is only the case if β= 3–4, although the off‐diagonal elements remain reasonably small for β= 5–6. For other values of β, however, we see that there exist off diagonal elements with absolute values greater than ∼0.1, which is a clear indication that the corresponding mode sets are not orthogonal. We also note that there exist values of β for which the modes are non‐orthogonal, but still have full rank.

5

Top: the logarithm of the value of the largest (solid line) and smallest (dashed line) diagonal elements of the Gram matrix R=EE as a function of the scale parameter β. Bottom: the logarithm of the largest off‐diagonal element of R as a function of β.

5

Top: the logarithm of the value of the largest (solid line) and smallest (dashed line) diagonal elements of the Gram matrix R=EE as a function of the scale parameter β. Bottom: the logarithm of the largest off‐diagonal element of R as a function of β.

3.2 Dual mode vectors

Since, for mode sets with β < 3 or β≳ 6, the mode vectors are not orthogonal, the corresponding dual mode vectors are not simply some multiple of the original modes. It is therefore of interest to investigate the form of the dual modes in these cases. As an illustration, in Fig. 6 we plot the first six dual mode vectors for the case β= 12. These should be compared with the corresponding original mode vectors plotted in Fig. 1. We see that the form of each dual is very different from the corresponding original mode vector. In particular, we note that the form of the duals is determined by the entire original mode set. As a result, the large spatial extent of the large‐k original mode vectors has an effect on the form of the low‐k dual mode vectors, even though the support of corresponding low‐k original mode vectors does not exceed the length of the image. Thus, for example, the k= 0 and k= 1 dual vectors are markedly different from the corresponding original mode vectors, even though the latter lie completely within the image length. For illustration, in Fig. 7 we also plot the dual mode vectors for the case β= 6, for which the original mode vectors are ‘marginally’ orthogonal. As one might expect, in this case the duals are very similar to the original modes.

6

The Gaussian–Hermite dual mode vectors graphic for k= 0, 1, … , 5 and β= 12.

6

The Gaussian–Hermite dual mode vectors graphic for k= 0, 1, … , 5 and β= 12.

7

The Gaussian–Hermite dual mode vectors graphic for k= 0, 1, … , 5 and β= 6.

7

The Gaussian–Hermite dual mode vectors graphic for k= 0, 1, … , 5 and β= 6.

Although clear from the discussion in Section 2.2, it is worth pointing out once more that the dual modes are not the basis set in terms of which an image is described. In the approach presented, the image is still represented as a linear combination of the original Gauss–Hermite (or shapelet) mode vectors. It is simply that the value of the coefficient ak in the linear combination is given by the scalar product of the image vector d with the kth dual mode graphic, rather than with the original mode vector ek. Thus, the advantageous properties of the Gauss–Hermite modes (or any other mode set under consideration) can still be used in the analysis of the resulting modal decomposition. From Fig. 6, however, it is clear that the dual vectors may not possess the same localization as the basis vector to which they relate. Consequently, for the k= 0 mode for example, a feature towards the edge of the field will affect the coefficients of the mode even though the original mode vector falls to zero there.

3.3 Decomposition of simple functions

Now that we have discussed the properties of the mode vectors and their duals for various values of the scale parameter β, it is of interest to investigate the effect of taking proper account of linear dependence and non‐orthogonality. To this end, we perform the modal decomposition of some simple 1D test functions using both the method advocated in Section 2.1 and the standard approach in which the coefficients in the decomposition are obtained simply by taking scalar products of the image vector with the original mode vectors (see e.g. Refregier 2003). The test functions considered, although simple, are relevant to the decomposition of astronomical images.

3.3.1 Uniform function

We begin by considering the modal decomposition of a uniform signal. The issue of accommodating (nearly) uniform background emission is clearly relevant in an astronomical context. The modal decomposition of this function was calculated, using both the standard method and the SVD method, for M= 20 modes and for integer values of the scale parameter in the range β= 1–20. In Fig. 8 we plot the resulting decompositions for the standard method (left‐hand column) and the approach advocated here (right‐hand column) for some representative values of β. These values have been chosen to correspond to mode sets for which the mode vectors are linearly dependent and non‐orthogonal, as demonstrated in Fig. 5. In Fig. 9, we plot the residual ε= |Ead| of the decompositions as a function of β for the standard method (solid line) and the SVD method (dashed line).

8

The uniform function (dashed line) and its modal decomposition (solid line) into M= 20 Gaussian–Hermite mode vectors with scale parameter β. The decomposition coefficients are obtained using the standard method (left‐hand column) and the SVD method (right‐hand column).

8

The uniform function (dashed line) and its modal decomposition (solid line) into M= 20 Gaussian–Hermite mode vectors with scale parameter β. The decomposition coefficients are obtained using the standard method (left‐hand column) and the SVD method (right‐hand column).

9

Decomposition residual ε= |Ead| as a function of β for the modal decomposition of the uniform function using the standard method (solid line) and the SVD method (dashed line).

9

Decomposition residual ε= |Ead| as a function of β for the modal decomposition of the uniform function using the standard method (solid line) and the SVD method (dashed line).

We see from Fig. 8 that, as one would expect, the decompositions produced by the two methods differ, in some cases significantly. It was confirmed, however, that for mode sets with β= 3 and β= 4, for which the mode vectors are linearly independent and orthonormal, the modal decompositions produced by the two methods coincide to within machine precision. In the cases illustrated in Fig. 8, we see that the SVD method provides a decomposition that is consistently better than that of obtained with the standard approach. In particular, we note that for a wide range of β values, the SVD method produces a decomposition that is visually indistinguishable from the input function, whereas the standard approach exhibits a pronounced ringing for all values of β. A quantitative description of the improved decomposition quality is given by Fig. 9. As expected, the SVD approach always produces the smallest decomposition residuals, indeed by many orders of magnitude for β≳ 7. We note, however, that for β < 7 both methods perform equally well to a good approximation. As the decomposition of an image is a linear process, the ability to describe accurately a uniform function enables better decompositions of discrete objects in a uniform background, which is clearly important in astronomical applications. We also note that, for the SVD method, the decomposition residual is a monotonically decreasing function of the scale parameter β. For the standard method, however, there is a shallow minimum in the residual at β= 7. It is interesting that, from Fig. 5, for this value of β the mode set is neither linearly independent nor orthogonal.

3.3.2 Top‐hat function

As our second illustration, we consider a top‐hat function of width equal to 20 units. Once again, this simple function is relevant to astronomical image analysis of, for example, saturated images of compact objects. The modal decomposition of this function was again calculated, using both the standard method and the SVD method, for integer values of the scale parameter in the range β= 1–20. In Fig. 10 we plot the resulting decompositions for the standard method (left‐hand column) and the SVD method (right‐hand column) for some representative values of β. The decomposition residuals are plotted in Fig. 11 as a function β for the standard method (solid line) and the method (dashed line).

10

As for Fig. 8, but for the top‐hat function of width 20.

10

As for Fig. 8, but for the top‐hat function of width 20.

11

As in Fig. 9, but for the top‐hat function of width 20.

11

As in Fig. 9, but for the top‐hat function of width 20.

We see from Fig. 10 that, once again, the two methods produce different decompositions for the values of β illustrated, and that the SVD method produces superior results. For both methods, however, the presence of only M= 20 modes leads to considerable ringing in the decompositions for almost all values of β. Nevertheless, it is interesting that for β= 1 the standard method is unable to produce a reasonable reconstruction, whereas an excellent decomposition is obtained over a limited extent of the function using the SVD method, the extent corresponding to the span of the mode set. The quantitative comparison of the two methods given in Fig. 11 once again shows that, by design, the SVD method always produces the most accurate decomposition. In this case, however, the difference between the two methods is not as great as it was for the decomposition of the uniform function. We also note that the residuals for both methods in this case are minimized at β= 3. At this point, the minimum residual is identical for the two methods, as the original mode set is linear independent and orthonormal to machine precision. For β= 2–10, both methods yield similar accuracies.

3.3.3 β‐model profile

Our final 1D illustration is that of a β‐model profile. Clusters of galaxies are often modelled (King 1972) as spherically symmetric, with an electron density profile of the form 

formula
where rc is the cluster core radius and b is a constant (this is usually denoted by β, but this has already been used in this paper to denote the scale parameter of the mode set). In this case, one easily finds that the projected thermal Sunyaev–Zel'dovich profile of the cluster is given by 
10
formula
where x the angular separation on the sky and λ= (3b− 1)/2. Assuming the standard value b= 2/3 gives λ= 1/2, and we use this value for our test function, together with a core length xc= 3.

The modal decomposition of this function was calculated, using both the standard method and the SVD method, for integer values of the scale parameter in the range β= 1–20. In Fig. 12 we plot the resulting decompositions for the standard method (left‐hand column) and the SVD method (right‐hand column) for some representative values of β. The decomposition residuals are plotted in Fig. 13 as a function β for the standard method (solid line) and the SVD method (dashed line).

12

As for Fig. 8, but for the β‐model profile with core radius xc= 3.

12

As for Fig. 8, but for the β‐model profile with core radius xc= 3.

13

As for Fig. 9, but for the β‐model profile with core radius xc= 3.

13

As for Fig. 9, but for the β‐model profile with core radius xc= 3.

From Fig. 12, we see again that the SVD methods produces superior decompositions for the values of β illustrated. For both methods, some ringing is observed in the decompositions, but this is much less pronounced for the SVD method. In particular, for large values of β, the standard method cannot accurately reproduce the narrow peak of the β model, whereas this is achieved by the SVD method. We also note that, for β= 1, the SVD method reproduces the central sharp peak very accurately. The quantitative comparison of the two methods given in Fig. 12 clearly shows the improvement obtained using the SVD method, which produces a residual significantly below that obtained using the standard method for β≳ 8. This is also the value of the scale parameter for which both methods yield their lowest residual, and the corresponding set of original mode vectors are close to linearly dependent and non‐orthogonal.

4 IMAGE DECOMPOSITION

We now consider the modal decomposition of real 2D astronomical images. These images were obtained by first using SExtractor (Bertin & Arnouts 1996) on the Hubble Deep Fields (HDFs; Williams et al. 1996, 1998). The convolution mask and detection parameters were adapted from those used by Massey et al. (2004). In particular, to allow the recovery of faint objects, we adopted a low signal‐to‐noise detection threshold, detect_thresh, of 1.3. Objects with class_star >97 per cent were discarded, as we wish to analyse only galaxies. The image was then segmented into small square ‘postage stamp’ regions around the remaining galaxies. The size of the images were set to 51 × 51 pixel. In this section, we investigate the decomposition of the HDF galaxy images into a set of 2D Gaussian–Laguerre modes.

4.1 Gauss–Laguerre mode functions

The 2D Gauss–Laguerre continuous mode functions are the simultaneous eigenstates of energy and angular momentum for the 2D isotropic quantum harmonic oscillator. Interestingly, these mode functions are also the solutions in polar coordinates to the paraxial wave equation in optics (see e.g. Goldsmith 1998). The relationship between the standard forms of these mode functions used in the above contexts is discussed in detail in the Appendix.

In the field of optical and quasi‐optical systems, it is customary (Murphy & Withington 1996; Withington, Murphy & Yassin 2000) to define the set of mode functions as 

11
formula
where x=r22 and L|m|p(x) are the standard associated Laguerre polynomials. The functions ψmp are conventionally called the Gaussian beam pm modes or simply the pm modes (see e.g. Goldsmith 1998) and form an orthonormal set over the infinite Euclidean plane. The radial index can take the values p= 0, 1, 2, … , ∞, and the azimuthal index m can take the values m= 0, ±1, ±2, … , ±∞. In practice, it is customary to truncate the mode set by imposing maximum values for p and |m|. Indeed, it is usual to choose |m|max=pmax, which leads to a total number of modes graphic. If f(r, θ) is real then clearly amp= (amp)*.

In the context of quantum mechanics, it is more usual to label the modes in terms of the energy quantum number n= 2p+ |m| and m, rather than p and m. Indeed, Refregier (2003) uses this convention to arrive at the polar shapelet modes 

12
formula
which also form an orthonormal set over the infinite Euclidean plane. The energy quantum number can take the values n= 0, 1, 2, … , ∞, and the azimuthal (angular momentum) quantum number takes the values m=−n, −n+2, … , n− 2, n, thereby giving n+ 1 values of m for each value of n. In practice, one truncates the mode set by imposing a maximum value for n, which leads to a total number of modes graphic. Once again, if f(r, θ) is real, then bmp= (bmp)*.

4.2 Properties of the mode sets

It is clear from the above discussion (and that given in the Appendix) that the two decompositions (11) and (12) use very different combinations of Gauss–Laguerre modes. The question thus arises as to the relative merits of the two mode sets in decomposing a 2D astronomical image. It is therefore of interest to investigate the properties of pixelized versions of these two mode sets.

As shown by Refregier (2003), by analogy with the criteria (9), the appropriate values of the scale parameter β and the maximum energy quantum number nmax for the polar shapelet mode set (12) are given by 

13
formula
for the decomposition of an image with features on scales ranging from θmin to θmax. The galaxy images to be decomposed are of size 51 × 51 pixels, and contain structure down to scales of around 3 pixel. Using the above criteria, we therefore choose nmax= 18. Thus, the total number of polar shapelet modes is 190. To allow a fair comparison between the polar shapelet and pm‐mode sets, one should arrange for the two sets to contain (as close as possible) the same total number of modes. Fortunately, by choosing pmax= 9, one can arrange for the pm‐mode set also to contain 190 modes. The corresponding pm mode and polar shapelet mode vectors are obtained by pixelizing the continuous mode functions on the same grid as the galaxy images. Each mode set contains only 190 modes, which is clearly far less than the number of pixels in the image, but need not be less than the number of degrees of freedom in some class of images. For an arbitrary image of infinite resolution the number of degrees of freedom and pixels will be the same and both decompositions can only represent an approximation to the image.

We may investigate the properties of the polar shapelet and pm‐mode sets in the same manner as we analysed the 1D shapelet modes. To this end, for each integer value of β in the range 0–20, we construct the mode matrix E for both the polar shapelets and pm modes. The rank r of both matrices was found to equal the number of modes (190) for all values of β under consideration, hence showing that both modes sets are linearly independent in all cases. To investigate whether the mode vectors in each set are orthonormal, we calculate the Gram matrix R=EE for each mode set for each value of β. For an orthonormal mode set R should equal the identity matrix. In Fig. 14 (top and middle), we plot the logarithm of the value of the largest and smallest diagonal element of R for the two mode sets. We see that, for both mode sets, the two curves coincide at log(Rii) = 0 only for β= 3 and 4, and so it is only for these values of the scale parameter that the mode vectors are of unit length. In the bottom panel of Fig. 14, we plot the logarithm of the largest off‐diagonal element of R for the polar shapelet (dashed line) and pm‐mode set (solid line). We see that, for each mode set, the largest off‐diagonal element is only below the machine precision for β= 3, and so it is only for this case that either mode set is orthogonal. We note that in Refregier (2003) and Massey et al. (2003), θmax was chosen to be approximately equal to the semimajor axis of the galaxies contained within images, a choice that corresponds to values of β associated with orthonormal bases.

14

Top: the logarithm of the value of the largest (solid line) and smallest (dashed line) diagonal elements of the Gram matrix R=EE as a function of the scale parameter β for the polar shapelet mode set. Middle: the same, but for the pm‐mode set. Bottom: the logarithm of the largest off‐diagonal element of R as a function of β for the polar shapelets (dashed line) and pm‐mode set (solid line).

14

Top: the logarithm of the value of the largest (solid line) and smallest (dashed line) diagonal elements of the Gram matrix R=EE as a function of the scale parameter β for the polar shapelet mode set. Middle: the same, but for the pm‐mode set. Bottom: the logarithm of the largest off‐diagonal element of R as a function of β for the polar shapelets (dashed line) and pm‐mode set (solid line).

4.3 Decomposition of HDF galaxy images

For the galaxy images to be decomposed, the result in equation (13) suggests one should set β= 14 for the polar shapelets; for comparison purposes we will also use this value of the scale parameter for the pm‐mode set.

To investigate the relative merits of the two mode sets, we first consider the decomposition of the single galaxy image shown in Fig. 15 (left). In the middle panel, we plot the decomposition residual using the standard approach for the polar shapelets (dashed line) and the pm modes (solid line) as a function of the total number of modes M in the set. As expected, for both mode sets, the decomposition residual decreases as more modes are used. We also see, however, that, for the same total number of modes, the pm modes yield a slightly lower decomposition residual than the polar shapelets. In the right‐hand panel of Fig. 15, plot the corresponding results obtained using the SVD approach. It is clear that, for any given value of M, the SVD approach outperforms the standard approach. We also see that, once again, the pm modes yield a lower decomposition residual than the polar shapelets. In Fig. 15 (middle and right panels) we also include an inset plot magnifying the region M= 0–21, which is the regime most commonly adopted for the decomposition of images of small weakly sheared galaxies with a low signal‐to‐noise ratio. We see that in this case, all four methods yield similar results.

15

Left: a HDF galaxy. Middle: the decomposition residual using the standard approach for the polar shapelets (dashed line) and the pm‐mode set (solid line) as a function of the total number of modes M in the set. Right: as middle panel, but using the SVD approach. The inset plots show a magnification of the region M= 0–21.

15

Left: a HDF galaxy. Middle: the decomposition residual using the standard approach for the polar shapelets (dashed line) and the pm‐mode set (solid line) as a function of the total number of modes M in the set. Right: as middle panel, but using the SVD approach. The inset plots show a magnification of the region M= 0–21.

The decomposition of five representative galaxy images into the pm‐mode set are shown in Fig. 16, using both the standard technique (middle column) and the SVD method (right‐hand column). As anticipated, as the mode set is not orthogonal, the two decompositions differ, markedly in some cases. For each galaxy image, however, we see that the SVD method produces a more faithful reconstruction. In particular, we note that the SVD method is more successful in reproducing the finer detail of the image. By contrast, the decompositions obtained using the standard approach contain structure only on larger scales. A quantitative comparison of the decomposition quality is given in Table 1, which lists the values of the residual ε of the decomposition. As expected, in each case the SVD method produces a more accurate decomposition than the standard method.

16

HDF galaxies (left‐hand column) decomposed into polar Gauss–Laguerre pm modes with pmax= 9 and β= 14 using the standard method (middle column) and the SVD method (right‐hand column).

16

HDF galaxies (left‐hand column) decomposed into polar Gauss–Laguerre pm modes with pmax= 9 and β= 14 using the standard method (middle column) and the SVD method (right‐hand column).

1

The residuals ε of the decompositions of the galaxy images shown in Fig. 16 for the standard method and the SVD method. The mode set used consisted of pm modes with pmax= 9, corresponding to M= 190.

1

The residuals ε of the decompositions of the galaxy images shown in Fig. 16 for the standard method and the SVD method. The mode set used consisted of pm modes with pmax= 9, corresponding to M= 190.

5 DISCUSSION AND CONCLUSIONS

We have discussed a general method for performing optimal image decomposition into a set of arbitrary digitized mode functions, which is an everyday problem in astronomy. Our approach may be applied straightforwardly even if the modes are linearly dependent, non‐orthogonal or incomplete, although in the last case the decomposition can only, in general, approximate the original image. By constructing the matrix E containing the digitized modes, the properties of the mode set can be easily obtained by performing a SVD. Moreover, having obtained the SVD, the optimal values for the coefficients in the linear decomposition can be obtained immediately. If desired, the optimal coefficients may also be obtained by constructing the set of dual modes, and performing scalar products of these duals with the input data vector. This should be contrasted with the standard method of taking scalar products of the original mode vector with the data vector. This latter approach yields optimal results only in the case where the digitized mode vectors form an orthonormal set. In particular, we note that the digitization process itself may lead to a mode set that does not inherit the properties of the continuous mode functions from which it is derived. Therefore, considerable care must be exercised when one performs decompositions using digitized versions of continuous mode functions that form orthonormal sets. Adopting the standard approach without first investigating the properties of the digitized modes can lead to unnecessarily inaccurate image decompositions.

We illustrate our general approach by first applying it to the decomposition of simple 1D functions into Gauss–Hermite (shapelet) modes. We show that, although the continuous shapelet mode functions form an orthonormal set on the entire real line, the corresponding digitized mode vectors can be linearly dependent and non‐orthogonal, depending on the relative values of the scale parameter β, the pixel size and the size of the image under analysis. We show that, for a wide range of these values, the method developed here produces decompositions that are significantly superior to those of the standard approach. In some cases, the decomposition residual for our method is many orders of magnitude lower than that for the standard technique.

We also illustrate our method by decomposing images of galaxies extracted from the HDFs into 2D Gauss–Laguerre modes. We consider both the polar shapelet approach where the modes are indexed using the energy and angular momentum quantum numbers n and m, and the pm modes used in optics, which are indexed by a radial number p and azimuthal number m. In both cases, although the set of continuous functions is orthonormal over the entire Euclidean plane, the corresponding digitized modes can be severely non‐orthogonal, depending on the relative sizes of the mode functions, the image size and the pixel scale. We show that the SVD method proposed here again outperforms the standard approach, yielding more faithful decompositions with lower residuals. We would therefore advise that in future applications of the shapelet decomposition method, the standard approach should be replaced with that presented here. We also find that the pm‐mode set provides consistently more accurate decompositions than the polar shapelets.

The decompositions presented in this paper used the SVD to provide an optimal decomposition for individual images. Frequently in astronomy, an analysis of the statistics of a collection of images is used to classify astronomical sources and properties. For such an application it may be desirable to use the duals method we have presented. In this case, the set of dual vectors can be calculated using a SVD, without reference to any image. The expansion coefficients for each image can then be very efficiently calculated by performing scalar products with the duals. Additionally, at this stage, a priori knowledge can be used (if desired) to constrain the duals to preferentially extract certain features from any image.

We conclude that the ability to work straightforwardly with linearly dependent, non‐orthogonal mode sets can be useful not just in accommodating digitization effects. In many applications, it may be more efficient to work with modes that are known a priori to have such properties. For example, in one or two dimensions, it can be prove useful to decompose images simultaneously in shapelet mode sets centred at a number of points in the image. The ‘multi‐centre’ shapelet bases will be discussed in a forthcoming paper. Finally we draw the readers attention to our own previous work investigating generalized bases or arbitrary completeness and orthogonality within the field of Terahertz optics (Berry et al. 2003a,b; Withington et al. 2004), including the description of second‐order statistics associated with partially coherent fields.

ACKNOWLEDGMENTS

RHB acknowledges the support of a PPARC studentship, the Cavendish Laboratory, the Cambridge Philosophical Society and Pembroke College, Cambridge. The authors thank Dr Phil Marshall for providing the images of HDF galaxies.

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Appendix

APPENDIX: PM MODES AND POLAR SHAPELETS

The Schrödinger equation for the 2D isotropic harmonic oscillator may be written in plane polar coordinates as 

formula
where we have set ħ=m= 1 and chosen our radial coordinate such that the potential is given by graphic. Seeking a separated solution of the form ψ(r, θ) =R(r)Θ(θ), one quickly finds the set of solutions 
formula
where L|m|p(x) is an associated Laguerre polynomial and A is a normalization constant. In addition to being energy eigenstates, these solutions are clearly also eigenstates of the angular momentum operator −i∂/∂θ with eigenvalue m. The angular momentum quantum number may take the values m= 0, ±1, ±2, … , ±∞, whereas the ‘radial’ quantum number may take the values p= 0, 1, 2, … , ∞. Fig. A1 shows the mode function distribution in the pm plane.

A1

The distributions of mode functions in the pm plane. The numbers in parentheses denote the value of the energy quantum number n for each mode. The dashed line indicates the usual geometry adopted in the pm plane for truncating the mode set; in this case pmax= |m|max= 3.

A1

The distributions of mode functions in the pm plane. The numbers in parentheses denote the value of the energy quantum number n for each mode. The dashed line indicates the usual geometry adopted in the pm plane for truncating the mode set; in this case pmax= |m|max= 3.

Interestingly, the functions (12) are also the solutions of the paraxial wave equation in optics (see e.g. Goldsmith 1998), where they are called the Gaussian beam pm modes or simply pm modes. These modes are often used to describe the electric field distribution in paraxial quasi‐optical systems. In this context, it is customary to truncate the mode set by imposing both a pmax value and a |m|max value. Moreover, it is usual to choose |m|max=pmax, so the truncated set of modes form a rectangular in pm space, as illustrated in Fig. A1. In this way, one obtains a total number of modes given by 

formula

It is straightforward to show that the energy of the mode ψmp is given by (reinstating ħħ for the moment) 

formula
where n is the energy quantum number. Indeed, in the context of the 2D harmonic oscillator, it is more usual to label the modes in terms of n and m, rather than p and m. In this way one arrives at the polar shapelet modes 
formula
where B is a normalization constant. The energy quantum number can take the values n= 0, 1, 2, … , ∞, whereas the azimuthal (angular momentum) quantum number takes the values m=−n, −n+ 2, … , n− 2, n; hence the energy level with quantum number n is (n+ 1)‐fold degenerate. As we have performed just a simple relabelling, the polar shapelets (12) and the pm modes have the same functional forms, and are directly related by ψmpm2p+|m|.

In Fig. A2 we show the distribution of modes in the nm plane. In Refregier (2003) and Massey et al. (2004), to perform the decomposition of 2D images, the mode set is truncated by imposing an nmax value, as illustrated in Fig. A2. In this way, one obtains a total number of modes given by 

formula

A2

The distributions of mode functions in the nm plane. The numbers in parentheses denote the value of the radial quantum number p for each mode and indeed may be a more intuitive index than n when investigating images due to p being an intrinisic measure of radial extent and scale. The dashed line indicates the usual geometry adopted in the nm plane for truncating the mode set; in this case nmax= 3.

A2

The distributions of mode functions in the nm plane. The numbers in parentheses denote the value of the radial quantum number p for each mode and indeed may be a more intuitive index than n when investigating images due to p being an intrinisic measure of radial extent and scale. The dashed line indicates the usual geometry adopted in the nm plane for truncating the mode set; in this case nmax= 3.

Footnotes

1
We note that this definition of the polar shapelet functions differs somewhat from that given in Refregier (2003) and Massey et al. (2003), which contain some typographical errors.