OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 25306–25320
« Show journal navigation

SVD for imaging systems with discrete rotational symmetry

Eric Clarkson, Robin Palit, and Mathew A. Kupinski  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 25306-25320 (2010)
http://dx.doi.org/10.1364/OE.18.025306


View Full Text Article

Acrobat PDF (2189 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

The singular value decomposition (SVD) of an imaging system is a computationally intensive calculation for tomographic imaging systems due to the large dimensionality of the system matrix. The computation often involves memory and storage requirements beyond those available to most end users. We have developed a method that reduces the dimension of the SVD problem towards the goal of making the calculation tractable for a standard desktop computer. In the presence of discrete rotational symmetry we show that the dimension of the SVD computation can be reduced by a factor equal to the number of collection angles for the tomographic system. In this paper we present the mathematical theory for our method, validate that our method produces the same results as standard SVD analysis, and finally apply our technique to the sensitivity matrix for a clinical CT system. The ability to compute the full singular value spectra and singular vectors could augment future work in system characterization, image-quality assessment and reconstruction techniques for tomographic imaging systems.

© 2010 Optical Society of America

1. Introduction

Medical imaging systems can be characterized by a linear system operator ℋ that maps a continuous object distribution through the system onto an image plane. The singular value decomposition (SVD) of ℋ provides an orthonormal set of basis functions that have numerous applications toward system characterization and image-quality assessment. For example, SVD data can be used to compute the null functions for an imaging system [1

1. H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991). [PubMed]

,2

2. H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

]. There have also been numerous publications that discuss the use of SVD in reconstruction methods [3

3. A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008). [CrossRef]

6

6. G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994). [CrossRef]

]. Additionally there is ongoing work investigating the use of singular vectors as channels for mathematical observer models [7

7. S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009). [CrossRef]

, 8

8. S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009). [CrossRef] [PubMed]

].

The forward model for an imaging system is often approximated by a sensitivity matrix H that maps voxels in object space to detector pixels in image space. To minimize the error in this approximation, it is desirable to have a large number of voxels to sample object space causing the column dimension N of the sensitivity matrix to be large. The row dimension of the sensitivity matrix is especially large in tomographic systems where M is the product of the number detector elements P and the number of collection angles J. The consequent large dimensions of H for tomographic medical imaging systems are problematic when using conventional SVD algorithms because they require adequate computer memory to compute and store HH or HH.

In this paper, we present a method to reduce the dimension of the SVD computation. We show that for tomographic systems with discrete rotational symmetry the dimension of the problem can be reduced by a factor equal to the number of collection angles. The theory and mathematics of our method are presented in Sections 2 – 5. In Section 6, we present results from a proof-of-concept experiment. These results verify that our reduced dimension algorithm produced the same data set as conventional SVD analysis. Finally, we show the singular value spectra and singular vector data that we obtained by applying the reduced dimension SVD to the sensitivity matrix for a simulated 3rd generation cone beam x-ray CT system.

2. The system operator and the symmetry operators

The symmetry operators we will be considering arise from rotations about an axis of the imaging system. If an object is described by a function f(r) of the three dimensional location vector r, and Rj is the matrix that describes a rotation by the angle 2πjJ about the symmetry axis, then an operator 𝒯j acting in object space is given by 𝒯jf(r)=f(Rj1r). The symmetry group in this case is ZJ, the cyclic group of order J. This group is represented in ℝ3 by the set of matrices {I, R1,..., RJ−1} which satisfy RjRk = Rj+k and R1J=I. These matrices can all be expressed in terms of R1 by Rj=R1j. Similarly this cyclic group is represented in object space by the set of operators {ℐ, 𝒯1,..., 𝒯J−1}, which satisfy the same multiplication rules as the rotation matrices.

Components of the data vector will be indexed by a vector m given by m = (p, j) = (px, py, j). The vector p identifies the location of a detector element in a given detector array, and the index j identifies the detector array. These J identical detector arrays are assumed to be spaced around the object at angles 2πjJ and to have identical apertures in front of them. If hp (r) is the detector sensitivity function for the detector located at position p on the detector with j = 0, then the detector sensitivity function for detector m is given by hm (r) = 𝒯jhp (r). The mean data vector has components given by
(g¯)m=g¯m=(f)m=Shm*(r)f(r)d3r,
(1)
which we will also write as = ℋf. If the number of detector elements in one detector array is P, then is a concatenation of J P-dimensional vectors j:
g¯=[g¯0g¯1g¯J1].
(2)
Thus ℋ is an operator from some function space U to data space V = ℝJP. We represent this symbolically by ℋ : UV. The decomposition of given above corresponds to the decomposition of V into an orthogonal direct sum of subspaces Vj, all of which are P-dimensional:
V=j=0J1Vj.
(3)
A different decomposition of V into subspaces arises from symmetry considerations and will be described below.

The components of the subvectors j can be thought of as the result of an operator ℋj acting on the object function:
(g¯j)p=g¯jp=(jf)p=S𝒯jhp*(r)f(r)d3r.
(4)
We will consider ℋj to be an operator that takes an object function and gives a vector in the subspace Vj of the full data space In other words, the result of applying this operator is the vector
jf=[00g¯j00].
(5)
Physically, the operator ℋj would be obtained by covering all of the apertures except the jth one. We will assume that the range of ℋj is all of Vj. We may then write the action of the system operator ℋ on an object function as
f=j=0J1jf,
(6)
with ℋj : UV and range(ℋj) = Vj.

Corresponding to the rotations of the object there are rotations of the data given by matrices Sj. The matrix S1 is described by the equation
S1g¯=[g¯J1g¯0g¯J2],
(7)
and the matrices Sj satisfy Sj=S1j. We may assume that the detectors are numbered in the clockwise direction so that ℋj = Sj0𝒯j. This equation says that we can get the projection on the jth detector array by rotating the object by an angle of 2πjJ, getting the projection on the 0th detector array, and then moving this P-dimensional data vector back to the correct location in the JP-dimensional vector .

The operator ℋ satisfies the symmetry condition Skℋ = ℋ𝒯k. In mathematical terms we say the operator ℋ intertwines the group representation {ℐ, 𝒯1,...,𝒯J–1} with the representation {I, S1,...,SJ–1}. Intuitively the symmetry condition tells us that rotating the object 2πjJ around the system axis gives the same result as rotating the ring of detectors through the same angle in the opposite direction. This condition places restrictions on the form of the system operator. These restrictions can be used to reduce the dimensionality of the SVD calculation for ℋ by at least a factor of J. Before we show this, we will introduce the inner products that we will use for object space and data space.

3. Inner products and adjoint operators

The SVD of any operator involves the notion of an adjoint operator. To define the adjoint of an operator, we need an inner product in the object space and in the data space. The inner product in object space will be defined by
(f,f)U=Sf*(r)f(r)d3r,
(8)
where S is a support region for all object functions and is invariant under the rotations in the symmetry group. For example, S could be a circular cylinder whose axis is the axis of the rotations in the group. In data space, we have the standard inner product
(g,g)V=mgm*gm.
(9)
We will have occasion to make use of complex object functions and data vectors, which is why the complex conjugate operation appears in these inner products.

We define adjoint operators 𝒯j and Sj by the equations (𝒯jf,f)U=(f,𝒯jf)U and (Sjg,g)V=(g,Sjg)V. The symmetry operators are unitary: 𝒯j=𝒯j1=𝒯j and Sj=Sj1=Sj. The first equality in each case is the unitary property, while the second follows from the definitions of the symmetry operators.

We can also define the adjoint of the system operator, ℋ, by the equation (ℋf, g)V = (f, ℋg)U. This operator is a map from data space to object space: ℋ : VU. In terms of the detector sensitivity functions, this adjoint operator is given by
(g)(r)=mgmhm(r).
(10)
This operator is also called the backprojection operator for the imaging system described by ℋ. We can use the the system operator given above to write
g=j=0J1jg=j=0J1𝒯j0Sjg.
(11)
Notice that we used the unitary property of the symmetry operators here. The adjoint operator j also maps data space to object space and its null space is the orthogonal complement to Vj. Symbolically, we write ℋ : VU and nullspace (j)=Vj. Next, we will consider a different decomposition of data space that arises from the symmetry group of the system.

4. Projection operators

In order to reduce the dimensionality of the SVD problem for ℋ, we need to decompose object functions into components that are eigenfunctions of the 𝒯j operators and data vectors into components that are eigenvectors of the Sj matrices. This process is essentially a Fourier decomposition with respect to the group of rotations. To this end define operators 𝒬k and Pk, for k = 0, 1,...,J – 1 via the equations [9

9. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).

]
𝒬kf=1Jj=0J1exp(i2πkjJ)𝒯jf
(12)
Pkg=1Jj=0J1exp(i2πkjJ)Sjg.
(13)
These operators satisfy the idempotent property: 𝒬k𝒬k = 𝒬k and PkPk = Pk. They are also Hermitian: 𝒬k=𝒬k and Pk=Pk. These two properties imply that these operators are orthogonal projections onto their ranges. If we apply a symmetry operator after a projection operator the result is
𝒯j𝒬kf=exp(i2πkjJ)𝒬kf
(14)
SjPkg=exp(i2πkjJ)Pkg.
(15)
This shows that the range of each projection operator consists of functions (or vectors) that are simultaneous eigenfunctions (or eigenvectors) of all of the symmetry operators. Further more, we have the decompositions
f=k=0J1𝒬kf
(16)
g=k=0J1Pkg.
(17)
The components in each of these decompositions are orthogonal to each other. If Ũj is the range of 𝒬j, and j is the range of Pj, then we have the corresponding orthogonal decompositions of object space and data space
U=j=0J1U˜j
(18)
V=j=0J1V˜j.
(19)
This suggests that we define the operators ℋ̃k by ℋ̃k = ℋ𝒬k = Pkℋ, where the second equality here follows from the symmetry property of ℋ. These operators are maps from object space to data space with the following ranges and null spaces: range (ℋ̃k) = k and nullspace (˜k)=U˜k. The system operator can be expressed as a sum of these operators:
=j=0J1˜j.
(20)
In this case, the decomposition of ℋ is in block diagonal form in the sense that ℋ̃j maps Ũk to k and vanishes on Ũj for jk.

The adjoint of ℋ̃k maps data space to object space and has the following properties: nullspace (˜k)=V˜k and range (˜k)=U˜k. The adjoint system operator also has block diagonal form:
=j=0J1˜j.
(21)
Then we find that the JP × JP matrix ℋℋ has block diagonal form as well:
=j=0J1˜j˜j.
(22)
Each block ˜j˜j in this decomposition maps j to j and vanishes on k for kj. This fact, together with the orthogonality of the decompositions of object and data space, implies that the SVD problem for the system operator reduces to finding the eigenvalues and eigenvectors for the J matrices ˜j˜j. These matrices map a P-dimensional space to itself and therefore the corresponding eigenvalue problems reduce to finding the eigenvalues for P × P matrices. The original JP×JP eigenvalue problem for the SVD of ℋ has now been reduced to a set of JP×P eigenvalue problems, which is more computationally tractable, especially for large values of J.

5. Reduction of dimension for SVD

We now want to find the eigenvalues and eigenvectors of the linear operator ˜k˜k. This operator is given by Pkℋ𝒬k = Pkℋ ℋ, where we have used the symmetry property of ℋ and the orthogonal projection properties of Pk. We may write this last form out as a triple sum:
˜k˜k=1Jj=0J1l0J1p=0J1exp(i2πkjJ)SjSl0𝒯l𝒯pH0Sp.
(23)
Now we let q = pl and re-index the sum over l to get
˜k˜k=1Jj=0J1q=0J1p=0J1exp(i2πkjJ)SjSpq0𝒯q0Sp.
(24)
Next we let r = j + pq and re-index the sum over j to arrive at
˜k˜k=1Jr=0J1q=0J1p=0J1exp(i2πk(rp+q)J)Sr0𝒯q0Sp.
(25)
By separating the exponential factor, we can separate the sums. The end result is ˜k˜k=J2Pk0𝒬k0Pk. We know that any eigenvector of this operator must lie in k, so we will write such a vector as Pkg in the eigenvalue equation: ˜k˜kPkg=λPkg. Since any vector in k may be reproduced from its values on the first detector, we may assume that g has the following form
g=[g000].
Now we look at the eigenvalue equation J2Pk0𝒬k0Pkg=λPkg and notice that, since 0 vanishes on V0 we must have JPk0𝒬k0g=λPkg. In this equation, J2 is replaced with J because Pk has a 1/J factor. These last two vectors are equal iff J0𝒬k0g=λg. This last equation can be regarded as a P × P linear system since the operator maps V0 to V0.

The result can also be formulated in terms of a lower dimensional SVD problem. We start with J0𝒬k0g=λg and write g0 = Zg so that Z is a P × M matrix. With g given as above, we also have g = Zg0. Our eigenvalue equation is now J0𝒬k0Zg0=λZg0. Multiplying both sides by Z, we get JZ0𝒬k0Zg0=λg0. Using the orthogonal projection properties of 𝒬k, we can write this as J(Z0𝒬k)(Z0𝒬k) g0 = λg0. Therefore, the eigenvalue problem for each k is equivalent to the SVD of the operator Z0𝒬k. Finally, note that Z0𝒬k = Zℋ𝒬k = ZPkℋ so that we could also do the SVD for these last two operators.

To make this calculation explicit, we define the operator 𝒜k by 𝒜k=JZ0𝒬k. Then we have the SVD equations for this operator in the data space for a single detector: 𝒜k𝒜kg0kl=λklg0kl. This is an eigenvector equation for a P × P matrix. The eigenvalues are real and non-negative. To get the singular vectors in the full data space, we use: gkl=JPkZg0kl. The J factor is there to preserve normalization. The singular functions fkl in object space are then given by: λklfkl=gkl. This expression can be expanded in terms of the single-detector system operator by using gkl=JPkZg0kl=J𝒬kZg0kl=J𝒬k0Zg0kl. This expression shows that we can generate the singular object functions by backprojecting a single detector singular vector with 0 and then applying the operator 𝒬k. This operator rotates the function through each of the J angles, multiplies each rotated function by a phase factor determined by k, and then sums. The symmetry properties of the resulting singular functions are determined by k.

6. Examples

A proof-of-concept experiment was conducted to compare the singular data from the reduced dimension SVD to the singular data from a standard dimension SVD computation. Each technique was used to calculate the SVD of a simulated 2-D parallel beam source x-ray CT system. In the absence of scattering, the system operator for an x-ray CT system can be linearized by making the approximations discussed in Barrett and Myers [2

2. H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

]. The discrete representation of this linear operator is the sensitivity matrix for the modeled system.

The sensitivity matrix, H, for the simulated x-ray CT system was computed using a discrete-to-discrete forward model to map from object space U to image space V. The object space was discretized into N = 256 × 256 voxels with an imposed circular field of view. The image space was measured with a P = 32 detector element camera collecting at J = 90 equally spaced angles over 360 degrees of rotation about the center of the field of view. As discussed in Section 5, the reduced dimension SVD therefore led to J SVD computations of the P × N matrices Ak=JZPkH. In comparison, the standard dimension SVD consisted of a single SVD computation of the JP × N matrix H. In both cases, the algorithm used to compute singular data for either the reduced dimension or the standard dimension matrix was the SVD function in the MATLAB software package [10

10. E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999). [CrossRef]

]. Computations for the proof-of-concept experiment were performed on an iMac computer equipped with a 3.2 GHz Intel Core i3 processor and 8GB of DDR3 memory. Calculation of all singular values and associated singular vectors up to the rank of H on this machine required 15.86 minutes using the reduced dimension SVD algorithm and 473.55 minutes using the standard dimension SVD algorithm. Our reduced dimension algorithm was therefore approximately 30 times faster than the standard dimension algorithm at computing the full SVD of the simulated sensitivity matrix. The speed of our algorithm could be further improved by trivially distributing the SVD computation of the Ak matrices onto the cores of a multi-core machine such as a Mac Pro computer.

Figure 1 shows the largest 16 singular values computed using the reduced dimension SVD and the standard dimension SVD. A slice through the central plane of the object space vectors for the 10 largest singular values is presented in Fig. 2.

Fig. 1 Plots showing the largest 16 singular values computed using the standard dimension SVD and the reduced dimension SVD. The spectra output by each SVD technique is equivalent.
Fig. 2 Images of slices through the central plane of the object space vectors associated with the 10 largest singular values for (a) the standard dimension SVD and (b) & (c) the reduced dimension SVD.

In Fig. 1 we can see that the some singular vectors occur in pairs with the same singular value (doublets), while others occur by themselves (singlets). The existence of this type of pattern would have been predicted from an analysis that made use of the full symmetry group of the system, which would include reflections as well as rotations [11

11. J. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph. D. Dissertation (University of Arizona, 1993).

13

13. P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

]. By replacing the cyclic group of rotations used in this work with the dihedral group of rotations and reflections, we can show that the spaces spanned by the singular vectors that share a singular value are predicted to be either one-dimensional or two-dimensional. This prediction follows from the fact that the irreducible representations of the dihedral group are either one-dimensional or two-dimensional [9

9. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).

].

In order to check the outputs from the reduced dimension algorithm we tested the orthonormality of the computed singular vectors. Orthonormality of singular vectors un can be expressed mathematically in terms of the inner product and the Kronecker delta as [2

2. H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

]
umun=δnm,n,m=1,2,N.
(26)
Shown in Table 1 and Table 2 are the magnitudes of the inner products for selected singular vectors that were computed using the reduced dimension SVD algorithm. The magnitudes of the inner products of singular vectors with the same index are unity while the magnitudes of the inner products of singular vectors with differing indices are close to zero. The error in the inner products is due to the estimations we made when simulating H for the proof-of-concept system. In our forward model, we used subvoxels and monte carlo techniques when quantifying the values of H at each collection angle. Consequently, our forward model does not strictly exhibit the property of discrete rotational symmetry discussed in Section 2. To further assess how this modeling error manifested itself in the SVD data, we compared singular vectors produced by the reduced dimension and standard dimension techniques. For this analysis we chose singlets u1, u6 and u15. We subtracted the singlet output by the reduced dimension SVD from the singlet output by the standard SVD resulting in an image of the error for the singlet. The results of this subtraction are shown in Fig. 3. The structure of these images shows that that there are contributions from additional singular vectors within each singlet that are a consequence of not preserving discrete rotational symmetry in our forward model. Note that the magnitude of the inner product between two singular vectors of differing indices that came from a reduced dimension SVD computation of the same Ak matrix (e.g. u1 and u6) are in fact zero to within the numerical precision of the computer.

Fig. 3 The difference between singular vectors output by the Standard SVD and Reduced SVD algorithms.

Table 1. The magnitudes of the inner products of singular vectors that were computed using the reduced dimension SVD. Shown in this table are |umun| for m = {1, 2,...,8} and n = {1, 2,...,16}

table-icon
View This Table
| View All Tables

Table 2. The magnitudes of the inner products of singular vectors that were computed using the reduced dimension SVD. Shown in this table are |umun| for m = {9, 10,...,16} and n = {1, 2,...,16}

table-icon
View This Table
| View All Tables

Fig. 4 An illustration of the geometry of a simulated clinical x-ray CT system for which we computed the SVD using the reduced dimension SVD algorithm. This system specifications were chosen to roughly approximate the Siemens Sensation 64 scanner ( Media 1, Media 2, Media 3, Media 4, Media 5, Media 6, Media 7, Media 8, Media 9, and Media 10).

Using a geometric forward projector, the sensitivity matrix for the simulated Siemens scanner was calculated. Applying the reduced dimension algorithm resulted in J SVD computations of P × N matrices. The singular data for a given P × N matrix was computed using a power methods algorithm [3

3. A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008). [CrossRef]

, 15

15. E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

]. Figure 5 shows the singular value spectra that was computed for the simulated Siemens x-ray CT system. Figure 6 shows the central 512 × 512 slice through the object space singular vector associated with the singular value spectra for singular value indices 1 – 10. Similarly, Fig. 7 and Fig. 8 show the central slice through the singular vectors associated with indices 88 – 92 and 175 – 179 respectively.

Fig. 5 A plot showing the singular value spectra for the simulated Siemens x-ray CT system. These data were computed using the reduced dimension SVD analysis.
Fig. 6 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 1–10.
Fig. 7 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 88–92.
Fig. 8 Images of slices through the central plane of the object space vectors for the modeled Siemens x-ray CT systems associated with singular value indices 175–179.

7. Conclusion

In this paper, we have shown that in the presence of discrete rotational symmetry, computation of the SVD for the imaging operator can be reduced by a factor equal to the number of collection angles for the system geometry. The results from our proof-of-concept experiment confirm that the reduced dimension SVD algorithm results in a decomposition that is analogous to that produced by a standard dimension SVD algorithm. However, data from the reduced SVD algorithm will be corrupted by approximations and estimates in the forward model that break the discrete rotational symmetry of the system operator. The proof-of-concept experiment also showed that the reduced dimension algorithm can significantly shorten the time required to compute an SVD. Using the same computing resources, the reduced SVD algorithm was able to calculate the singular value spectra and associated singular vectors up to the rank of our tested system matrix in 15.86 minutes compared to 463.55 minutes for a standard dimension SVD computation which is a factor of 30 times faster.

Our algorithm is especially useful for tomographic medical imaging systems in which the system matrix has large dimensions resulting in memory issues for a standard desktop computer running a SVD computation. As we have demonstrated, we were able to apply the reduced dimension SVD technique to the simulated system matrix of a 3rd generation clinical x-ray CT system resulting in the singular values and singular vectors associated with 180 values from the singular value spectra. If we were to run our algorithm for a longer time, we could compute more terms from the spectra yielding a data set that could be used to quantify image-quality assessment metrics such as the null space and measurement space for the imaging system [2

2. H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

]. Our reduced dimension SVD algorithm can therefore be used for tomographic medical imaging systems to output the higher order terms of the singular value spectra that would require extensive computing resources using standard SVD.

Acknowledgments

The authors thank Dr. Harrison Barrett and Dr. Lana Volokh for their helpful discussions on the topic of singular value decomposition. This research was supported by NIBIB/NIH grants RC1-EB010974 and P41-EB002035.

References and links

1.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Prog. Clin. Biol. Res. 363, 211–226 (1991). [PubMed]

2.

H. Barrett and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).

3.

A. K. Jorgensen and G. L. Zeng, “SVD-Based evaluation of multiplexing in multipinhole SPECT systems,” Int. J. Biomed. Imaging 2008, 769195 (2008). [CrossRef]

4.

Y. Hsieh, G. Zeng, and G. Gullberg, “Projection space image reconstruction using strip functions to calculate pixels more natural for modeling the geometric response of the SPECT collimator,” IEEE Trans. Med. Imaging 17(1), 24–44 (1998). [CrossRef] [PubMed]

5.

G. Zeng and G. Gullberg, “An SVD study of truncated transmission data in SPECT,” IEEE Trans. Nucl. Sci. 44(1), 107–111 (1997). [CrossRef]

6.

G. Gullberg and G. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels,” IEEE Trans. Nucl. Sci. 41(6), 2812–2819 (1994). [CrossRef]

7.

S. Park and E. Clarkson, “Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds,” J. Opt. Soc. Am. A 26(11), 59–71 (2009). [CrossRef]

8.

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28(5), 657–668 (2009). [CrossRef] [PubMed]

9.

M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).

10.

E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999). [CrossRef]

11.

J. Aarsvold, “Multiple-pinhole transaxial tomography: a model and analysis,” Ph. D. Dissertation (University of Arizona, 1993).

12.

J. Aarsvold and H. Barrett, “Symmetries of single-slice multiple-pinhole tomographs,” in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673–1677. [CrossRef]

13.

P. Varatharajah, B. Tankersley, and J. Aarsvold, “Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors,” in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184–1188.

14.

S. Steckmann, M. Knaup, and M. Kachelrieß, “High performance cone-beam spiral backprojection with voxel-specific weighting,” Phys. Med. Biol. 54(12), 3691–3708 (2009). [CrossRef] [PubMed]

15.

E. Isaacson and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

OCIS Codes
(110.2960) Imaging systems : Image analysis
(110.3000) Imaging systems : Image quality assessment
(110.6955) Imaging systems : Tomographic imaging

ToC Category:
Imaging Systems

History
Original Manuscript: August 13, 2010
Revised Manuscript: November 11, 2010
Manuscript Accepted: November 11, 2010
Published: November 19, 2010

Citation
Eric Clarkson, Robin Palit, and Matthew A. Kupinski, "SVD for imaging systems with discrete rotational symmetry," Opt. Express 18, 25306-25320 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25306


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Prog. Clin. Biol. Res. 363, 211-226 (1991). [PubMed]
  2. H. Barrett, and K. Myers, Foundations of Image Science (John Wiley and Sons, 2004).
  3. A. K. Jorgensen, and G. L. Zeng, "SVD-based evaluation of multiplexing in multipinhole SPECT systems," Int. J. Biomed. Imaging 2008, 769195 (2008). [CrossRef]
  4. Y. L. Hsieh, G. L. Zeng, and G. T. Gullberg, "Projection space image reconstruction using strip functions to calculate pixels more "natural" for modeling the geometric response of the SPECT collimator," IEEE Trans. Med. Imaging 17(1), 24-44 (1998). [CrossRef] [PubMed]
  5. G. Zeng, and G. Gullberg, "An SVD study of truncated transmission data in SPECT," IEEE Trans. Nucl. Sci. 44(1), 107-111 (1997). [CrossRef]
  6. G. Gullberg, and G. Zeng, "A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential radon transform using natural pixels," IEEE Trans. Nucl. Sci. 41(6), 2812-2819 (1994). [CrossRef]
  7. S. Park, and E. Clarkson, "Efficient estimation of ideal-observer performance in classification tasks involving high-dimensional complex backgrounds," J. Opt. Soc. Am. A 26(11), 59-71 (2009). [CrossRef]
  8. S. Park, J. M. Witten, and K. J. Myers, "Singular vectors of a linear imaging system as efficient channels for the bayesian ideal observer," IEEE Trans. Med. Imaging 28(5), 657-668 (2009). [CrossRef] [PubMed]
  9. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover Publications, 1989).
  10. E. Anderson, Z. Bai, and C. Bischof, LAPACK Users’ Guide (Society for Industrial Mathematics, 1999). [CrossRef]
  11. J. Aarsvold, "Multiple-pinhole transaxial tomography: a model and analysis," Ph. D. Dissertation (University of Arizona, 1993).
  12. J. Aarsvold, and H. Barrett, "Symmetries of single-slice multiple-pinhole tomographs," in Conference Record of the 1996 IEEE NSS/MIC (IEEE, 1997), vol. 3, pp. 1673-1677. [CrossRef]
  13. P. Varatharajah, B. Tankersley, and J. Aarsvold, "Discrete models and singular-value decompositions of single-slice imagers with orthogonal detectors," in Conference Record of the 1998 IEEE NSS/MIC (IEEE, 1999), vol. 2, pp. 1184-1188.
  14. S. Steckmann, M. Knaup, and M. Kachelrieß, "High performance cone-beam spiral backprojection with voxel-specific weighting," Phys. Med. Biol. 54(12), 3691-3708 (2009). [CrossRef] [PubMed]
  15. E. Isaacson, and H. Keller, Analysis of Numerical Methods (Dover Publications, 1994).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Supplementary Material


» Media 1: MOV (40 KB)     
» Media 2: MOV (55 KB)     
» Media 3: MOV (57 KB)     
» Media 4: MOV (66 KB)     
» Media 5: MOV (53 KB)     
» Media 6: MOV (56 KB)     
» Media 7: MOV (71 KB)     
» Media 8: MOV (59 KB)     
» Media 9: MOV (70 KB)     
» Media 10: MOV (59 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited