WO2008087433A1

WO2008087433A1 - Signal transformation and correction techniques

Info

Publication number: WO2008087433A1
Application number: PCT/GB2008/000177
Authority: WO
Inventors: Bernd Matthaus Hofer; Wolfgang Drexler; Boris Ladislaus Wolfgang Povazay
Original assignee: University College Cardiff Consultants Ltd
Priority date: 2007-01-18
Filing date: 2008-01-18
Publication date: 2008-07-24
Also published as: GB0700987D0

Abstract

The invention relates to the correction of the spectrometer signal in Spectral-Domain Optical Coherence Tomography. The output of the spectrometer is treated as a set of sub-band signals according to filter bank theory and is fed to a synthesis filter bank to reconstruct a corrected OCT signal. In particular, the frequency response of the CCD camera is corrected for. The synthesis filter bank is calculated after a calibration of the frequency response of the element of interest.

Description

SIGNAL TRANSFORMATION AND CORRECTION TECHNIQUES

The invention relates to techniques for treating a signal occupying a position in a signal space defined by a base and which has been transformed into a spectral representation by detection means so as to transform the spectral representation into a signal in said space.

Figure 1 illustrates a conventional spectrometer-based, frequency domain optical coherence tomography (OCT) apparatus 1. Light source 10 is a broadband laser (e.g. a femtosecond laser) and emits light beam E₁. At the beam splitter 12, a part of beam E₁ is diverted towards a fixed reference mirror 14 as beam E_r whilst the remainder of beam E₁ passes through the splitter 12 as beam E_s which proceeds towards the sample 16 that is being investigated by the OCT apparatus 1. Light travelling to the sample 16 from the beam splitter 12, and vice versa, passes through a suitable arrangement of scanning optics 18 that directs beam E_s to a desired site in the sample 16. Beam E_r is reflected from mirror 14 as beam E_r(τ) and passes through the splitter 12 to form one part of beam E_d . Light returning from the sample 16 forms beam E_s which is diverted by the splitter 12 to form the other part of beam E_d . Beam E_d is incident on a diffraction grating 20 which disperses the frequencies in beam E_d over a CCD line camera 22. The outputs of the

CCDs in the line camera 22 are digitised and provided to a post-processing stage 24. The post-processing stage 24 is arranged to produce a depth scan from the signals provided by the CCDs of the line camera 22 by techniques that are well understood in the art and which rely on fast Fourier transformation (FFT) algorithms. The post-processing stage 24 may be a PC but all that matters is that the post-processing stage 24 is endowed with computing power sufficient to perform the requisite calculations within an acceptable time frame. The depth scan is a vector of reflectivity values lying along the z axis within the sample 16 at the site selected by the scanning optics 18. Depth scans can be produced for a number of sites over the sample 16 to build up a tomogram of the sample 16, which can be shown on a suitable display device 26, such as an LCD screen.

The invention is defined by the attached claims to which reference should now be made. By way of example only certain embodiments of the invention will now be described by reference to the accompanying drawings, in which:

Figure 1 is a schematic representation of a known OCT system;

Figure 2 is a schematic representation of an OCT system according to an embodiment of the present invention;

Figure 3 is a schematic representation of a filter bank;

Figure 4 is a schematic representation of a variant of the OCT system shown in Figure 2;

Figure 5 is a plot of line camera CCD response versus frequency for three neighbouring CCDs within a line camera; and

Figure 6 is a schematic representation of a further variant of the OCT system shown in Figure 2.

Figure 2 illustrates an OCT apparatus 28 according to an embodiment of the invention. Elements carried over from Figure 1 to Figure 2 retain the same reference numerals and, for brevity, the description of their functions and constitution shall not be repeated. In OCT apparatus 28, the post-processing stage 24 has been replaced with post-processing stage 30. In essence, stage 30 processes the signals from the CCDs of the line camera 22 in a different manner (compared to stage 24) to produce the depth scans. Stage 30 relies on filter bank theory rather than FFT techniques to produce the depth scans, in a manner that will now be described.

Descriptions of fundamentals of filter bank theory can be found in, for example, P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, New York (1993), P. P. Vaidyanathan Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications: A Tutorial, Proceedings of the IEEE, Vol. 78, No. 1 January 1990 and M Vetterli et al, Wavelets and Filter Banks: Theory and Design, IEEE Transactions on Signal Processing, Vol. 40, No.9 September 1992. A variety of tools for description and proper calculation of filter banks are available, and those that are chosen to be employed in a given scenario depend on the filter bank model that is being considered and on the type of filters used. ^■

Figure 3 illustrates a filter bank framework 32. A signal z is supplied to a bank of i parallel band pass filters (BPFs) 34-1 to 34-i, which bank is termed an analysis filter bank 36. Each BPF in bank 36 has a different pass band. The signal values produced by the BPFs 34-1 to 34-i are labelled S₁ to Sj and these values are often termed sub-band signals. Taken together, the sub-band signals constitute a vector p such that p = [s_l s₂ s₃ S₁]. The sub- band signals S₁ to si are supplied to respective BPFs 40-1 to 40-i, each BPF having a different pass band. The BPFs 40-1 to 40-i constitute a synthesis filter bank 42. The outputs of the BPFs 40-1 to 40-i are then combined to produced signal _£_, which is an approximation to signal z.

If one assumes that the BPFs 34-1 to 34-i and 40-1 to 40-i are critically sampled (i.e. the number of sub-bands, i, equals the number of samples within z and each sub-band signal consists of just one sample - for the avoidance of doubt, the invention extends to systems that are not critically sampled) with time-invariant finite-impulse-responses (FIRs), the pass bands (transfer functions) of the filters in the analysis bank 36 can also be presented via a convolution matrix (so-called analysis matrix A) and, likewise, the pass bands of the synthesis bank 42 can be presented as a so-called synthesis matrix S. Given a set of transfer functions the corresponding impulse responses can be constructed via a variety of methods, e.g. one can use a frequency-sampling-based FIR design. If the aim is to make J_ match z, then ideally it would be arranged that SA=I (I being the unity matrix) or, in other words, that S=A^"1 or that S=A⁺ with A⁺ being the pseudo inverse of A . Matrix A can be understood as the mathematical representation of the physical system that acquires z, namely its spectral or transform domain representation via sub-band vector p. : The synthesis matrix represents the post-processing stage that tries to rebuild the vector z as close as possible to match the original within set limitations, which finally results in z_.

The outputs of the CCDs of line camera 22 of OCT apparatus 28 can be considered as a set of sub-band signals for a filter bank of the type shown in Figure 3, the signal z being taken to be the depth profile of the sample 16 at the site specified by the scanning optics 18.

Accordingly, if matrix S can be estimated, then z , which is depth scan estimating depth profile z , can be estimated by calculating S p . Of course, this requires the estimation of A so that S can be calculated in the form A^"1 or in the form of some other matrix that maximises the similarity between _z_and the original reflectivity profile z . An example of a scheme for estimating A will now be outlined.

Figure 4 shows the OCT apparatus 28 reconfigured for the purpose of estimating matrix A. The broadband source 10 has been switched out of the optical path leading to the beam splitter 12 and in its place a tuneable, narrowband laser 44 has been switched in. Moreover, the optical path to the sample site is closed off since this is not required for calibration purposes. The beam E₁ produced by laser 44 is deflected to mirror 14 by splitter 12 from whence it returns to the splitter 12, through which it passes, to arrive at grating 20 as beam E _d . The frequency width of laser 44 is specified such that its beam, upon dispersion by grating 20, will illuminate approximately one (or a number of that order of magnitude) CCD of the line camera 22. The tuning range of the laser 44 is specified so that its beam, upon dispersion from grating 20, can be scanned over the whole extent of the CCD line constituting camera 22. The frequency of laser 44 is then varied in a series of steps sufficient to cause the beam dispersed by grating 20 to sweep from one end of the CCD line to the other. At each step of the frequency of laser 44, the values of the outputs of the CCDs are recorded by the post-processing stage 30. These values are refined by static background subtraction, noise cancellation and energy preservation and other such standard techniques. The recorded CCD outputs thus refined establish a two dimensional table of CCD output values, the table values being indexed by CCD position (within the camera's CCD line) and output frequency of laser 44. This table is an estimate of the transfer function description of matrix A. That is to say, an estimate of the transfer function of CCD responsiveness versus frequency for a given CCD in the camera is provided by the set of values in this table that is indexed by the CCD position in question.

Figure 5 shows three rows of such a table, each row corresponding to a different CCD in the CCD line, plotted on the same axes. The three rows shown are each indexed by CCD position and correspond, respectively, to the transfer functions for the 64^th, 65^th and 66^

CCDs in a line camera of undisclosed length.

The table need not be populated by tuning the laser 44 to sweep the dispersed beam over the entirety of the CCD line. The laser may be tuned to a plurality of discrete frequencies or swept over just a part of the tuning range needed to span the CCD line. In such circumstances, only part of the table is populated by measurement. The remaining entries in the table can then be created through interpolation or similar techniques.

Next, a frequency sampling procedure is applied to the transfer functions making up the table in order to estimate the impulse responses constituting matrix A. In cases where only the magnitude transfer function is accessible, suitable phase estimation procedures can be applied, e.g. complex valued FIR functions can be constructed via the Hubert transform.

Once matrix A has been estimated, the optimum matrix S can be deduced. Either by simple inversion, which is only possible in a very limited number of cases or by, for example, using a least squares minimisation approach to find an instance of S that minimises the quantity SA - 1 . It will however be apparent to the skilled person that other known techniques are available for deducing S such that it at least approximately satisfies the relation SA=I. To obtain a stable solution via the synthesis matrix S suitable windowing is applied on S as well as regularization procedures within the least squares algorithm for calculation of S.

Figure 6 illustrates an OCT apparatus 47 that is a variant of the OCT apparatus 28 of Figure 2 that can be used for calculating matrix S. It, will be noted that in Figure 6 fixed mirror 14 has been replaced with a mirror 46 that can be moved along the path extending from that mirror to the splitter 12. Moreover, the OCT apparatus 47 is provided with a fixed mirror 48 that can be switched into the optical path between the splitter and the scanning optics 18. In Figure 6, the OCT apparatus 47 is shown with the mirror 48 switched into the optical path and as a result the scanning optics 18 and the sample zone are not shown, in order to maintain clarity. A scheme for calibrating OCT apparatus 47 and deducing its filter bank representation analysis matrix will now be described. The position of mirror 46 is stepped over a range during the calibration process (although its position is fixed in the post-calibration measurement mode) and for each position of the mirror 46, the response of the line of CCDs in camera 22 is recorded by the postprocessing stage 30. As in the Figure 4 arrangement, these values are refined by background subtraction, noise cancellation and energy preservation and other such standard techniques. Thus, a two dimensional table of CCD response values can be built up, indexed by CCD position (i.e. within the line camera) and mirror position (i.e. the position of mirror 46). Each position of the mirror 46 gives rise to a depth scan z. Each of these depth scans is known because the mirror 46 positions are known (since the mirror is moved by a calibrated drive mechanism) and the reflectivity profile of the mirror (which should be a narrow peak or pulse) is known. Therefore, an optimized solution for synthesis

2 matrix S can be constructed by solving the optimization problem argmin SP-Z .Here,

the columns of Z are the known depth scans z, and the columns of P are the sub-band vectors p that correspond to the depth scans.

Two methods of estimating S have now been described in detail. Having arrived at an estimate for S, depth scans of sites in actual samples can be performed. To do so, the vector p describing the sub-band signals representing the refined outputs of the camera

(i.e. refined by background subtraction, noise cancellation and energy preservation and other such standard techniques) when the scanning optics 18 targets a desired site in the sample 16 is processed by stage 30 to yield a depth scan z for the site in question using the relation Z = S p . Depth scans from various sites in sample 16 can be assembled in known fashion to create a tomogram of the sample.

It will be apparent to the skilled reader that the embodiments described above recover spatial domain information about the sample 16, i.e. a depth scan, from information provided in the frequency domain, i.e. the sub-band signals constituting the outputs of the CCDs of the line camera 22. In other words, matrix S effects a transformation between the frequency and spatial domains.

The techniques described above for translating between information in two different domains can be introduced to other types of system such as time-encoded, frequency domain OCT systems, where broadband source 10 in Figure 2 is replaced by a tuneable narrowband light source and the grating 20 is replaced by one ore more detectors with broadband sensitivity. In this kind of apparatus, the spectrum is acquired as a function of time, rather than space. However, the spectrum can still be post-processed according to the filter bank approach.

The principles of the invention can also be used to calibrate systems intended to transform information from one domain to another.

For example, consider a spectrometer for producing a frequency domain spectrum from a time varying optical input signal (e.g. Fourier transform spectroscopy). The input signal can be regarded as a vector z where the spectrometer constitutes an analysis filter bank that can be represented by a matrix A such that the spectrometer performs the operation Az to produce a set of sub-band signals that can be represented by a vector p and which constitute the spectrum of z as measured by the spectrometer. If the signal z is a test signal with a known spectrum that should manifest as a set of sub-band signals y , then it is

2 possible to deduce a matrix M that minimises the quantity M p — y Matrix M so estimated can then be multiplied with vector p to produce a corrected or calibrated spectrum during a live measurement on some other form of z .

For example, a CT scanner will produce a set of sub-band signals containing frequency domain information about spatial attributes of a sample and it is possible to characterise the CT scanner and estimate an analysis matrix A that produces the sub-band signals and then deduce a synthesis matrix S (e.g. by the least squares approach mentioned above) which can be used to transform the frequency domain information in the sub-band signals into spatial domain information for the production of tomograms.

As a further example, consider a simple digital camera with a lens system transferring information from its object plane to a two-dimensional CCD array located at the Fourier plane of the lens system. Without imaging aberrations the two dimensional Fourier transform of the obtained image would represent the real image and could be directly computed. A known test pattern can be sited at the object plane, it being known what the response of the CCD array should be if the lens system was aberration free. The actual response of the CCD array can be represented as a two dimensional matrix P, in effect a set of sub band signals. The desired, i.e. perfect, response can be represented by a two dimensional matrix Y and, again, a matrix M can be deduced such that MP=Y, again by finding an instance of M that minimises the quantity MP- Y . The matrix M can thereafter be employed in the corrected calculation of images captured by the camera.

IQ the foregoing description, the filters utilised in the filter bank treatment are of the FIR type. It will of course be appreciated that filters of the infinite impulse response (IIR) type or indeed of other types could have been used instead. It is also possible to recalculate the synthesis filter bank from time to time to counter non-linear effects that might arise within - a given system.

It is also possible to adjust the synthesis filter bank to correct for dispersion and other effects arising from material (e.g. vitreous humour) in the path leading to the sample (e.g. a retina). This can be achieved by altering the matrix S representing the synthesis filter bank using a blind deconvolution approach.

Referring back to Figure 3, it is possible to subject the outputs of BPFs 40-1 to 40-i to respective all pass filters prior to the combination of these signals to produce z , as part of a scheme for compensating for bulk dispersion to improve image sharpness. The all pass filters adjust for the dispersive effect that broadens peaks due to material dependent phase distortion of the multiple spectral components constituting the peak. These phase distortions arise from different propagation speeds of electromagnetic waves with unequal wavelengths. Known algorithms for improving image sharpness can be used to configure the parameters of the all pass filters to combat this dispersion.

Thus, various arrangements have been described in which filter bank analysis is used to link a signal, e.g. a vector representing a reflectivity versus depth profile, in a given signal space, that space being represented by a base or basis vector set, to a spectral representation, e.g. a spectrometer output.

Claims

1. A method of signal processing involving a signal occupying a position in a signal space defined by a base , and which has been transformed into a spectral representation by detection means, the method comprising treating the spectral representation as a set of sub-band signals according to filter bank theory and feeding the sub-band signals into a synthesis filter bank that is configured, to transform the spectral representation into a signal in said signal space base.

2. A method according to claim 1, wherein the synthesis filter bank is configured to reconstruct said signal.

3. A method according to claim 1 or 2, wherein the signal is a depth 'scan and the spectral representation is obtained from an object by an optical coherence tomography system.

4. A method according to claim 3, wherein the optical coherence tomography system generates the spectral representation by probing the object with a broad-band, low coherence light source.

5. A method according to claim 3, wherein the optical coherence tomography system generates the spectral representation by probing the object with a narrow-band, swept-frequency light source.

6. A method according to any one of claims 1 to 5, wherein the synthesis filter bank contains all pass filters and the method comprises adjusting the all pass filters to control the sharpness of the signal as transformed back into said signal space by the synthesis bank.

7. A method of designing a synthesis filter bank for use with a signal occupying a position in a signal space defined by a base and which has been transformed into a spectral representation by detection means, the method comprising stimulating the detection means with a known signal in said space to produce a spectral representation and constructing the synthesis filter bank .such that it transforms the spectral representation into a signal in said signal space base using the frequency domain response as a set of sub-band signals as inputs for the synthesis filter bank.

8. A method according to claim 7, wherein the known signal comprises a series of positions of moveable reflecting means.

9. A method of designing a synthesis filter bank for use with a signal occupying a position in a signal space defined by a base and which has been transformed into a spectral representation by detection means, the method comprising constructing an analysis filter bank that models the transformation of signals from said space to said spectral representation and calculating from the analysis filter bank a synthesis filter bank that will transform a spectral representation into a signal in said signal space base.

10. A method according to claim 9, wherein the step of calculating a synthesis filter bank comprises calculating by a least squares approach a matrix that represents the synthesis filter bank and which is an inversion or pseudo-inversion of the a matrix representing the analysis filter bank.

11. A method of determining a correction for a system that transforms a signal occupying a position in a signal space defined by a base into a spectral representation, the method comprising providing the system with a known signal in said space, treating the spectral representation produced by the system as a set of sub-band signals emerging from an analysis filter bank representing the system and determining the corrective matrix required to alter said set into a form representing the spectral representation that the system would make to signal if the system were perfect.

12. A method according to claim 11, wherein the system is an imaging system, the signal is a known object at the object plane of the imaging system and the spectral representation is the manifestation of the object at the imaging system's Fourier plane.

13. Signal processing apparatus for processing a spectral representation of a signal, wherein the signal occupies a position in a signal space defined by a base, the spectral representation arises from transformation of the signal by detector means and the apparatus comprises a synthesis filter bank to which the spectral representation is applied as a set of sub-band signals and which is configured to transform the spectral representation into a signal in said signal space base.

14. Apparatus according to claim 13, wherein the synthesis filter bank is configured to reconstruct said signal.

15. Apparatus according to claim 13 or 14, wherein the signal is a depth scan and the spectral representation is obtained from an object by an optical coherence tomography system.

16. Apparatus according to claim 15, wherein the optical coherence tomography system generates the spectral representation by probing the object with a broad-band, low coherence light source.

17. Apparatus according to claim 15, wherein the optical coherence tomography system generates the spectral representation by probing the object with a narrow-band, swept-frequency light source.

18. Apparatus according to any one of claims 13 to 17, wherein the synthesis filter bank contains all pass filters and the apparatus further comprises means for adjusting the all pass filters to control the sharpness of the signal as transformed back into said signal space by the synthesis bank.

19. Apparatus for designing a synthesis filter bank for use with a signal occupying a position in a signal space defined by a base and which has been transformed into a spectral representation by detection means, the apparatus comprising means for stimulating the detection means with a known signal in said space to produce a spectral representation and means for constructing the synthesis filter bank such that it transforms the spectral representation into a signal in said signal space base using the frequency domain response as a set of sub-band signals as inputs for the synthesis filter bank.

20. Apparatus according to claim 19, wherein the apparatus further comprises interferometric measuring equipment including a moveable reflecting means and the known signal comprises a series of positions of the reflecting means.

21. Apparatus for designing a synthesis filter bank for use with a signal occupying a position in a signal space defined by a base and which has been transformed into a spectral representation by detection means, the apparatus comprising means for constructing an analysis filter bank that models the transformation of signals from said space to said spectral representation and means for calculating from the analysis filter bank a synthesis filter bank that will transform a spectral representation into a signal in said signal space base.

22. Apparatus according to claim 21, wherein the means for calculating a synthesis filter bank is arranged to calculate by a least squares approach a matrix that represents the synthesis filter bank and which is an inversion or pseudo-inversion of the a matrix representing the analysis filter bank.

23. Apparatus for determining a correction for a system that transforms a signal occupying a position in a signal space defined by a base into a spectral representation, the apparatus comprising means for providing the system with a known signal in said space, and means for treating the spectral representation produced by the system as a set of sub-band signals emerging from an analysis filter bank representing the system and for determining the corrective matrix required to alter said set into a form representing the spectral representation that the system would make to signal if the system were perfect.

24. Apparatus according to claim 23, wherein the system is an imaging system, the signal is a known object at the object plane of the imaging system and the spectral representation is the manifestation of the object at the imaging system's Fourier plane.