Speckle noise reduction in optical coherence tomography of paint layers

Speckle noise reduction in optical coherence tomography of paint layers Michael Hughes, 1, * Marika Spring, 2 and Adrian Podoleanu 1 1 School of Physical Sciences, University of Kent, Canterbury, CT2 2DE, UK 2 The National Gallery, Trafalgar Square, London, WC2N 5DN, UK *Corresponding author: mrh22@kent.ac.uk Received 3 September 2009; revised 19 November 2009; accepted 25 November 2009; posted 1 December 2009 (Doc. ID 115660); published 21 December 2009 We present and characterize a sequential angular compounding method for reducing speckle contrast in optical coherence tomography images of paint layers. The results are compared with postprocessing methods, and we show that the compounding technique can improve the speckle contrast ratio in B-scans by better than a factor of 2 in exchange for a negligible loss of resolution. As a result, image aesthetics are improved, thin layers become more distinct, and edge-detection algorithms work more efficiently. The effect of varying the angular scan size and number of averages is investigated, and it is found that a degree of statistical correlation between speckle patterns exists, even for relatively large changes in angle of incidence. Angular compounding is also performed on three-dimensional data sets and compared with a method whereby en face slices are averaged over depth. 2009 Optical Society of America OCIS codes: 110.4500, 170.4500, 030.6140. 1. Introduction Since it was first reported in 1991 [1], optical coherence tomography (OCT) has developed into an important technique in the biomedical field, finding a range of applications from ophthalmology to dermatology, dentistry, and endoscopy [2]. More recently, its potential as a useful tool for the investigation of historical artworks and monitoring of conservation processes has been explored in a number of studies [3,4]. Several avenues of interest have been developed, including imaging of underdrawings [5] and paint and varnish layer structure [4,6], dynamic monitoring of laser ablation of varnish layers [7] and of LIBS spectroscopy [8], and studies of canvas deformation in response to changes in relative humidity [9]. It has been noted [10 12] that OCT images suffer from a form of noise known as speckle. This manifests as a seemingly random, but temporally stable, pattern of light and dark spots where one would expect a uniform shade of gray. In the examples that 0003-6935/10/010099-09$15.00/0 2010 Optical Society of America follow, it can be observed that speckle is a significant problem in OCT imaging of the stratigraphy of paint layers, obscuring layer boundaries and other features that may be of interest. For an inexperienced end-user, speckle may also reduce the aesthetic appeal of the image as well as imply the presence of structure that does not exist. In this paper, we propose a method of speckle reduction, based on angular compounding, that is suitable for art applications. The technique is relatively simple, with a low cost of implementation. A comparison is made with software postprocessing approaches such as median filters and averaging over depth layers. An assessment is then made of the suitability of these methods for different applications in art imaging. 2. Speckle Noise in Optical Coherence Tomography In its simplest form, OCT consists of a Michelson interferometer with a mirror placed in one arm (the reference arm) and the sample under study in the other (the sample arm). The useful signal, I PD, is then generated from the time averaged cross correlation product of the reference arm field, E R, and the portion of 1 January 2010 / Vol. 49, No. 1 / APPLIED OPTICS 99

the sample arm field backscattered from the sample, E S [13]: I PD RehE S E R i: ð1þ The depth resolved signal is extracted from this by analysis either in the time or frequency domain. For the OCT signal to contain useful information about the depth structure of the sample, the signal received should be deterministically related to its backscattering-with-depth profile [10]. This is not the case if the sample beam loses spatial coherence while propagating through the sample medium. If the sample is turbid and contains multiple scattering centers, a class that unfortunately covers almost all OCT applications (including art), the beam experiences random delays due to multiple forward and backscattering events [10]. The scattering introduces phase changes along the wavefront that, providing the variations are smaller than the coherence length, leads to self-interference within the sample beam. The resultant localized regions of destructive and constructive interference are exhibited as a seemingly random variation in the electric field of the backreflected sample beam, and hence in the intensity of the final image. It may be tempting for an inexperienced user of OCT to interpret the speckle pattern as a representation of the structure of the sample. Indeed certain optical methods, such as speckle interferometry [14], make use of speckle patterns to improve resolution, and the possibility of inferring scatterer density from OCT speckle has been discussed [15]. However, the speckle pattern in OCT cannot, by definition, be considered a direct representation of subresolution spatial structure. A wider definition of speckle may include both signal-carrying speckle, which is responsible for most of the OCT signal, and signal-degrading speckle, which is generated by multiple scattering from out of focus centers [10]. If we restrict our discussion to only the signal-degrading speckle, then we can state that, in general, this speckle serves no useful purpose in OCT imaging. It is a form of multiplicative that should be removed in order to obtain higher quality images. The statistical characteristics of speckle in OCT have been analyzed previously [11,12,16], and the results will not be repeated in detail here. However, a key result of theoretical analyses of the expected properties of OCT speckle in turbid media is that they predict, for a fully developed speckle pattern, an image intensity distribution, PðIÞ, that follows a Rayleigh distribution of the form [16] PðIÞ ¼ I σ 2 exp I2 2σ 2 ; ð2þ where σ is the standard deviation and PðIÞ is defined for I ½0; Þ. Experimental results have suggested that the speckle pattern for biological tissue and other media follows this distribution to a good approximation [10,16]. In order to perform a quantitative assessment of the reduction of speckle noise, it is necessary to define a method by which the amount of speckle in an image may be determined. In keeping with some previous reports [2,16] we define the speckle contrast ratio, SCR, within a given area of the image as the ratio of the standard deviation of the intensity in that area, σ I, to the mean intensity in that area, hii: SCR ¼ σ I hii I : ð3þ It should be noted that, in some reports [11,17,18], an alternative measure, speckle signal-to-noise ratio (SNR) is used; this is the inverse of the SCR [2]. p Given ffiffi that the mean of a Rayleigh distribution is σ π 2, and the variance is 4 π 2 σ2, then it is straightforward to show that the expected SCR for a fully developed Rayleigh-distributed OCT speckle pattern is 0.52 [10,16]. A. Removal of Speckle Noise Postacquisition image processing can be used to reduce speckle contrast; approaches range from simple linear filters (for example, mean and Gaussian) and median filters [19] to wavelet analysis and adaptive smoothing [10]. Although often successful at reducing speckle, these methods typically lead to some loss of information including resolution. A number of hardware solutions have been suggested including use of multimode fibers [20]. Other approaches are based on averaging, taking advantage of the fact that the average of N uncorrelated p speckle patterns has been shown to exhibit a ffiffiffiffi N reduction in speckle contrast [2]: SCR AV ¼ p 1 ffiffiffiffi : ð4þ SCR ORIGINAL N For in vivo biomedical imaging, sample motion between the acquisition of successive scans may be sufficient to decorrelate the speckle patterns [21], although this can introduce motion artifacts and, in any case, is not generally applicable to art imaging. We must therefore seek to introduce speckle pattern diversity between scans by some other means. One approach, frequency compounding, exploits the reduced correlation between the speckle generated at different wavelengths, but unfortunately involves binning regions of the source bandwidth [10] and hence drastically worsening the axial resolution, or else effectively having multiple interferometers [16]. Angular compounding, which makes use of the decorrelation between scans acquired for different angles of incidence of the sample beam, is a more attractive option since resolution is not so adversely affected. Early attempts used parallel detection at multiple angles [18,22], but later the high frame rate of modern spectral OCT systems was exploited to 100 APPLIED OPTICS / Vol. 49, No. 1 / 1 January 2010

allow for sequential acquisition [17]. The latter method used a galvanometer scanner to provide a high angular scan speed. However, it required the use of a long galvanometer mirror to perform the transversal scanning, and hence would be difficult to incorporate into a three-dimensional (3D) scanning system. A more recent parallel method [23] involves encoding the angle of incidence of different components of the sample beam in the Doppler shift induced by decentering the sample beam on the galvanometer mirror. However, this method suffers from the drawback that the transversal resolution is reduced, since only a portion of the beam aperture is used for each image. Given that there is little need for very high speeds in art imaging, we have devised an alternative angular compounding arrangement more suited to this application. This method permits a larger variation in the angle of incidence as well as the despeckling of cubes. It shares some similarities with the method of Ref. [11], where a translating prism was used to alter the angle of incidence. However, the method that we report here operates at significantly higher speeds and, as discussed below, is compatible with 3D imaging. Fig. 1. Schematic of SS-OCT with angular compounding: SS; swept source; DC; fiber directional coupler, L; lens (f ¼ 4 cm); TS; translation stage; GS; galvo scanner head, ADC; analog to digital converter; DSP; digital signal processing (and image display). The change of the angle of incidence by movement of the translation stage is shown in the inset (bottom right). 3. Swept Source Optical Coherence Tomography System All speckle reduction measurements were performed using an inhouse OCT system, shown in Fig. 1. The system was illuminated using a tunable source (Axsun Technologies) with a central wavelength of 1310 nm, a sweep range of 100 nm, and a sweep rate of 48 khz. Light from the source was divided between the sample and reference arms using a 20=80 broadband fiber coupler. The numerical aperture (NA) of the sample arm optics was approximately 0.05, providing a theoretical transversal resolution of 16 μm. The reference arm was then recombined with backscattered light from the sample arm in a second, 50=50 coupler. The difference between the two outputs from this coupler (identical except for a π phase shift in the cross-correlation signal) was acquired using a dual balanced photodetector receiver. This signal was then high-pass filtered to remove residual dc voltage (due to imperfections in the balancing) and transferred to PC memory via the analog to digital converter (ADC) circuit of a frame-grabber (Bitflow Alta). Once each spectral sweep was acquired in memory, dc terms were removed by subtracting a reference spectrum. This was obtained by averaging all the component spectra of each B-scan. Since each spectrum is modulated at difference frequencies (due to the different backreflectors at different transversal positions within the sample) the spectrum modulation (channeled spectrum) is washed out, leaving only the reference and sample arm dc powers and any autocorrelation terms, as well as any transversal position independent signal from the sample arm (due to backreflections from lenses or fiber tips) [24]. Since the output wavenumber (k) of the swept source did not vary linearly in time, each spectrum required resampling, which was performed by linear interpolation. The interpolation coefficients were determined essentially by inverting the phase progression per pixel in a channeled spectrum generated by placing a strong, single surface reflector in the sample arm. Following interpolation, each spectrum was Fourier transformed to generate a single A-scan. Each of these A-scans was acquired in 10 μs (a 50% duty cycle). B-scans were acquired by driving a galvanometer (the x scanner ) in the sample arm with a triangular waveform at 45 Hz; this yielded an acquisition time for a 512 line B-scan of 11 ms (with a frame rate of 45 Hz). 3D cubes were generated by driving a second galvanometer ( y scanner ) with a linear sawtooth ramp at 0:08 Hz; the acquisition time for a 512 512 pixel cube was therefore 11:4 s. The coherence length (6 db both ways) was measured to be 20 μm at an optical path depth of 100 μm. The sensitivity at the same depth was 95 db for a sample illumination of 1:7 mw, measured using a mirror in the sample arm and 48 db of calibrated attenuation. There was a sensitivity fall with depth of 6 db in 1:8 mm. This can be attributed to a number of factors, including the finite linewidth of the source, finite bandwidth of the photodetector, finite sampling rate of the ADC, and inaccuracies in the numerical linearization [25]. A. Angular Compounding As can be observed in Fig. 1, the galvanometer scanning head was mounted on a motorized translation stage, allowing translation perpendicular to the optical axis. Such a translation alters the angle of incidence of the sample beam on the target but does not change its transversal position, or the optical path length, providing the target is at focus (shown in the inset of Fig, 1). The advantage of this method over a previously reported sequential method [17] is that the maximum angle obtainable is not dependent 1 January 2010 / Vol. 49, No. 1 / APPLIED OPTICS 101

on the size of the galvanometer mirror, and depends ultimately only on the diameter of the sample lens. It also permits, as described later, the acquisition of despeckled cubes. The principal disadvantage is a lower maximum speed of the angular scan due to the use of a translation stage; this is of little concern for art studies but may limit the method s applicability for in vivo imaging. Other methods could be used to translate the sample arm beam, such as a prism [11]. However the method of Ref. [11] suffers from the same drawback as Ref. [17], in that a transverse scanning galvanometer is not translated with the beam, and hence the use of large angles of incidence requires either a large galvanometer mirror or slow lateral scanning. The report made here provides a significant speed increase over the method of Ref. [11]. The maximum employed displacement of the translation stage was 5 mm. This produced a variation in the sample beam angle of 7. A set of 45 B- scans collected at a range of 7 toþ7 (in approximately 0:3 angular steps) could be attained in 1 s, but if a faster frame rate was required, then a smaller number of angular steps could be obtained over a smaller range. However, the turnaround time of the stage limited the acquisition rate of each set of anglevaried B-scans to approximately 2 Hz. Each frame was individually processed, and then, in order to compensate for any misalignment, image registration was performed using an intensity based subpixel method [26]. Due to the ease of alignment of the system (the most crucial being that the stage translation is perpendicular to the optical axis) this step did not generally produce visibly different results and could be omitted if shorter processing time was required. As has been reported with similar angular compounding techniques [17], as one moves away from the focal plane in the target, the beams with different angles of incidence no longer converge at the same point. There is therefore a loss of transversal resolution with depth, and it is necessary to accept a tradeoff between usable depth range and degree of speckle reduction. For this purpose, we use the reasonable (if somewhat arbitrary) criterion that the degree of fanning of the beams is acceptable as long as its transversal extent does not exceed the transversal resolution (on the surface and at focus), given by used. The maximum usable depth range, R max,is therefore R max ¼ 1:22λ 0n NAΔθ i ; ð7þ when the focal plane is placed halfway through the depth range. For the system NA of 0.05 and central wavelength of 1300 nm, and a sample of refractive index ¼ 1:5, the usable depth range is shown in Fig. 2. In practice, the situation was somewhat improved from that shown, both because of aberrations leading to a larger transversal spot size and a falloff of transversal resolution with depth due to the finite depth of field of the sample lens. Given that the samples used in this study generally did not have interesting features over a depth range greater than 200 μm, it was possible to use the full angle range of 14. 4. Results In order to perform a quantitative analysis of the speckle contrast improvement, an area of paint containing yellow ochre in linseed oil (Italian golden ochre, reference number 4022, supplied by Kremer Pigmente), applied on a chalk and glue priming layer on a Teflon (PTFE) support, was studied in cross section/b-scan (Fig. 3). The paint was highly scattering and hence provided a relatively homogenous, structure-free, speckle pattern suitable for a quantitative analysis. To assess the speckle contrast, 10 similar, apparently homogenous, speckled rectangular areas (10 50 pixels in size) were selected and the average of the SCRs for these areas was taken to be the SCR of the image. Since there was some degree of subjectivity in selecting these areas, all values should be considered to be indicative only. It should be noted that although the images are displayed in log scale, the SCR was calculated based on the linear values. Δx ¼ 0:61λ 0 NA ; ð5þ where λ 0 is the central wavelength and NA is the numerical aperture of the sample arm optics. The spread of the beams, Δd, at a distance from the focal plane of R, is Δd ¼ R tanðn sin Δθ i Þ RnΔθ i ; ð6þ where Δθ i is the range of angles of incidence (in air) and where the small angle approximation has been Fig. 2. Change of maximum usable depth range with angle of incidence range. For imaging of thin paint layers, very large angle ranges are acceptable. 102 APPLIED OPTICS / Vol. 49, No. 1 / 1 January 2010

Fig. 3. B-scan of yellow ochre in linseed oil, applied on a chalk/ glue priming on a Teflon (PTFE) support. Due to the highly scattering nature of the yellow ochre, the priming layer is not discernible. (a) single B-scan, (b) 3 3 median filter applied, (c) average of 40 scans with no angular variation, and (d) angle of 40 scans with angle of incidence varied over 14. Scale bar is 1 mm in air in both the lateral and axial directions. The error quoted for the SCR values is the standard error of the SCR for the 10 measurement areas. Since there may have been some variation in the speckle pattern statistics (or indeed intrinsic intensity variations) between the areas, the error is not quite equivalent to the standard error as commonly understood; it does, however, provide a reasonable estimate of the variation of SCR across the areas. For a single B-scan, as shown in Fig 3(a), the SCR was calculated to be 0:47 0:02. This is comparable to SCRs measured for biological tissue and close to the theoretical value of 0.52 for the Rayleigh distribution of Eq. (2). It can therefore be assumed, in the case of this sample, that there are sufficient scattering centers within a coherence volume to generate a fully developed speckle pattern. Hence it can be expected that despeckling procedures will exhibit the same reduction in SCR as for images of highly scattering biological tissue. Application of a median filter with a 3 3 kernel, shown in Fig. 3(b), reduced the contrast ratio to 0:29 0:02, but obviously at the expense of image resolution. Median filters with 4 4 and 6 6 kernels (not shown) improved the SCR to 0:23 0:02 and 0:20 0:02, respectively, but further degraded the resolution. Linear filters, such as mean and Gaussian filters, produce a similar reduction in SCR but without the edge retaining advantages of median filters. More complex filters, such as the rotating kernel transform and wavelet analysis, should perform better, but were not investigated here. Forty B-scans of the same sample were then acquired without changing the angle of incidence of the sample probe beam. The median average of these images is shown in Fig. 3(c). The SCR improved slightly from 0:47 0:02 for the single B-scan to 0:35 0:05, probably due to the averaging of instabilities in the imaging system and reduction of other noise forms. A further set of 40 images were then acquired, with the angle of incidence ranging from 7 to þ7 in approximately equal steps (allowing for the small angle approximation). Again, the median average was taken and is shown in Fig. 3(d). In this case, since the speckle patterns of the averaged scans now had a degree of decorrelation, a much greater improvement in SCR to 0:17 0:03 can be observed. The improvement is therefore better than that obtained from a 6 6 kernel median filter. The effect of varying the number of averaged scans is shown in Fig. 4. In each case, the scans were approximately evenly distributed over an angle range of 14. It can be observed that the improvement in SCR with larger numbers of averages is significantly less than would be expected from the square root rule in Eq. (4) (shown on the plot by a dotted line). This can most likely be attributed to incomplete statistical decorrelation between the speckle patterns. One cause of the residual correlation is likely to be the finite diameter of the sample arm beam on the lens. This means that each scan will be formed from the coherent summation of the signal returned from a finite range of angles of incidence. When a small change in the angle of incidence is made, the new distribution of the spot on the lens will overlap the previous distribution to some extent. The angle ranges Fig. 4. Effect of number of averages on improvement in speckle contrast ratio. For each number of averages, the averaged frames were evenly spaced in angle over 14 total angular change. The improvement expected by the square root rule in Eq. (4) is shown by the dotted line, suggesting a degree of speckle pattern correlation between the frames. 1 January 2010 / Vol. 49, No. 1 / APPLIED OPTICS 103

will therefore not be fully independent, and there will be some correlation between the speckle patterns. Decreasing the sample arm beam diameter would counteract this limitation, but at the expense of lateral resolution. This hypothesis is confirmed somewhat by Fig. 5, which shows how the SCR from averaging a set of 8 images varies with the maximum scan angle. In each case, the angle step between successive frames was approximately constant. It can be observed that below an angle range of approximately 6 (0:75 per scan) there is a sharp rise in SCR, serving to highlight the importance of a large scan range in angular compounding. The method was then applied to a sample exhibiting a visible layer structure. The test plate consisted of a Teflon (PTFE) sheet primed with chalk bound in glue, onto which an area of yellow ochre in linseed oil had been applied, with an area of smalt in linseed oil (dark, supplied by L. Cornelissen & Son) adjacent to it. This panel has been used in previous OCT studies [3,27], and both photographs and B-scans are available in the literature. The B-scans in Fig. 6 are from a line crossing a boundary between the two types of paint, where the smalt slightly overlaps the yellow ochre. As the smalt is less scattering than the yellow ochre, the chalk ground can be seen beneath it at the left of the image. The effect of increasing the number of averaged scans is shown; as before, these were distributed evenly over 14 of angular variation. Successive improvement in both general image aesthetics and in layer definition (enlarged section) can be observed up to approximately 10 or 12 averages. Beyond this point, any further improvement is minor. There is no observable degradation of resolution in the averaged images; indeed, the improved boundary delineation shown in the enlarged section suggests that, in one sense, the effective resolution (i.e., that which is normally limited by speckle noise) has improved. Fig. 6. Despeckling of B-scan of Teflon (PTFE) sheet primed with chalk bound in glue, onto which an area of yellow ochre in linseed oil had been applied (right part of image), with an area of smalt in linseed oil adjacent to it (left part of image). The improvement in image quality with increasing number of frames acquired at different angles (equally spaced over 14 ) is shown. The number of frames averaged in each case is indicated on the left. The zoomed in region to the right shows the dramatic improvement in boundary sharpness. Scale bar is 1 mm. Some applications in art imaging may benefit from automated or semiautomated image segmentation; for example, the automatic identification of layer boundaries is useful both as an objective measure of whether a distinction between two layers can be detected, and also in allowing quantitative measurement of layer thickness to be made. Figure 7(a), which shows the results of applying a Cauchy filter (low threshold ¼ 0:1, high threshold ¼ 0:3, σ ¼ 1) to the single B-scan from Fig. 6, demonstrates how Fig. 5. Effect of angular scan size on improvement in speckle contrast ratio. The number of averaged frames was 8 in each case, distributed approximately evenly in angle within the angular scan size. Fig. 7. Cauchy edge detection filter (low threshold ¼ 0:1, high threshold ¼ 0:3, σ ¼ 1) applied to image from Fig. 6: (a) filter applied to single scan (top of Fig. 6), (b) filter applied to single scan with 3 3 median filter, and (c) filter applied to average of 40 angular scans (bottom image in Fig. 6). The despeckled image (c) generates a vastly improved result. 104 APPLIED OPTICS / Vol. 49, No. 1 / 1 January 2010

speckle in the image can confuse such algorithms, leading to a large number of spurious edges that obscure real features. Applying a 3 3 kernel median filter prior to edge detection [Fig. 7(b)] has little beneficial effect. However, despeckling of the image by angular compounding (median average of 40 frames over 14 ) dramatically improves the performance of the filter, as can be observed in Fig. 7(c). In borderline cases, speckle reduction may also help a human user to discern thin boundaries. An example is shown in Fig. 8, a scan of a thin layer of ultramarine over lead white, both bound in linseed oil. In both the normal [Fig. 8(a)] and despeckled [Fig. 8(b)] images, the thicker ultramarine layer to the right of the image is visible, although it is clearer in the despeckled image. The much thinner layer to the left of the image is visible only on the despeckled image, albeit at the limit of resolution. A. Effect on Resolution From the preceding images it can be ascertained that the angular compounding method does not significantly degrade transversal resolution within the relatively thin layer of interest, albeit in this particular case of relatively low NA sample arm optics. To determine the effect on axial resolution, the coherence envelope was measured at two depths, 300 μm and 2 mm, both with and without angular compounding, by placing a single surface reflector in the sample arm. The result (Fig. 9) shows that there is essentially no loss of axial resolution at either depth. A small axial shift in the image was also recorded at 2 mm, likely an artifact due to the image registration procedure. B. Speckle Reduction in Three-Dimensional Tomograms Certain art applications, particularly studies of artists underdrawings, require the collection of 3D data sets from which perpendicular (en face) slices can be extracted. Reducing speckle in cubes by angular compounding is challenging, as significant tilt and other distortions are introduced when the angle of incidence is varied. These can be corrected for by image registration, but this will reduce the field of view (since some frames will rotate out of the window), and there may be a loss of resolution due to errors in the registration. Fig. 9. Effect of angular compounding on axial resolution. There is no discernible loss of resolution at either 300 μm depth or 2000 μm depth. For applications such as viewing of underdrawings, alternative despeckling methods are possible. Given that features such as underdrawings may not have a complex depth structure, it is possible to reduce speckle and improve contrast by averaging over a judiciously chosen range of depth slices [6]. This method is compared to angular compounding in Fig. 10. In Fig. 10(a), a single en face slice extracted from a 9 9 5 mm (512 512 pixel) cube acquired for a single angle of incidence (0 ) is displayed, showing a line of underdrawing made with carbon black paint underneath a layer of lead white in linseed oil only (top) and azurite in linseed oil over the lead white layer (bottom). The SCR in the lead white only region is 0:46 0:01. In Fig. 10(b), the result of averaging en face slices from three cubes acquired Fig. 8. B-scans of thin layer of ultramarine in oil over lead white in oil (a) without and (b) with despeckling. Scale bar is 1 mm. Fig. 10. Speckle reduction on en face slices extracted from cubes of A-scans generated using SS-OCT: (a) slice from single cube, (b) slice from average of 3 cubes at different angles of incidence, (c) 4 4 median filter on single slice, and (d) z-axis average of 25 slices from a single cube. Scale bar is 1 mm. 1 January 2010 / Vol. 49, No. 1 / APPLIED OPTICS 105

Fig. 13. Metal point underdrawing under lead white and azurite layers (left of each image) and lead white only (right of each image): (a) single en face slice and (b) average of 20 slices over 250 μm depth. Scale bar is 1 mm. Fig. 11. Effect on speckle contrast ratio of increasing the number of averaged en face slices (from Fig. 10) selected evenly from a depth range of 375 μm. at angles of incidence of 2, 0, and þ2 is shown. Since the angular scanning was slow in comparison to the transversal scanning, it was much more efficient to acquire in this way rather than assembling a despeckled cube from successive despeckled B- scans. The speckle contrast ratio has decreased to 0:33 0:01, which is in line with expectations. A very slight blurring can be observed. The total acquisition time for three cubes was 36 s, although processing time was considerably longer. A 4 4 median filter was applied to the original slice and can be seen in Fig. 10(c). As expected, the result is a visual blurring of the image, and so the improvement in SCR to 0:16 0:01 does not greatly increase the appeal of the image. Figure 10(d) shows the speckle reduction (SCR ¼ 0:15 0:01) and improvement in image appearance obtained by averaging 25 en face slices over a range of approximately 300 μm. The penalty is a loss of depth information, but in cases such as this where the primary purpose of the depth selection is to remove Fig. 12. Effect on speckle contrast ratio of increasing depth range from which 8 evenly spaced en face slices (from Fig. 10) are selected for averaging. a top layer obscuring a deeper feature, this is of little concern. The improvement in SCR using the depth averaging method suggests a large degree of speckle pattern decorrelation between slices. This is confirmed in Fig. 11, which shows the improvement in speckle with number of averaged slices. As for Fig. 4, the improvement expected from Eq. (4) is shown by a dotted line. Although the data points do not lie on the line, indicating that some correlation remains, the situation is better than that shown in Fig. 4. This is confirmed by Fig. 12, which shows the effect of increasing the depth range over which 8 slices are averaged (i.e., increasing the differential depth between successive slices). It can be seen that only a small improvement is obtained by increasing the differential depth beyond 150 μm (equivalent to 18 μm) per slice, and hence there is no clear benefit to further degrading the axial resolution by doing so. This result is reasonable as 18 μm is of the order of the source coherence length. The benefit of speckle reduction in the viewing of underdrawings goes beyond improving image aesthetics in cases where the underdrawing is faint and difficult to see. An example of such a situation is shown in Fig. 13, where a thin metal-point underdrawing is underneath a layer of lead white in oil and a layer of azurite in oil. Averaging 20 slices over a depth of 250 μm improves the visibility of the underdrawing considerably. Clearly, if the underdrawing was painted on an uneven surface (i.e., it was not a single plane), then there would be an additional benefit to averaging over depth. 5. Conclusion Angular compounding has been shown to reduce speckle contrast in biological samples in a number of reports, and this study has confirmed its applicability to scans of paint layers. Whereas biological applications may be more suited to methods that use a galvanometer to change the angle of incidence, or parallel detection schemes, the general lack of motion artifacts in art imaging means that the method described here is appropriate, particularly as it allows for a potentially larger maximum angle of incidence. With an acquisition time of 1 s for 45 averages, the method is fast enough to be convenient 106 APPLIED OPTICS / Vol. 49, No. 1 / 1 January 2010

to use in real investigations. It performs much better than median filtering and is particularly beneficial when edge detection algorithms are to be used. Averaging 40 scans over an angle of incidence range of 14 reduces the SCR in a B-scan of yellow ochre in linseed oil from 0:47 0:02 to 0:17 0:03; by comparison a 6 6 median filter was required to achieve a speckle SCR of 0:20 0:02. It is possible to achieve a comparable reduction using as few as 12 scans over the same angular range, but reducing the angular range dramatically lowers the performance due to the increased statistical correlation between the speckle patterns. For despeckling of cubes and, by extension, en face slices, our angular compounding method has been shown to be effective but slow, with averaging of 3 cubes reducing the SCR from 0:46 0:01 to 0:33 0:01 in an en face slice of lead white. 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