Phase discontinuity predictions using a
machine-learning trained kernel
Firas Sawaf* and Roger M. Groves
Aerospace Non-Destructive Testing Laboratory, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2600 GB, Delft, The Netherlands
*Corresponding author: f.sawaf@tudelft.nl
Received 28 April 2014; revised 26 June 2014; accepted 17 July 2014; posted 18 July 2014 (Doc. ID 210938); published 15 August 2014
Phase unwrapping is one of the key steps of interferogram analysis, and its accuracy relies primarily on the correct identification of phase discontinuities. This can be especially challenging for inherently noisy phase fields, such as those produced through shearography and other speckle-based inter-ferometry techniques. We showed in a recent work how a relatively small10 × 10 pixel kernel was trained, through machine learning methods, for predicting the locations of phase discontinuities within noisy wrapped phase maps. We describe here how this kernel can be applied in a sliding-window fashion, such that each pixel undergoes 100 phase-discontinuity examinations—one test for each of its possible positions relative to its neighbors within the kernel’s extent. We explore how the resulting predictions can be accumulated, and aggregated through a voting system, and demonstrate that the reliability of this method outperforms processing the image by segmenting it into more conventional10 × 10 nonoverlapping tiles. When used in this way, we demonstrate that our10 × 10 pixel kernel is large enough for effective processing of full-field interferograms. Avoiding, thus, the need for substantially more formidable computational resources which otherwise would have been necessary for training a kernel of a significantly larger size. © 2014 Optical Society of America
OCIS codes: (100.5088) Phase unwrapping; (100.4996) Pattern recognition, neural networks; (120.6165) Speckle interferometry, metrology; (120.2650) Fringe analysis; (120.4290) Nondestructive testing.
http://dx.doi.org/10.1364/AO.53.005439
1. Introduction
Optical measurement has become a valuable non-contact full-field tool for metrological imaging [1] of dynamic and composite materials undergoing mechanical loading. Shearography [2], for instance, is an effective interferometric speckle technique [3] that is well suited for the direct measurement of strain components in a nondestructive setting [4], and continues to be a fertile field of research in this domain [5]. Images produced by an interferometric speckle capture system are inherently noisy. These
images, often called interferograms, are made up of pixels whose intensities are proportional to the phase of the corresponding speckle. Phase informa-tion is naturally wrapped onto the range −π to π. The reconstruction of the unknown multiples of 2π is called phase unwrapping, and thus plays an essential role in the image processing required for interferogram analysis—ultimately leading to the extraction of the measurand data, examples of which are an object’s shape [6] and features [7], vibration modes [8], and strain components [9].
Despite numerous and diverse approaches by workers in the field [10–13], through the application of a wide variety of techniques [14–17] to this problem, reliable phase unwrapping remains a key
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challenging task [18–20] within the overall image processing stages required for interferogram analysis [21].
Statistical pattern recognition techniques [22] are benefiting from renewed enthusiasm [23], and re-ceiving substantial attention, both in the theoretical and practical domains of new frontiers [24], often labeled big data [25], machine learning [26], and data science [27]. Interferogram analysis [28] can be viewed as a specialized domain in the wider class of machine vision. Neural networks are a subclass of machine learning techniques, which were applied to phase unwrapping in various applications in the past, such as prefiltering wrapped phase maps [29], and phase unwrapping of simplified small scale phase maps [30] and more extensive field interfero-grams [31].
A common issue highlighted by many of these works is that neural networks required an excess of time-consuming training on a considerably large number of wrapped phase maps [32]. Nevertheless, machine learning lends itself particularly well to solving machine vision problems [33] in cases where traditional image processing operators produce far inferior results, compared to those that can be gleaned by a human expert in a relatively straight-forward manner [34].
Therefore, striving to overcome the aforemen-tioned challenges remains a worthwhile endeavor which we pursued in a recent work [35], where we showed how a machine learning system can aid the phase unwrapping process by predicting the locations of phase discontinuities within noisy interferograms, through the application of a neural network by training a10 × 10 pixel kernel to predict the locations of phase discontinuities within noisy wrapped phase maps. While this relatively small 10 × 10 kernel served well in demonstrating the potential and viability of this approach, we go further in this work by showing that the reach of a10 × 10 pixel kernel can be extended to a larger effective neighborhood size of19 × 19 pixels.
A key and interesting aspect of this finding is that a machine-learning trained10 × 10 pixel kernel can be indeed large enough for effective processing of full-field interferograms, when applied through the various methods and techniques we introduce in this work. This in turn means that it is possible to avoid the need for substantially more formidable computa-tional resources that otherwise would have been necessary for training a kernel of a significantly larger size, thus helping to establish that future works can be focused mainly on improving the reli-ability of predictions produced by the10 × 10 kernel underpinning the19 × 19 effective neighborhood. 2. Approach
In this section we state the main objectives of this work, after providing some background to the topol-ogy and properties of the neural network employed. We also introduce a set of computer simulated
images, which are used in subsequent sections for demonstrating, in practical terms, the ideas and methods being discussed.
A. Neural Network Employed
In general terms, a neural network can be viewed as a mapping function, which maps a set of input parameters in a problem domain, to output parame-ters in the solution domain. The mapping function itself may well be highly nonlinear and complex, and often does not have any known direct analytical solutions. To achieve this, the neural network needs to contain at least one hidden layer of neurons, in addition to the set of neurons in the output layer. For this reason, this configuration, Fig. 1, is often called the two-layer neural network, in reference to the two layers of adaptive weights [36].
The two-layer neural network is of particular interest due to its well established property of being capable of mapping any continuous function in this manner [37]. In our case, the neural network is constructed using 100 input nodes, with each node representing the pixel intensity of the10 × 10 noisy wrapped phase map kernel. The neural network maps these inputs onto a corresponding set of 100 output nodes, each representing the outcome of predicting whether its input pixel counterpart is a location where a phase discontinuity is present. The prediction is of binary nature, in that each pixel in the input is predicted to be either a phase discon-tinuity location or not. The neural network is ini-tially trained through a set of examples, to which the solution is known. This is called supervised learning [38]. Once the training phase is complete, the neural network can be used for solving similar problems in the application domain.
We created a set of 120,000 training examples by generating 300 computer-simulated 200 × 200 pixel full-field wrapped phase maps, containing various amounts of additive random noise. These were then subdivided into training examples, each represent-ing a10 × 10 pixel image portion. The 300 wrapped phase map training examples themselves were computer-generated based on a simulation of a phase distribution field by scaling and translation of a
Fig. 1. Two-layer neural network. We employed the hyperbolic tangent (tanh) as the activation function.
Gaussian distribution. Random noise was then added to this field, and then fed into sine and cosine functions, which in turn were fed into an arctangent function, generating a simulated noisy wrapped phase map. The process was repeated multiple times, choosing random amounts of scaling and translation of the Gaussian distribution, and random amounts (i.e., variance) of noise. The wrapped phase fields were then split into the desired portion size, i.e.,10 × 10 pixels in this case, and randomly shuffled. The pseudorandom number generator used throughout this process drew samples from a standard normal distribution. More details on the description of this technique can be found in [35].
The accuracy of the mapping function, imple-mented through a two-layer network, is partly dependent on the diversity of the training examples, in the extent to which these examples are represen-tative of the problem domain.
The effectiveness of the mapping function is also dependent on the number of neurons in the hidden layer. The larger the number of neurons in the hidden layer, the more degrees of freedom the net-work has for fitting to the training examples. Our network contains 3200 neurons in the hidden layer. There are 100 input nodes, which naturally matches the number of pixels in the 10 × 10 pixel image sections on which the network operates. For this work, we used the hyperbolic tangent (tanh) as the sigmoid activation function [39], which offers some practical advantages [40] over the logistic sigmoid [41] function.
B. Motivation and Main Objectives
The correct identification of phase discontinuities in the naturally noisy speckle-based wrapped phase maps is both challenging and central to reliable phase unwrapping. It is so vital because, in order to unwrap a phase map by restoring the correct multiples of2π, it is essential to be certain of the lo-cation of the phase jumps, or else risk introducing er-rors to the measurement process by mistaking noisy features for actual phase wraps. There are many other aspects of interferogram phase unwrapping, such as the underlying object’s shape discontinuities, fringe termination points, and localized phase un-wrapping error propagation to spatially adjacent neighborhoods.
Nonetheless, the sole focus of this work, and that of [35] on which it builds, is the central issue of correct identification of phase discontinuities, based on statistical pattern recognition and machine learning techniques. The aim, at this stage, is not so much to present a new and complete end-to-end phase unwrapping method but rather to specifically exam-ine ways in which it might be possible to improve the reliability in which phase discontinuity locations can be discerned in inherently noisy wrapped phase map speckle fields.
In our neural network, each input node is con-nected to each neuron in the hidden layer, and those
in turn are each connected to the output layer’s neu-rons. In our configuration, this makes for 640,000 (i.e., 2 × 100 × 3200) connections. Each of these con-nections has a weight associated with it, which is finely optimized through multiple iterations for each of the 120,000 training examples. In practice, all of this translates into a large number of floating point operations. Even when running on modern multicore computer machines, a training regime structured in this manner can consume many days’ worth of continuous 100% CPU utilization.
When deciding on how to allocate computational resources, there are many competing concerns. On one hand, we would like to increase the kernel size. At the same time, we would like to improve the performance of the mapping function by increasing both the number of training examples and the num-ber of the neurons in the hidden layer.
In this work, we explore the merits of an approach that maintains a relatively small10 × 10 kernel size. Rather than attempting to extend our work in [35] by striving to match the size of typical full-field 512 × 512 or even 256 × 256 pixel interferograms, we choose instead to focus on improving the reliability of the kernel, by the virtue of extending its reach be-yond the immediate 10 × 10 neighborhood. We do this by accumulating multiple predictions per pixel, by sliding the 10 × 10 kernel smoothly in one-pixel increments, both horizontally and vertically, gather-ing predictions across a larger effective neighborhood of19 × 19 pixels. The way in which this is achieved is described in detail in subsequent sections, which show that this alternative avenue for improving the reliability of the predictions produced by the ker-nel makes it possible to fix its size to10 × 10 pixels, thus directing computational resources toward in-creasing the number of training examples and the number of neurons in the hidden layer.
C. Demonstrative Examples
To illustrate the various ideas and methods pre-sented in this work, we use the computer-simulated wrapped phase map, shown in Fig. 2, throughout the sections that follow. This example was con-structed so that it contained a range of phase discon-tinuity features and densities, and levels of noise,
Fig. 2. (a) Computer-simulated wrapped phase map used as a test subject throughout this work. (b) Noisy wrapped phase map; noise was added to the underlying object’s surface. (c) Actual phase discontinuity locations were recorded for performance comparison purposes.
representative of those typically observed in practi-cal optipracti-cal systems. In addition to this computer-generated example, we also show in subsequent sections how, in principle, our kernel can be applied to nonsimulated optically obtained interferograms, Fig. 11.
While having two specific examples, one computer generated and the other optically captured, serves to provide some insight, it is more appropriate to quan-tify performance against a relatively large number of cross validation set of examples. This is presented in the form of Fig. 13, which also highlights ways in which performance improvements can be sought. 3. Sliding-Window Method
It is possible to produce phase discontinuity predic-tions for the full-field image by simply partitioning it into tiles, matching the size of the10 × 10 kernel, as shown in Fig.3(a).
One drawback to this approach is the need for a mechanism for reconciling the resulting tiles into a cohesive image, and making adjustments at the boundaries to avoid introducing spurious phase discontinuity endpoints.
Moreover, the reliability of the predictions would not benefit from the additional information present further afield within the regions of neighboring tiles, unless yet another mechanism is put in place for this purpose.
These effects can be seen in Fig.4(a). When phase discontinuity pixels fall near the boundary of the
kernel, tiling induces artifacts which can be demon-strated by processing the simulated noise-free phase map, Fig. 4(b), which serves to show more clearly where these gaps fall, Fig.4(c).
The cause for the appearance of these artifacts can be understood, as illustrated in Fig.5, by considering how larger structures passing tangentially to a tile’s kernel would evade detection, due to lack of sufficient contextual information within the processing kernel. By contrast, using a sliding window for processing the interferogram overcomes many of these chal-lenges. By using this approach, the 10 × 10 kernel is moved along horizontally and vertically in one-pixel increments, Fig. 3(b). This makes it possible for a larger neighborhood of pixels to contribute the information they contain to the process of produc-ing the predictions.
The effective size of the combined neighborhoods of the kernel, as it slides through all possible positions, is a 19 × 19 pixel area, Fig.6(a). In other words, by employing the sliding-window mechanism, the phase discontinuity predictions produced for each pixel are based on its own intensity, along with the intensities of its 360 neighbors. This allows each pixel in the full-field interferogram to be examined up to 100 times, once for each possible location relative to its neigh-bors within the kernel, Fig.6(b).
Fig. 3. Illustration of how the kernel can be applied to a full-field image, (a) by tiling and (b) through a sliding window, which is moved, horizontally and vertically, by one pixel at a time. Actual kernel size used is10 × 10 pixels. 5 × 5 kernel size is shown in this figure for illustration.
Fig. 4. (a) Predictions made by nonoverlapped tiling method. Ac-tual phase discontinuity locations are shown in (b) for comparison. (c) artifacts of tiling can be seen even when applied to the noise-free phase map.
Fig. 5. Phase discontinuity location, which is intersected by a tile in a tangential manner, can escape detection due to the lack of sufficient contextual information within the tile’s kernel.
Fig. 6. (a) The pixel at the center of the overlapping kennels re-ceives prediction verdicts from a combined effective neighborhood of19 × 19 pixels. For illustration, a 5 × 5 kernel is shown, resulting in an effective neighborhood of9 × 9 pixels. (b) Pixels sufficiently farther from boundary of full-field receive the full number of verdicts, i.e., 100 verdicts for a10 × 10 kernel.
A. Aggregating Predictions Through a Voting Ballot We described in the previous section how applying the processing kernel in a sliding-window fashion is advantageous, one of its benefits being the ability to produce multiple predictions per pixel location. Predictions thus produced need to be aggregated, to reach a final single verdict, stating whether a given pixel is a location for a phase discontinuity or not.
We propose a voting system for aggregating the predictions produced by the sliding window, where each pixel needs to exceed a boundary line, for total number of positive votes received, before it can be confirmed to be a phase discontinuity location. Choosing a suitable boundary line can be viewed as a calibration step that can be performed against a set of computer-simulated cross-validation inter-ferograms, while taking into account the optical setup and the application at hand.
The boundary needs to be selected carefully so that it is high enough not to permit excessively large amount of information, which would result in bridg-ing the areas in between phase discontinuities. Equally, the boundary needs to be low enough so that phase discontinuity information does not become punctured and disjointed. This can be seen in Fig.7, which shows phase discontinuity locations desig-nated by various selections of boundary lines. B. Benefits of Increased Effective Kernel Size
The benefits of the sliding-window method can be seen in Fig.8(a), which shows a gray-scale represen-tation of the accumulated number of times each pixel has been identified to be a phase discontinuity location.
It is also possible to see, Fig. 8(c), that using the sliding-window method has the added advantage of eliminating tiling artifacts.
There are additional benefits to the increased effective size of the combined neighborhoods of the
kernel, Fig. 6(a), which spans a 19 × 19 pixel area. Each pixel location now receives up to 100 predic-tions, and there is a larger effective neighborhood area contributing information to the overall verdict of whether a given pixel site is a phase discontinuity location or not.
These benefits are demonstrated in Fig.9, which shows that the larger the effective area is increased to, the better the tiling artifacts are overcome.
Another benefit of the sliding-window mechanism is that, the smoother the pixel increments by which the kernel is passed over the image to produce predictions, the larger the number of predictions received per pixel. This effect of increased number of predictions can be seen in the sub-captions of Fig. 9, which also show that the gains in both the maximum and mean number of prediction per full-field pixel come at the expense of increased process-ing time. It is worth notprocess-ing that at even the19 × 19 pixels effective neighborhood, which is the maximum extent possible to achieve with an underlying10 × 10 pixel kernel, the overall duration for processing a 200 × 200 pixel full-field noisy phase map is still under 20 s when running on a midrange laptop ma-chine: Lenovo Ideapad Z580, Intel Core i5 3210 M 2.5 GHz processor, 2 cores, 8 GB RAM, 1 TB hard disk drive.
By contrast, the training phase of the10 × 10 ker-nel was considerably more time consuming, taking more than 24 h worth of server-grade computational resources in the cloud, further details on which are available in [35]. The point that may be worth bear-ing in mind, in terms of performance efficiency, is that the training phase is a one-off operation.
Once the kernel has been trained, it can, from there on, be applied directly using much lower speci-fication machines, and performs within timeframes comparable to more conventional image processing convolutional operators.
As well as overcoming tiling artifacts, increasing the effective size of the kernel also improves the
Fig. 7. Phase discontinuity locations designated by applying a range of boundary lines. The boundary is applied to the accumu-lated number of times each pixel was predicted to be a phase discontinuity location.
Fig. 8. Gray-scale representation of the accumulated number of times each pixel was predicted to be a phase discontinuity location, when visited by the sliding window, is shown in (a). Actual phase discontinuity locations are shown in (b) for comparison. Applying sliding window to noise-free phase map can be seen in (c), showing that tiling artifacts are largely overcome overall, and completely so in areas further than the kernel’s width (10 pixels) away from full-field boundary.
overall reliability of the predictions. Intuitively speaking, this can be stated by saying that 100 votes are better than one. In other words, as each pixel is receiving more predictions, thanks to the sliding-window mechanism, the resulting aggregated verdict is more likely to be reliable.
These reliability improvements can be assessed using Matthews correlation coefficient (MCC), which is also known as the phi coefficient, which in turn is a variation on Pearson’s chi-square test [42]. The MCC measure takes into account the accuracy of the verdicts in terms of not only true positives and true negatives, but also false positives and false negatives.
We examined the improvements obtained via the sliding-window mechanism, in conjunction with the voting ballot method, against a range of effective neighborhood extents and across all possible voting boundary lines, Fig. 10, which served to confirm,
in an objective and quantitative manner, the signifi-cance of the improvements which can be thus achieved.
C. Viability Against Nonsimulated Phase Maps
In this section we discuss the viability of using our machine learning approach to the problem of reliable identification of phase discontinuities, when applied not only to computer-simulated wrapped phase maps, as shown thus far, but also when applied to real-world phase fields obtained using an actual optical system application.
We describe some of the practical issues involved, and take a glimpse by showing how, in principle, our kernel can be applied to nonsimulated optical phase maps, even though the kernel itself was trained against computer-generated phase field simulations exclusively.
Ideally, the neural network should be trained us-ing examples as representative as possible of those typically encountered in the target optical measure-ment system whose output wrapped phase maps are to be analyzed. In practice, however, this may not always be possible for many practical reasons. For example, even if it could be arranged for the neces-sarily large number of real, and therefore noisy, wrapped phase maps to be recorded using the optical setup, the actual location of the phase discontinuities cannot be known with the same degree of certainty as that of computer-simulated phase fields. Moreover,
Fig. 9. Gray-scale representation of accumulated predictions, with increasing effective combined neighborhoods’ size. Subcap-tions read as follows: effective combined neighborhoods’ size in pix-els, processing duration formatted as seconds: milliseconds, maximum number of predictions received by any pixel within the full-field image, mean number of predictions per pixel rounded to three significant figures. Bottom subplot shows the mean num-ber of prediction received by pixels in full-field phase map, for a given effective combined neighborhoods’ size.
Fig. 10. Measuring improvements in prediction performance ob-tained by increasing the effective combined neighborhoods’ size through the sliding-window mechanism. Mathews correlation co-efficient (MCC) score across all possible voting boundary lines. Subplots axes: MCC on vertical axis, and voting boundary line of absolute number (as opposed to percentage) of votes received on horizontal axis. Subplot captions format: effective combined neighborhoods’ size in pixels, maximum MCC score in subplot.
expert human input would in all likelihood be re-quired, compounding the laboriousness of such an endeavor.
Furthermore, at the present juncture in this inves-tigation into this machine learning approach, it is not entirely clear just how many training examples are needed before satisfactorily reliable phase disconti-nuity predictions can be realized. It is for these reasons why it might be considered more appropriate to confine this investigation, into machine learning avenues, to computer-generated interferograms at this point in time.
Nonetheless, and even at this relatively early stage of development, it is reassuring to see, Fig.11, that our computer simulation trained kernel can be directly applied to a real optical phase field [43], the
likes of which the kernel had never encountered during its training phase.
D. 10 × 10 Quantifying Kernel Nonaggregated Performance
Assessing the performance of the kernel in terms of the quality of the predictions it produces, in order to better investigate ways in which it can be improved, is something that we have examined primarily in a recent work [35], in which we introduced a new measure called the MCC% score.
The MCC% measure quantifies the performance of predictions, while also taking into account false positives and false negatives. It has the added advan-tage of allowing for the performance to be visualized in a manner that provides intuitive insight into ways in which the performance can be improved. The MCC % is obtained by multiplying the classification threshold, as a percentage figure, with the corre-sponding MCC score achieved by comparing the pre-dictions of the kernel against a set of cross-validation examples. For this purpose, we used a set of 16,000 computer-simulated examples,10 × 10 pixels each.
The computer simulation method used for creating these cross-validation examples is the same as that used for generating the training examples. However, two distinct computer simulation runs where performed for randomly generating the full-field interferograms from which the training and cross-validation10 × 10 pixel example phase map portions were obtained. Details of this process are available in [35].
The classification threshold is the figure against which the excitation level of the neurons in the out-put layer of the neural network, Fig.1, is compared, Fig. 12. The MCC% plot, Fig.13, shows the perfor-mance of the 10 × 10 kernel underlying the sliding window described in this work, which employed an 85% threshold for classification.
The gap in performance, between that against the training and cross-validation sets, indicates that
Fig. 11. (a) Mechanical sample is a cylinder that is 400 mm long, 190 mm in diameter, and 5 mm in wall thickness [43]. (b) Cylinder was pumped with oil and mounted in a shearography optical setup. (c) Applying the10 × 10 kernel to the optically obtained noisy wrapped phase measuring 604 × 1024 pixels. The aggregated kernel’s predictions are shown (d) in gray-scale representation. Applying a voting ballot boundary line is shown in (e).
Fig. 12. MCC% measure is defined as: MCC% = classification threshold% * MCC. The excitation level of the neural network’s output layer is compared against the threshold, where output levels equal to or exceeding the threshold are classified as phase discontinuity locations. A nominal 50% threshold (corresponding to zero on the vertical axis) is typical, though higher (or lower) values can be used to demand more confidence in (or relax) classification predictions.
improvements can be attained by increasing the number of training examples used for training the network. This gap shows that more training exam-ples are needed to allow the kernel to better general-ize the knowledge present in the training examples to new problems never encountered during training.
4. Discussion and Future Work
The main focus of this work was to ascertain whether a10 × 10 kernel would be large enough to process a full-field interferogram. At first sight, it may have been tempting to assume that such a relatively small kernel would be merely a useful tool for simply illus-trating, in principle, how machine learning tech-niques can be applied to the problem at hand. We have shown, however, that a 10 × 10 kernel can in fact be large enough for application to full-field phase maps.
Nevertheless, it is worth bearing in mind that, fun-damentally, the raw performance of the underlying kernel itself is paramount. Furthermore, it would be worth looking into the possibility of applying the sliding window in a multipass fashion, each pass utilizing a classification threshold selected from a wider range of values, rather than, e.g., 85% per se. To facilitate a direct comparison between perfor-mance against simulated phase fields versus those obtained in real optical setups, both qualitatively and quantitatively, it would be interesting for a fu-ture work to make use of a more rigorous simulation of speckle phase fields [44]. This would both over-come the issue of generating a large number of train-ing examples whose actual phase discontinuity locations are known a priori with certainty, and go a long way to ensuring that the computer-generated simulations are as representative as possible of the practical optical setup for the target application at hand.
Another avenue worthy of investigation is revisit-ing the techniques employed durrevisit-ing the initial train-ing phase. One approach here would be to utilize a boosting [45] mechanism to train a range of rela-tively weak classifiers in a successive manner, result-ing in better quality of predictions durresult-ing the training phase, which would be envisaged to trans-late into better generalizations to the problem domain.
5. Conclusion
We reported on how a10 × 10 machine learning ker-nel can be used for identifying phase discontinuities across a full-field 200 × 200 pixel interferogram. The machine learning neural network, which has 3,200 neurons in its hidden layer, was trained with 120,000 training examples.
We demonstrated how improvements were
obtaining by employing a sliding-window mecha-nism, thus aggregating up to 100 predictions for each pixel within the full-field interferogram, by compar-ing the number of positive predictions to a given voting majority boundary line.
We showed that choosing the voting boundary line can be viewed as a calibration step, and described the important criteria for the selection of such a boun-dary for a given application. We also showed that better results can be obtained by aggregating phase discontinuity predictions in this way, when compared to results generated by a more direct nonoverlapped tiling of the interferogram. This overcomes many of the limitations imposed by the relatively small size of a 10 × 10 kernel, and makes it possible for a wider effective neighborhood of19 × 19 pixels to contribute to the predictions, thus avoiding the necessity for in-creasing the size of the kernel to be trained. A great benefit of doing so is to allow for the computational budget to be allocated to other aspects of the overall kernel training procedure. This helps to demonstrate that it is viable for future works to focus primarily on improving the raw predictions generated by the10 × 10 kernel, now that we have shown how predictions produced by such a kernel can be sufficiently suitable for processing a full-field interferogram.
Firas Sawaf would like to thank Delft University for kindly making available to this research various facilities at the Aerospace Non-Destructive Testing Laboratory, and for the continued collaborations with members of this group at the Faculty of Aerospace Engineering.
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