Limited view X-ray tomography for dimensional 1 measurements

6 The growing use of complex and irregularly shaped components for safety-critical applications has increasingly led to the adoption of X-ray CT as an NDE inspection tool. Standard X-ray CT methods require thousands of projections, each regularly distributed evenly through 360 ◦ to produce an accurate image. The time consuming acquisition of thousands of projections can lead to signiﬁcant bottlenecks. Recent developments in medical imaging driven by both increasing computational power and the desire to reduce patient X-ray exposure have led to the development of a number of limited view CT methodologies. Thus far these limited view algorithms have been applied to basic synthetic data derived from simple medical phantoms. Here, we use experimental data to rigorously test the capability of limited view algorithms to accurately reconstruct and precisely measure the dimensional features of an additive manufactured sample and a turbine blade. Our ﬁndings highlight the importance of prior information in producing accurate reconstructions capable of signiﬁcantly reducing X-ray projections by at least an order of magnitude. In the turbine blade example a dramatic reduction in projections from 5000 to 24 was observed while still demonstrating the same level of accuracy as standard CT methods. The ﬁndings of the study also suggest the importance of sample complexity and the presence of sparsity in the X-ray projections in order to maximise the capabili-ties of these limited algorithms. With the ever increasing computational power limited view CT algorithms oﬀer a method for reducing data acquisition time and alleviating manufacturing throughput bottlenecks without compromising image accuracy and quality.

Thus far these limited view algorithms have been applied to basic synthetic data derived from simple medical phantoms. Here, we use experimental data to rigorously test the capability of limited view algorithms to accurately reconstruct and precisely measure the dimensional features of an additive manufactured sample and a turbine blade. Our findings highlight the importance of prior information in producing accurate reconstructions capable of significantly reducing X-ray projections by at least an order of magnitude. In the turbine blade example a dramatic reduction in projections from 5000 to 24 was observed while still demonstrating the same level of accuracy as standard CT methods. The findings of the study also suggest the importance of sample complexity and the presence of sparsity in the X-ray projections in order to maximise the capabilities of these limited algorithms. With the ever increasing computational power limited view CT algorithms offer a method for reducing data acquisition time and alleviating manufacturing throughput bottlenecks without compromising

Introduction 8
Modern engineering is increasingly utilising complex components. Turbine 9 blades, for example, feature complex cooling channels and highly optimised 10 curved surfaces, and the rise of additive manufacturing has given huge poten-11 tial for extremely complex shapes. Such shapes present significant inspection 12 challenges to traditional NDE techniques, as these features can obscure defects 13 or manufacturing errors. X-ray computational tomography (CT) is one of the 14 few technologies capable of non-destructively measuring both the external and 15 internal features of a component [1]. Numerous CT approaches exist, but within 16 industry they commonly consist of a static X-ray source and a movable detector 17 which is perpendicular to the source. The sample to be CT scanned is placed on 18 a movable disk which rotates through 360 • allowing multiple X-rays projections 19 to be captured. Standard X-ray CT methods require thousands of projections, 20 each regularly distributed evenly through 360 • to produce an accurate image 21 [2, 3]. Once an accurate tomographic image is generated it can be used to as-22 sess the specimen for flaws, quality control and undergo dimensional analysis 23 by comparison with CAD [e.g., 1]. 24 One of the major downsides of X-ray CT is the time consuming data acqui-25 sition process which can lead to significant bottlenecks. To alleviate these bot-26 tlenecks in throughput companies may be forced to purchase additional X-ray 27 CT capability at great cost or reduce individual X-ray exposure times lowering 28 the signal-to-noise ration and image quality. Spurred by the ever increasing (e.g., material properties and geometry) to aid in the reconstruction is usually 59 available in NDE applications but is often difficult to accomplish. 60 An alternative method to filtered back projection imaging is to relate the 61 measured projection data to a set of unknown image pixels via a set of alge-62 braic equations [2, 10, 14]. The measured amplitude of a monochromatic X-ray 63 through an object is given [2] where I is the measured X-ray intensity at the detector, I 0 is the intensity 65 of the monochromatic X-ray source and r µ(x, y)ds is the ray-path integral 66 through the object with radiographic attenuation µ(x, y). Equation (1) maybe 67 discretised and re-written as 68 −log where i indicates the pixel number, a i is the weighting of each pixel based on 69 the length of the X-ray raypaths crossing each pixel and ν is the attenuation where b ∈ R m are the X-ray projections, A ∈ R m×n is a matrix of pixel weights 72 which relates the image to the data projections and is often called the projection 73 matrix and x ∈ R n are the image pixels (Fig. 1).
The second reconstruction procedure considered is the SIRT which performs  134 For the tomographic reconstruction problem we reorder the vector of image 135 pixels x such that it now represents a 2D array i.e., the image x ∈ R r×c where 136 r and c are the total numbers of row and column pixels respectively. The TV 137 norm of the discretised image may therefore be defined as [e.g., 10, 12,16]: where, i = 1 . . . r and j = 1 . . . c.

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Numerous algorithms have been developed in the recent years which min-  approximation must be used [9, 10, 12]: where δ > 0 is small. The selection of δ is vital for an accurate reconstruction

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The method of Sidky et al. [12] combines the ART algorithm with a TV 157 minimisation step to solve the following optimisation problem: in an iterative manner. The first stage of the method is to minimise Ax = b in a 159 least squares sense using a single iteration of ART (Algorithm 1). It should be 160 9 noted that the inclusion of the optional positivity constraint is applied at the 161 end of each ART iteration as opposed to after each row iteration as described 162 in Algorithm 1. The next step in the method is to minimise the TV of the ART 163 reconstruction using a fixed step, γ, gradient descent method. This two step 164 process is repeated for a fixed number of iterations (Algorithm 3). Throughout 165 this manuscript this algorithm will be referred to as as ART-TV.

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Algorithm 3 ART-TV 7: 10: for j = 1, . . ., N grad do ⊲ Gradient TV minimisation 13: The second TV reconstruction algorithm introduces the TV as a regularisa-167 tion term into a least squares problem and the goal is to minimise: where α > 0 is the regularisation parameter that controls the weighting between if k > 1 then 8: ⊲ Barzilai and Borwein step 9: 16: proving the rate of convergence and being less affected by ill-conditioning than  The experimental X-ray CT data for a turbine blade were acquired using a   with positivity and SIRT (Fig. 6). For the edge detection analysis a Gaussian 283 kernel of 9 pixels with a standard deviation of 2 pixels was deemed optimal. with the degradation in the reconstructed images (Fig. 5) and an increase in 307 error for the dimensional analysis (Fig. 8).

Additive manufactured sample 309
For the additive manufacturing test case a simple example was manufac- where the X-ray sampling is limited.

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To qualitatively assess the capability of the various algorithms to provide 341 accurate dimensional assessments of the sample thickness, measurement were 342 made along a cross-section through the sample (Fig. 10). For the dimensional       sampled reconstruction and the second dataset was generated using a circular 440 mask to reduce the object's complexity such that it closely mimics that of the 441 turbine blade (Fig. 14). We note that in the simplified synthetic example .

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The reconstruction of the circular synthetic dataset did not exhibit the 462 smearing artefacts seen in the fully segmented example (Fig. 14). The cross-   in such a way that each successive hyperplane is orthogonal to the previous one.

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In a practical sense this amounts to ordering the projection data so that the 513 current projection data is as independent as possible compared to the previous 514 projection [21][22][23]. A number of approaches have been proposed in the ordering 515 of the rows of A [e.g., 22, 23], however similar convergence acceleration may be 516 achieved by a randomised ordering scheme [21].

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Much work has been conducted over the past decade on applying compres-519 sive sensing methods to medical X-ray CT as a way of reducing patient radiation 520 exposure. These methods have yet to be applied or rigorously tested to indus-521 trial X-ray CT where reduced data acquisition times are desirable to improve 522 manufacturing throughput. This study surveyed and rigorously tested the ca-