Back-projection Filtration Image Reconstruction Approach for Reducing Out-of-plane Artifacts in Laminography

Because of incomplete depth information in computed laminography, 3D reconstructed images are contaminated by out-of-plane artifacts, or ripple artifacts along the depth direction. Furthermore, these issues become more severe whenever printed circuit boards (PCB) are imaged due to the presence of high density materials, such as ball grid array, chips, etc. In this reasearch, in order to remove these artifacts, we proposed the use of the voting strategy. This method is a weighted back-projection which gives less weight to outliers, or odd values. However, standard analytical reconstruction methods, such as filtered back-projection (FBP), cannot fully judge and remove the outliers. Thus, we proposed to combine this strategy with the back-projection filtration (BPF) reconstruction method, which first reconstructs the derivative of the projection image, and then filters the reconstructed image by the Hilbert transform. Ripple artifacts were shown to have been dramatically reduced by using the voting strategy in the BPF algorithm.


Introduction
Computed laminography (CL) is an x-ray imaging modality that acquires a finite number of projections from a tilted angle and reconstructs 3D images with algorithms similar to those used in computed tomography (CT).Nonetheless, whenever a large plane object, such as a printed circuit board (PCB), is being imaged, the conventional CT geometry cannot fully cover the information along the parallel direction because x-rays are highly attenuated by the thickness of the traversing direction of the PCB.The high attenuation usually causes significant photon starvation and noise.The tilted angle scan in CL can solve theses problems, but only to a certain degree.[1] CL also offers some disadvantages.Generally, PCB contains high density materials such as ball grid array and chips.These cause the out-of-plane artifact or ripple artifact along the depth direction which is perpendicular to large plane objects [2].Those artifacts disrupt the detection of defects by contaminating out-of-plane structures.Currently, there are great efforts and several kinds of solutions to reduce this artifact in the field of limited angle CT and tomosynthesis.[3,4] Unfortunately, most objects like PCBs attenuate the x-ray with much higher degree than anatomical structures do.Also, high density regions in PCB are closely assembled each other making the artifact more pronounced in the image.Thus, most of the methods used in the anatomical tomography are not effective due to such reasons.Among these various methods, we suggest the use of one effective method called voting strategy.[5] It sorts out the outlying data and changes them to average value.It can be used in the filtered backprojection (FBP) reconstruction algorithm.However, the back-projection filtration (BPF) reconstruction algorithm can strengthem the effect of the voting strategy because BPF presents the differentiated back-projection (DBP) image and edge information in the DBP image is convenient to vote.[6] 2 Methods

Phantom and Geometry
PCB was cut with containing a compressed ball grid array and rectangular semiconductor chips (see Figure 1).It was projected by 140keV x-ray with three parallel line detectors so as to generate a parallel fan beam geometry.Three source positions were chosen to make enough angular variation.The distance between them was 7.4mm, and the tilted angle was 27.85 degree because source to object distance was 14mm.Therefore, nine parallel fan beam projections were totally acquired.Making 2D projections from 1D detectors were achieved by moving objects linearly in the direction perpendicular to detector and parallel to the ground.The direction where object moves, the direction of detector form, and the way up and down form the xyz coordinate system, respectively.The uv coordinate system of the projection data is parallel to the xy plane.In this research, 400mm line detectors having 9216 pixels were used, and source to detector distance was 271mm.Figure 2 describes the overall geometry.This has more advantages than using flat panel detectors in terms of projecting time and cost.

Reconstruction Algorithm
CL is a kind of oblique CT, so the conventional CT reconstruction algorithms can be used with some modification.The analytical FBP algorithm is common and basic, but the BPF algorithm can strengthen the voting strategy, specifically judging the outliers.The critical difference is the DBP derived from differentiated projections.In DBP, outlier detection in the voting strategy can be implemented for the edges of structures, but not the blob regions.Because of such reason, it has the advantage of not being affected by the gradation of the blob regions.After voting and weighted back-projection, the Hilbert transform changes the DBP image into a final reconstructed image.The Hilbert transform actually performs filtering; therefore, the theoretically same image as the FPB is calculated.The BPF algorithm has a so-called PI-line that is in the direction of the Hilbert transform.Because the geometry was not full angle scan, we used virtual PI-line segments parallel to the detector, which are the same as circular conebeam virtual PI-line segments.Following is the equation for BPF reconstruction.
, where is the reconstructed image, is the projection data, is the angle which parameterizes source position and represents the object moving direction, ̂ is the unit vector of the ray direction, ⃗⃗ ′ is the point on the PI-line segments, ⃗⃗ is the source position, and ℋ is the Hilbert transform.
, , | ̂ is the differentiated projection which used for the voting strategy.Figure 3 describes the flow of the BPF reconstruction with the weighted back-projection using the voting strategy.

Weighted back-projection with voting strategy
By voting which value is appropriated for a specific image voxel from contributing projection data pixels, the outliers that give the odd value to that voxel can be identified.The mean and standard deviation can be derived from the contributing projection data and a Gaussian distribution can be built.The Gaussian weighting term from the th projection data pixel in the th projection to th image voxel is following.
, where , is the value that is contributed from ℎ data to th image voxel, is the mean of the data contributing to th image voxel, is the standard deviation of them.Weighting is applied by following manners.
This is performed for each image voxel.This process works with differentiated projections when we reconstruct a DBP image.

Results
We scanned PCB samples containing a ball grid array and semiconductor chips.Using a conventional BPF algorithm, ripple artifacts evidently contaminated these high density structures in the image.After incorporating the voting strategy in the BPF algorithm, the significant reduction of artifacts can be observed in the reconstructed slice image (see Figure 4).When the ball grid array is in-plane structure, the ripple artifacts by the boundaries of the semiconductor chips are removed.In addition, when the ball grid array is out-of-plane structure, the ghost artifacts of the ball grid array are removed.

Discussion
The voting strategy can be applied without Gaussian weighting, which means that outlying data is removed and substituted by the average of contributing pixels.Then there may be some unnatural discretization in complex structures of the reconstructed image.It implies that the degree of the Gaussian weighting must be optimized.The outlying data, which are far from the average value, have to be considered how far they are and how much they are weighted.It can be parameterized by multiplying some coefficient to standard deviation.Currently, because the parameter is heuristically determined and it is constant for every voxels, there is a limitation to get artifact-free image when various structure and contrast exist in the image.For example, when there are metals and materials having attenuation coefficient near water, if one wants to remove the ripple artifacts by metal and set the parameter high enough, the artifacts by the structure in the water-like material is excessively reduced.Because it destructs the structure severely, we have to compromise with deciding the parameter value.
In addition, BPF can release such a problem in some degree.The gradation characteristic in simple back-projection is the actual reason why out-of-plane artifacts are shown as ripple or spread.Whereas DBP does not use the blob area of the projections for voting, but uses edge or gradient information.Therefore, the voting is applied in smaller area generally, which reduces the spread of structures.If the projection angle is larger, those gradations and overlapping decrease, and the voting is more powerful.However, there is a problem to use the BPF and the voting in the differentiated projection domain, which is a contrast problem.Because ripple artifacts are generally considered as low frequency components corresponding to the contrast, artifacts will be regenerated if we add such components for high contrast.Such very low frequency components have very low gradient in the spatial domain.Small gradient has low pixel value when it is differentiated, and the voting cannot consider it important factor.

Figure 1 :Figure 2 :
Figure 1: Projected PCB which contains a ball grid array and semiconductor chips

Figure 3 :
Figure 3: The flowchart of BPF reconstruction algorithm with the weighted back-projection using the voting strategy

Figure 4 :
Figure 4: Slice of BPF reconstructed images of PCBs.The ball grid array is in-plane structure in the top images and out-outplane structure in the bottom images.Without weighted back-projection (LEFT), with weighted back-projection using voting strategy (RIGHT)

Figure 5 :
Figure 5: Slice of BPF reconstructed images of PCBs.The ball grid array is in-plane structure in the top images and out-outplane structure in the bottom images.Without weighted back-projection (LEFT), with weighted back-projection using voting strategy (RIGHT)