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This video presents a new approach 

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to reconstruction of Computed Tomography 

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data driven by geometric techniques.

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Computer tomography is one 

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of the most popular modalities 

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because its capability to visualize 

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internal organs of the body 

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with high resolution.

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In Computed tomography an unknown

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object is reconstructed from projection images.

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To acquire the projection images, 

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we have an x-ray source 

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and an image detector. 

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Then, data are obtained by

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exposing a patient or an

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object to a beam of x-rays,

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which can be understood as 

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a set of rays penetrating

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a weighted region.

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Then we move our imaging system 

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to a different position and 

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generate an additional projection image. 

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We follow this rotation and 

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acquire many projection images 

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up to a thousand different positions. 

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Here you see an example of 

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projection images of a rabbit. 

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We used 220 positions 

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to capture these images.

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These projection data are then 

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reconstructed using standard techniques 

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based on the Radon Transform.

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The volume is reconstructed by

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back projecting the acquired data. 

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The summation is performed

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all projection angles.

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Here you see how a 

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slice of rabbit data is reconstructed.

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The Nyquist sampling criteria limit 

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techniques based on the Radon transform.  

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Therefore, these techniques provide 

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reduced image quality when 

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using a reduced number of

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projections. But using fewer 

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projections reduces radiation dose

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and reduces incidence of

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cancer in patient population.  

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The literature shows that 

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high radiation dose as used 

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in standard CT can lead to cancer. 

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Compressed sensing presents a new 

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method to capture and represent compressible

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signals at a sampling rate significantly 

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below the Nyquist rate. The main 

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idea behind this new approach can 

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be summarized as an alternative to 

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directly reconstructing a signal

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or an image. The projection 

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data are transformed into a sparse version.

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Then the signal or image is 

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reconstructed in a dimensionality-reduced 

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space by minimizing the L1 - distance.

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To use this method, the 

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signal to-be-reconstructed needs 

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to have sparse-like attributes,

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which standard medical images do not have.

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We observed that standard CT 

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techniques converge quickly when the 

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intensity of all voxels is similar. 

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This occurs because the value 

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placed in each voxel along each

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penetrating ray will be close to its

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actual intensity when all voxels 

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have similar intensity.

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Additionally, we observed that when

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plotting the entropy as a function of the 

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number of projections used in the 

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reconstruction a plateau is visible 

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even after a small number of projections.

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Thus, it should be possible to 

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reconstruct high quality images from 

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a limited number of projections.

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Based on these observations, our 

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reconstruction strategy resembles a

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geometric compressed sensing approach. 

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As the sparse transform, we

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are handling each region of the

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to-be-reconstructed image as separate entity.

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Let's review our approach step by step.

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We have an unknown object.

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A radon-based technique is used

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to generate a poor quality 

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reconstruction. To obtain a rough

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segmentation of our object-of-interest, 

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we make use of models or priors.

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In order to make use of our earlier 

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observation, we have to classify the rays 

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that penetrate the object.

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As you see here, the rays 

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that penetrate the object are 

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shown in red, and the complement 

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of these rays is shown in green. 

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To classify our rays, we reduce 

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the problem from 3D to 2D. 

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Each plane of rays is handled separately.

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Now lets take a closer look 

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at one of our 2D arrangements.

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In the 2D plane, we have 

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a set of objects obtained from 

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the rough segmentation, surrounded

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by lower density regions called 

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complements. To separate the reconstruction

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of the objects from that of 

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the complements, we first approximate

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the rough boundary of the objects by 

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simple polygons and then treat the polygons as obstacles. 

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To efficiently classify all the rays, 

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we use a point-line duality transformation 

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on the set of boundary segments

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of the polygons. Each vertex of 

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the polygons transforms to a dual

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line and each ray maps to a point in

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the dual space. The set of dual lines

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forms an arrangement, which partitions the 

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space into a set of convex cells.

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Each cell contains the set of rays 

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intersecting the same subset of boundary

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segments of the polygons, that is, 

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all rays traveling through the hourglass

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defined by the set of segments. All 

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types of rays can be identified by sweeping the

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arrangement in an online fashion.

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To report all the cells and rays in 

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a more efficient way, we use 

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the topological peeling algorithm due to its 

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optimal time and space complexities.

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Let's go back into 3D.  

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We have shown how to optimally classify the 

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rays for one plane by using geometric techniques. 

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In our algorithm, we perform 

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this for all projection images.

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For the remainder of this presentation, 

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two projections will continue to represent

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the vast number of images that are actually used.

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We reconstruct from the outside

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inward in a layer-by-layer fashion. 

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First, we reconstruct the outer region 

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by using the green rays. Next, 

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we project the reconstructed outer region. 

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These resulting data are subtracted 

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from the original projections. 

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Second, we reconstruct the outer object 

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by using the red rays. We project

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the reconstructed object by

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using the remaining rays. These

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resulting data are subtracted from the projection space.

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Finally we reconstruct the remaining 

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object by using the yellow rays.

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In the end, we have produced a 

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reliable reconstruction from 

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a reduced number of projections.

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Now to our results:

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Here you see real data from our 

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rabbit study. To the left is 

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a rough segmentation of the object-of-interest.

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The center object shows the reconstruction

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from a reduced number of projections obtained 

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using standard techniques.

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On the right you see our reconstruction. 

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Compared to the center image that is

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calculated from the same number of projections, 

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our result on the right has a significantly improved image quality.

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We thank you for your time and 

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interest from the University at Buffalo.