Image Processing And Analysis Of Ct Scan Images Biology Essay

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Medical Imaging is an important aspect of image processing and has a wide usage. In this project, we will focus on creation of CT-images which are noiseless and have high spatial and intensity detail.

CT images can be reconstructed analytically or iteratively. The analytic methods, e.g. filtered back-projection, are known to be computationally inexpensive and highly accurate. Iterative reconstruction seems of high interest since better dose usage is expected.

We are going to focus on the reconstruction of the image by discussing various types of CT-machines and the algorithms regarding to them.

Introduction

The basic concept behind this project is the reconstruction and analysis of images that are taken from any sort of a Multi-slice Helical Computed Tomography Machine. The

Importance of the System

The purpose of the construction is to reconstruct images that are constructed by the CT so that they are interpreted in a better why by the radiologists.

There are two features of the system:

The system can be used as a simulator for a Multi-Slice Helical Cone-Beam CT-Machine.

The system can be used to reconstruct image for any valid geometry of the machine of the following type:

Parallel Beam CT

Fan Beam CT

Flat-Bed Detectors

Arc-Bed Detectors

Axial Cone-Beam CT

Helical Cone-Beam CT

Problem Statement

The reconstruction of medical images is an inverse mathematical problem. However, these problems cannot be arithmetically solved. Therefore, we need to find alternative methods to reconstruct these images.

In case of Computed Tomography, the images, when acquired, are usually some complex transformation of original image. The simplest case of this is the radon transformation. However, the problem we are going to tackle is the reconstruction from a much more complex sort of data. The data that needs to be constructed is from a multi-slice helical CT, which will be discussed in the Chapter 7 of this thesis.

Research objectives

The objective of this project is to develop a system that is able to reconstruct raw data acquired from a CT-machine and give the user an ability to manipulate the reconstructed image using the windowing technique functions. The intended purpose of writing this thesis is to study, implement and improve the techniques to be used for the purpose of reconstruction of the images.

Methodology

To reconstruct an image from radon transform, we make use of iterative and algebraic recursive algorithms. Therefore, we are going to implement an iterative algorithm, FBP, on the most basics of types and extend it to match the requirements of the better technologies. Also, we'll discuss SSRB technique used to reduce the complex data to a simpler form using an exact algorithm.

Thesis Contribution

The thesis contributes in the field of CT reconstruction. In this thesis, we are going to discuss the geometry of different generations of CT-machines and then implement an efficient enough algorithm for each generation which can perform reconstruction of images.

Literature Survey

Medical Imaging

Medical imaging is a set of non-invasive techniques and processes used to create images of the human body. There are different techniques for producing medical images such as magnetic resonance imaging (MRI), X-ray computed tomography and ultrasound. All of the before mentioned techniques are used to capture images which are help full in diagnosis of many diseases and body disorders such as bone fracture and therefore are widely used for clinical purposes.

The basis of medical imaging can be said to be the solution of inverse mathematical problems. In case of X-ray tomography, image reconstruction or iterative reconstruction is process that is used to reconstruct 2-D or 3-D images from the amount of the x-ray attenuated at a certain point at a certain angle.

Tomography

Tomography is a non-invasive technique letting the visualization of the internal of an object without the superposition of over- and under-lying structures.

In conventional radiographs, the over and under-lying organs were superimposed on the same film. When compared to tomographic techniques, we don't use have to face such problems. The reason is that, we get images in slices of precise thickness and therefore we don't have to deal with extra information in the image.

Tomography is done by different modalities. Each tomographic modality measures a different physical quantity [1]:

X-Ray Tomography: The number of x-ray photons transmitted through the patient along individual projection lines.

Nuclear medicine: The number of photons emitted from the patient along individual projection lines.

Ultrasound diffraction tomography: The amplitude and phase of scattered waves along a particular line connecting the source and detector.

The thing to do in all the above mentioned cases is to generate the estimation, from these measurements, of the distribution of a particular physical quantity in the object. The quantities that can be used as data for reconstruction are:

X-Ray Tomography: The distribution of linear attenuation coefficient in the slice being imaged.

Nuclear medicine: The distribution of the radiotracer administered to the patient in the slice being imaged.

Ultrasound diffraction tomography: the distribution of refractive index in the slice being imaged.

Computed Tomography - CT

Computed tomography (CT) is a medical imaging method that uses tomography done with the help of computational power. CT produces a bulk of data that can be later be manipulated in order to demonstrate various bodily structures based on their ability to block the X-ray beam through "Windowing".

Earlier the images were generated in the axial planes, orthogonal to the

z-axis of the body; now-a-days the machines allow the volume of captured data to be represented in various planes or as 3D structures.

Image Reconstruction

The goal of image reconstruction is to retrieve back the information that has been lost or destroyed during imaging process itself. Image data can also be lost during imaging process when the data has been corrupted by noise or in case of blurring which is known as an option for undesirable images.

Medical imaging reconstruction systems improve every day, however the aim of all of this development is to reduce noise in projection and makes images with better resolution and contrast.

Noise in projections can produce some images that have lower resolution, contrast and brightness. Noise can also be produced by sensors and detectors itself. Noise of image can be considered as an undesirable part of image that makes pixel transparency and recognition difficult.

CT Machine and Concepts

This topic describes the basics of a CT machine and the concepts behind its working. Also, it discusses different types of geometries for which we have to reconstruct images from raw data.

Working of a CT Machine

Computer Tomography basically uses a computer and a rotating x-ray device to create detailed, cross-sectional images, or slices, of organs and body parts.

A CT machine resembles a large, square doughnut. A flat "patient couch" is situated in the circular opening, which is about 24 to 28 inches in diameter. The patient lies on the couch, which can be moved up, down, forward, and backward to position the patient for imaging.

The CT scanner itself is a circular, rotating frame with an x-ray tube mounted on one side and a banana-shaped detector mounted on the other. A fan-shaped beam of x-rays is created as the rotating frame spins the x-ray tube and detector around the patient. For each complete rotation, one cross-sectional slice of the body is acquired.

As the scanner rotates, the detector takes numerous snapshots called "profiles." Typically, about 1,000 profiles are taken in one rotation. Each profile is analyzed by computer, and the full set of profiles from each rotation is compiled to form the slice-a two-dimensional image.

These profiles are basically the count X-ray photons that are received at the other end i.e. the detector.

Machine Geometry

The geometry of a CT-Machine is not just the distance between the detector and array but it accounts for a much more complex combination of features that are used for a proper reconstruction of an image. Under this heading we are going to discuss the general geometry of a CT-machine which is going to be described more specifically in their related topics.

The following are the attributes that are used for defining the geometry of a CT-Scan machine.

Start of the Orbit

Orbit Size (180-degree for short scans and 360 for fan)

Number of samples

Angular Samples

Vertical Samples

Horizontal Samples

Sample Spacing

The Distances between

Source and Detector (dsd)

Source and Isocenter (dso)

Isocenter and Detector (dod)

Focal Point and Source (dfs)

Collimation

Since X-Rays are produced in every direction. A collimation device is used to focus them to some ROI.

A collimator is a device that narrows a beam of particles or waves. To "narrow" can mean either to cause the directions of motion to become more aligned in a specific direction (i.e., collimated or parallel) or to cause the spatial cross section of the beam to become smaller.

The tube or source collimators are used in the x-ray source tube and determine the section thickness that will be utilized for a particular CT scanning procedure. When the CT technologist selects a section thickness he or she is determining tube collimation by narrowing or widening the beam. A second set of collimators located directly below the tube collimators maintain the width of the beam as it travels toward the patient. [2]

A final set of collimators called post-patient or pre-detector collimators are located below the patient and above the detector. The primary responsibilities of this set of collimators are to insure proper beam width at the detector and reduce the number of scattered photons that may enter a detector.

Hence, Collimation, denoted by D(mm) is the thickness of the image slice, attained during tomography, on the z-axis.

Pitch

The pitch of a helical scan refers to the ratio of the table translating distance per gantry rotation to the thickness of the individual x-ray beam. Let p and s be the helical pitch and table translating distance per gantry rotation, respectively. In the single slice CT, the x-ray beam thickness is determined by the beam collimation setting. The helical pitch for single slice CT is defined in equation 3-1. [2]

Figure Single Slice Helical CT z-Sampling Pattern

The figure above gives help in visualizing the table translation distance per gantry rotation s, in the however for a multi-slice CT is quite different and will be later described in Section 3.3.

Parallel-Beam CT

The Parallel Beam CT is the simplest and the earliest implementation of the Computed Axial Tomography. The term CAT-Scans was used in the early days of this technology because it took scans with axial rather than helical rotations.

Machine Geometry

The parallel beam of X-rays is translated in a linear motion across z-axis to obtain a projection profile. The couple is then rotated about the patient isocenter by approximately 1 degree, and another projection profile is obtained. This translate-rotate scanning motion is repeated until the source and detector have been rotated by 180 degrees. [3]

The highly collimated beam provides excellent rejection of radiation scattered in the patient; however, the complex scanning motion results in long scan times. This geometry was used by Hounsfield in his original experiments [Hounsfield, 1980] but is not used in modern scanners.

Figure Geometry of Parallel Beam CT

Radon Transformation

When the data is acquired from the CT-scan machine it is not an image. Instead it is an attenuation map called 'sinogram' which shows the amount of attenuation on every rotation angle.

The sinogram actually is the Radon Transformation of the original image therefore we have to calculate the inverse transformation of the retrieved raw data. Equation (4 - 1) is the formulation of radon transform.

However, since the inverse of the transformation is not arithmetically possible. We make us of iterative and recursive algorithms to solve this inverse problem.

Following is an image of a sinogram of a Shepp-Logan Phantom.

Description: C:\Users\Tariq Bashir\Desktop\sin.jpg

Figure Parallel-Beam Sinogram of a Shepp-Logan Phantom

Fan-Beam CT

The technique of CT evolved and instead of using a source which released parallel X-rays we opted to use the one which minimized the dosage of X-rays and still give better results.

Therefore, a new technique was opted to do so. A point source was used for this type of CTs and a flat array of detectors was placed opposite to it. Since the rays were released in more of a fan like structure the name opted for this type of CT was the Fan-beam.

Flat-Bed

A modern tomograph does not create parallel beams but a set of rays diverging from one and the same point, the X-ray source. See Figure 4. The source moves in the (x,y)-plane along a circle of radius R . At projection angle β it is positioned at the coordinate (−D cos β, −D sin β ).

Figure Geometry of Fan Beam flat-bed-

Arc Bed

Faster scans required the elimination of translation motion and the use of smoother and simpler pure rotational motion. This goal is accomplished by widening the x-ray beam into a fan beam encompassing the entire patient width and using an array of detectors to intercept the beam. This design, characterized by linked tube-detector arrays undergoing only rotational motion, is known as third-generation geometry.

The early Arc-bed CT scanners, installed in late 1975, could scan in less than 5 s; current designs can scan as quickly as one third of a second for cardiac applications.

The arc-bed fan-beams were a better option than the flat-bed fan-beam CTs. The reasons are obvious as they could be: [9]

Dose Efficiency

Scatter

Spatial Resolution

Certain characteristics that the 3rd Generation Arc-bed Fan-beam scanners had, which helped it give better results than the previous generations of CT-machines were:

Wide x-ray fan beam encompassing patient

Wide detector array to encompass fan beam

Rotation motion only (no translation)

Figure 5, shows us the geometry of the 3rd generation machines.

Figure Fan-beam CT with Arc-bed

Multi-Slice Helical CBCT

CT examination times were dominated by inter-scan delays. After each 360° rotation, cables connecting rotating components (x-ray tube and, if third generation, detectors) to the rest of the gantry required that rotation stop and reverse direction. Cables were spooled onto a drum, released during rotation, and then respooled during reversal. Scanning, braking, and reversal required at least 8-10 s, of which only 1-2 were spent acquiring data. The result was poor temporal resolution (for dynamic contrast enhancement studies) and long procedure times.

Eliminating inter-scan delays required continuous (nonstop) rotation, a capability made possible by the low-voltage slip ring. A slip ring passes electrical power to the rotating components (e.g., x-ray tube and detectors) without fixed connections. The idea is similar to that used by bumper cars; power is passed to the cars through a metal brush that slides along a conductive ceiling. Similarly, a slip ring is a drum or annulus with grooves along which electrical contactor brushes slide. Data are transmitted from detectors via various high-capacity wireless technologies, thus allowing continuous rotation to occur. A slip ring allows the complete elimination of inter-scan delays, except for the time required to move the table to the next slice position. However, the scan-move-scan sequence (known as axial step-and-shoot CT) is still somewhat inefficient. For example, if scanning and moving the table each take 1 s, only 50% of the time is spent acquiring data. Furthermore, rapid table movements may introduce "tissue jiggle" motion artifacts into the images.

Machine Geometry

The data acquisition geometry for helical CT is similar to that of spiral CT: the source/detector assembly rotates about a rail along which the patient bed moves at constant speed (head to toe).

Although detector surfaces might be constructed from multiple curved detector rows, in this paper we assume the area detector is flat. We also consider that the field of view is an imaginary cylinder of radius r, centered on the scanner axis ( z -axis). To simplify the exposition, the z -axis will now be referred to as the vertical direction. Relative to the patient, the detector and x-ray source rotate together in a helical fashion with the center of the detector at the same z -position as the source.

The source to z -axis distance is R>r, the pitch of the helix is P, and the source to detector distance is D. We use to represent the CB measurements where (u,v) refers to detector locations with the v -axis parallel to the rotation axis z . The parameter is an angle which indicates the position of the source. When increases by , the source moves vertically upwards by P .

Figure Machine Geometry for Helical CB CT

Methodology and Results

The chapter discusses in details the methodology that has been used for the reconstruction of the CT-Images. The chapter moves step by step following the generations of the machine from PBCT to Multi-Slice Helical CBCT.

Parallel Beam CT

The first and earliest of the CT-Machines was the Parallel beam CT. Thereby for the complete construction of a system we need to go through the basics of concepts and thus this topic will enable us to understands the basics of our system.

Reconstruction Algorithm

This method utilized the advantages of Fourier slice theorem and it implies that slice of the two dimensional Fourier transforms of the image correspond to the Fourier transforms of one dimensional projections which are used to reconstruct the image.

Four things that led to this method:

Direct back-projection results in blurred image

Could filter (de-convolve) resulting 2-D image

Linear systems theory suggests order of operations unimportant

Fourier Slice Theorem

The attenuation co-efficient of X-Rays in computed tomography is calculated by the well-known formula given in

With taking the natural logarithm of the attenuation, a line integral value is obtained

Figure The Fourier Slice Theorem. Left: A parallel beam projection p of object f(x,y). Right: The one-dimensional Fourier transform of the projection is found along a radial line of the two-dimensional Fourier transform of the object.

Hence, X-ray measurements may be considered as line-integration values after the simple manipulation in (4 - 3).

Consider the line L belongs to a set of parallel lines constituting a projection as shown in Figure 7. At projection angle and detector position t, the equation (4 - 3) can be written as equation (4 - 2).

The projection values can be drawn in a Cartesian space. Edholm and Jacobson [7] have named this space a 'sinogram'. The reason for such name is because each point in the object f(x,y) contributes to sinusoidal curve in this space.

The Fourier slice theorem states that the values of the one-dimensional Fourier transform of a parallel projection is found along a radial line in the two-dimensional Fourier transform of the object as shown in Figure 7.

Filtering Techniques

The name 'Filtered Backprojection', suggests that the technique implies usage of filtration. As described earlier, different filters can be applied to the projections formed for the image. You can see in the table given below that the resultant image formed without the filtering is highly erroneous.

The filtration techniques make the image a lot better than the one that is made without filtering. This can be seen by the fact that the SSIM index in all the techniques applied is approx. 1 and the RMSError is also quite low. However, another problem still remains present i.e. Image Quality. The image is quite perceivable yet for the purpose of medical imaging it would be better if it can be improved more. The filtration technique at the moment gives 0.2 out of 1 in the Quality Index.

Hamming

Hann

Blackman

Kaiser Filter

Ramp ( Ram - Lak)

Tukey FilterDescription: C:\Users\Tariq Bashir\Desktop\untitled.jpg

Figure Plot of applied Filters

These filters have been applied by applying different windows to the ramp filter. However, there are other filters that are used for this purpose.

Table Result of Filtering Techniques

Filter name

SSIM Index [5]

RMSE

Quality Index [6]

PSNR

Ram - Lak

1.0000

0.0021

0.2070

74.8588

Tukey

0.9999

0.0034

0.1967

72.7472

Kaiser

1.0000

0.0022

0.2061

74.5572

Blackman

1.0000

0.0030

0.2009

73.3717

Hann

0.9999

0.0034

0.1967

72.7472

Hamming

0.9998

0.0042

0.1914

71.8944

None

0.0072

2.6681e+003

5.3263e-004

13.9027

Images were reconstructed for 512x512 pixels phantom and Nearest-Neighbor interpolation based on projections from 1o to 180o on increment of 1o.

SSIM = Structural SIMilarity (SSIM) index

RMSE = Root Mean Square Error

PSNR = Peak Signal-to-Noise Ratio

Figure 9 shows the sinogram in figure 3 after application of the Ram-Lak filter.

Description: C:\Users\Tariq Bashir\Desktop\sinogram.jpg

Figure Parallel-Beam Sinogram with Ram-Lak Filter

Back-Projection

After the application of filter on the sinogram, the image can be reconstructed in the spatial domain using back-projection algorithm.

A single profile is back projected to give a dark stripe across the entire image plane Figure 10(a). As we scan the phantom from many directions and back project the ray profiles onto the image plane Figure 10(b), an image of the radio-dense dot, albeit a poor one, begins to resolve Figure 10(c). As the number of projections increases, the quality improves but some blurring will always remain in the image.

C:\Users\Tariq Bashir\Desktop\NM19_3.gif

Figure Backprojection in Computed Tomography

Interpolation Techniques

Following are a few techniques used for interpolation during the Back Projection algorithm. These techniques are already implemented in MATLAB reconstruction function.

Table Result of Interpolation Techniques

Technique name

SSIM Index [5]

RMSE

Quality Index [6]

PSNR

Nearest

1.0000

0.0021

0.2070

74.8588

Linear

1.0000

0.0019

0.2076

75.3235

Spline

1.0000

0.0016

0.2098

76.2390

pchip / Cubic

1.0000

0.0017

0.2090

75.8992

Cubic spline

1.0000

0.0016

0.2093

76.0255

Images were reconstructed for 512x512 pixels phantom and Ram-Lak filter based on projections from 1o to 180o on increment of 1o.

SSIM = Structural SIMilarity (SSIM) index

RMSE = Root Mean Square Error

PSNR = Peak Signal-to-Noise Ratio

Following is an image reconstructed after successful back-projection and complete interpolation. Since, the algorithm is very efficient there is no loss of data and an image of a high PSNR and low RMSE is reconstructed.

Description: C:\Users\Tariq Bashir\Desktop\im.jpg

Figure Reconstructed Shepp-Logan Phantom from Parallel-beam Sinogram using Spline Interpolation

Fan-Beam CT

This topic discusses the reconstruction of both flat and arc detector bed type CT machines.

Flat-bed Geometry

The method for reconstruction of data for machines with flat-bed geometry is discussed in detail in this topic.

Reconstruction Algorithm - FBP

The image above shows us to characteristics of the fan-beam CT. The equation (4 - 1) can be re-written as [8]

Where, U is the weighting factor to be applied

Figure Zoom of part of sinusoid showing the points calculated according to equation (4 -1)

When the discretized radon transform given in equation (4 - 1) is used the nearest pixel is selected near the resulting co-ordinate. Figure 12 shows the gaps between the selected points in a reconstructed image.

Interpolation

To improve image quality, we use the interpolation methods so that we can recover any loss of pixel data that is not being provided by enough projections. The procedure to obtain a linear interpolation is to compute a weighted combination of the two adjacent pixels according to proximity of them to the sinusoid. Figure 13 is an illustration of the adjacent pixels incorporated while using interpolation techniques. [8]

Backprojection is then given by:

Figure Zoom of part of sinusoid showing the points calculated using interpolation techniques

Weighting Factors

The weights u and v are calculated by

Reconstructing a Phantom Image for a Flat-Bed Scanner

The geometry defined for generating the phantom:

Table Machine geometry defined for generating phantom for flat-bet fan-beam CT

Description

Value

Radial Samples (detector array)

280

Angular Samples (rotation)

400

Ray Spacing (detector spacing)

4 mm

Orbit

360 degrees

Following is the geometry defined for Image acquisition:

Description

Value

Horizontal Resolution

256 pixels

Vertical Resolution

252 pixels

Pixel size

2 mm

Given is the phantom that has been generated using ellipsoids for the given image geometry.

Figure The Phantom Image Used, x

The following is the sinogram generated for the above given phantom image generated.

Figure Sinogram for Phantom, sino

Figure Reconstructed Image using FBP, recon is the image that has been reconstructed using the algorithm based on equation (5 - 2).

Figure Reconstructed Image using FBP, recon

Figure Error, xtrue - recon is the error that has occurred due to deficiencies in the current technique. The noise outside the body is permissible but the noise inside the circle effects the judgment of a radiologist or at least makes the work difficult to do so.

Figure Error, xtrue - recon

Figure Plot for the true and reconstructed image vertically and horizontally through the center shows us the plot for intensity levels of two lines passing through the center of the body. There are two plots in each image in the following figure. We can see that there are noise fluctuations in the following images which are expected to be corrected in the next generation.

Figure Plot for the true and reconstructed image vertically and horizontally through the center

Figure 19 shows the images reconstructed using the Ram-Lak, Hanning and Hamming filters and the results are displayed in the same order in Table 4. The circular patterns that exist outside the phantom body are called artifacts. These ones are particularly called the Ring Artifacts.

Please keep in mind that the image has been manipulated for the printing purposes by adjusting brightness and contrast.

recon range: [-0.796588 19.0358]

1

128

1

126

Figure Reconstructed Images using Different Filters

Table 4 shows the quality measurements for the reconstructed images with the original phantom image.

Table Noise for Flat-bed fan beam CT

Filter name

SSIM Index [5]

RMSE

PSNR

Ram - Lak

0.9997

0.0191

65.3657

Hann

0.9997

0.0198

65.1907

Hann50

0.9978

0.1368

56.8048

Hann75

0.9993

0.0490

61.2633

Hann80

0.9994

0.0405

62.0953

Hamming

0.9997

0.0190

65.3670

We can observe, in the result given in the table, how the filtration acts on the images. The images with high artifacts having high objective measures: For example, the Ram-Lak. However, the images with low artifacts are shown to have low objective measures while having a better subjective quality, which is actually as important as or even more important than the objective quality.

Following is a set of images reconstructed by varying the images reconstructed by the same method but with varying number of projections (40, 80, 120…600). The effect of this can be seen clearly in figure 20.

Figure Image Reconstructed by Varying No. of Angular Samples

The graph in Figure 21 is based upon the images in figure 20. The graph is plotted for the Peak Signal to Noise Ratio for each image. It is visible that with the increase in projections there is an exponential increase in the PSNR value of the image.

The reason for this is that we are not using any prediction or a method for recovery of data that is lost because of low-sampling rate. So, when the projections are increased the quality and the efficiency automatically increase.

The graph in Figure 22 is also based on the images in Figure 20 and is plotted for the Root Mean Square error. We can deduce that the error sharply reduces with the increase in the projection angles.

Figure PSNR for images with angular samples on X-axis

Figure RMSE for images with angular samples on X-axis

Arc-Bed Geometry

The method for reconstruction of data for machines with arc-bed geometry is discussed in detail in this topic.

Reconstruction Algorithm - FBP

Since the geometry of both flat-bed and arc-bed CT's is almost similar when looking at them with the sense of reconstruction. The algorithms for both are almost similar. The only thing that differs now is that the detector array now has a focal point and that now instead of measuring the rays equidistantly we have to sample those using equiangular detectors.

Therefore, to reconstruct the image we have to use different weighting factors.

We can use the same equation (4 - 1) and modify the value of U.

Where, β is the angle of projection and;

R is the source offset value

Reconstructing Phantom Image for an Arc-bed Scanner

The geometry defined for generating the phantom:

Table Machine geometry defined for generating phantom for arc-bet fan-beam CT

Description

Value

Radial Samples (detector array)

888

Angular Samples (rotation)

984

Ray Spacing (detector spacing)

1 mm

Orbit

360 degrees

Following is the geometry defined for Image acquisition:

Description

Value

Horizontal Resolution

512 pixels

Vertical Resolution

504 pixels

Pixel size

1 mm

Given is the phantom that has been generated using ellipsoids for the given image geometry.

Figure The Phantom Image Used, x

The following is the sinogram generated for the above given phantom image generated.

Figure Sinogram for Phantom, sino

Figure 22 is the image that has been reconstructed using the algorithm based on equation (6 - 1).

Figure Reconstructed Image using FBP, recon

Figure 23 is the error that shows the difference between true and reconstructed image. As stated before, the noise outside the body is permissible but the noise inside the circle effects the judgment of a radiologist or at least makes the work difficult to do so. However, we can see that unlike the previous reconstruction algorithm, this one has significantly reduced the noise inside the borders of the image.

Figure Error, xtrue - recon

Figure 24 shows us the plot for intensity levels of two lines passing through the center of the body. There are two plots in each image in the following figure. One thing is evident that the plots completely overlap each other in the positions inside the body however there is a huge noise outside the body which affects the calculated numbers.

Figure Plot for the true and reconstructed image vertically and horizontally through the center

Figure 25 shows the images reconstructed using the Ram-Lak, Hanning and Hamming filters and the results are displayed in the same order in Table 6. As in the previous case, t

Please keep in mind that the image has been manipulated for the printing purposes by adjusting brightness and contrast.

Figure Reconstructed Images using different Filters

Table 6 describes the quality factors in the Figure 28 us about the noise in the image.

Table Noise for Arc-bed fan beam CT

Filter name

SSIM Index [5]

RMSE

PSNR

Ram - Lak

0.9996

0.0210

64.9415

Hann

0.9996

0.0298

63.4277

Hann50

0.9975

0.1527

56.3274

Hann75

0.0622

0.9990

60.2282

Hann80

0.9992

0.0528

60.9377

Hamming

0.9996

0.0210

64.9426

Just as the previous example, we can observe how the filtration acts on the images. The images with high artifacts having high objective measures: and the images with low artifacts are shown to have low objective measures while having a better subjective quality.

Following is a set of images reconstructed by varying the images reconstructed by the same method but with varying number of projections (40, 80, 120…600). The effect of this can be seen clearly in figure 20.

Please keep in mind that the image has been manipulated for the printing purposes by adjusting brightness and contrast.

Figure Image Reconstructed by Varying No. of Angular Samples

The graph in Figure 21 is based upon the images in figure 20. The graph is plotted for the Peak Signal to Noise Ratio for each image. It is visible that with the increase in projections there is an exponential increase in the PSNR value of the image.

The reason for this is that we are not using any prediction or a method for recovery of data that is lost because of low-sampling rate. So, when the projections are increased the quality and the efficiency automatically increase.

The graph in Figure 22 is also based on the images in Figure 20 and is plotted for the Root Mean Square error. We can deduce that the error sharply reduces with the increase in the projection angles.

Figure PSNR for images with angular samples on X-axis

Figure RMSE for images with angular samples on X-axis

Multi-Slice Cone-Beam Helical CT

Reconstruction Algorithm - SSRB

Conceptually, the method converts the CB data set to a stack of fan-beam sinograms , each associated with one horizontal (trans-axial) z -slice of the ROI. Once the fan-beam sinograms are built, reconstruction is performed for each z -slice using 2D filtered backprojection (FBP).

Overview of Technique

The CB-SSRB algorithm consists of two steps:

Fix the z -sampling of the 3D image, and determine, for each CB source point, the multi-fan that will be estimated from the projection . The z -slices of the multi-fan are within the distance of the source point.

Process each CB projection sequentially as follows:

For each relevant z -slice, calculate one complete fan-beam projection using equations (7 - 1) and (7 - 2) with linear interpolation in.

For each fan-beam projection of step

Multiply the ray-sums by the weight of equation (7 - 3) with .

Apply the classical fan-beam filtering and backprojection steps to the single weighted projection available for each z -slice from step (b).

Because each fan-beam projection lies on exactly one multi-fan, the z -slices will be

correctly reconstructed after all CB projections are processed. Full use of the CB data will not be achieved unless the spacing between the z -slices is smaller than R=D times the thickness of the detector rows.

Rebinning

The fan-beam geometry is defined to match the CB geometry, so for each z -slice, the (virtual) fan-source lies at a distance R from the z - axis and the linear fan-beam detector array is aligned along the rows of the area detector at the distance D from the fan-source. [10]

The rebinning step only involves the vertical direction. Each fan-beam ray-sum is estimated from a single oblique (CB) ray-sum located in the vertical plane containing the fan-beam ray. The CB source for this oblique ray-sum is the nearest source location lying directly above or below the fan-source. Define M as the mid-point of the intersection of the fan-line with the ROI. The oblique ray selected to approximate the fan-beam ray is the one passing through M. Mathematically, the rebinning equation is

Where,

A direct implementation of the CB-SSRB method would be to build and process all fan-beam sinograms after CB data collection is completed. Our approach involves reordering operations so that each CB projection is processed sequentially. This reordering is based on the observation that each CB projection is used to build a stack of complete fan-beam projections, called a multi-fan projection. Each multi-fan projection occupies the z -slices within the distance d of the CB source.

Parker Weighting

The short-scan case in fan-beam computed tomography requires the introduction of a weighting function to handle redundant data. Parker introduced such a weighting function for a scan over π plus the opening angle of the fan. In this article we derive a general class of weighting functions for arbitrary scan angles between π plus fan angle and 2π (over-scan).

These weighting functions lead to mathematically exact reconstructions in the continuous case. Parker weights are a special case of a weighting function that belongs to this class. It will be shown that Parker weights are not generally the best choice in terms of noise reduction, especially when there is considerable overscan. We derive a new weighting function that has a value of 0.5 for most of the redundant data and is smooth at the boundaries. [11]

… (7 - 3)

Image Reconstruction using FBP

After the virtual short-scans have been weighted, we can reconstruct the sinograms with the help of FBP algorithm mentioned in Chapter 6.2.

The geometry defined for generating the phantom:

Table Machine geometry defined for generating phantom for Helical CBCT

Description

Value

Vertical Samples

32

Angular Samples

3625

Distance from Focal Point to Center

0

Pitch

.53125

Initial Z-Position

-50.5494

Total Orbit

1326.2195

Initial Orbit Position

109.12139

Table Image geometry defined for generating phantom for Helical CBCT

Description

Value

Vertical Samples

32

Angular Samples

3625

Distance from Focal Point to Center

0

Pitch

.53125

Initial Z-Position

-50.5494

Total Orbit

1326.2195

Initial Orbit Position

109.12139

Figure The Phantom Image Used, x

Figure Sinogram for Phantom, sino

Figure 29 is the phantom that has been generated using ellipsoids for the given image geometry and the projections from the phantom have been generated in figure 30 using the attributes given in table 7.

Figure 31 is the set of sinograms that has been constructed using the algorithm based on equation (7 - 1).

Figure Sinogram constructed using SSRB, sino

Figure 32 depicts the projections and their parts that have been used for rebinning the projections to the sinograms in Figure 31.

Figure Projections used in SSRB. Black for unused projections and White for used

Sinogram after application of Parker weighting becomes like the following image

Figure Sinograms after applying Parker Weight

The images that are reconstructed from sinograms above are shown in fig. 31

Figure Reconstructed Image using FBP, xssrb

Figure 32 is the error that shows the difference between true and reconstructed image. As stated before, the noise outside the body is permissible but the noise inside the circle effects the judgment of a radiologist or at least makes the work difficult to do so. However, we can see that unlike the previous reconstruction algorithm, this one has significantly reduced the noise inside the borders of the image.

The following image is mapped from [-500 : 500] grayscale levels. The white and black spots show error but we can see that it is really minute and while creating an objective measure of the image we can clearly see that such errors exist only at the edges of the body and are really unimportant.

Figure Error, xtrue - xssrb

Figure 33 shows us the plot for intensity levels of a line passing on the z-axis and through the body.

Figure Intensity levels at the z axis of the phantom object. Blue plot shows the intensity in xtrue. Red plot show the intensity in xssrb.

One of the few problems with SSRB is that it uses Parker Weighting. Since Parker weighting doesn't depend upon the Orbit starts of the short-scans, it is therefore affected by the sampling rate. Figure 40 show a few performance measures based on the changing of pitch from 0.3 to 1.3.

Figure Square Points represent Mean Absolute Error (MAE). Circles represent Peak Signal to Noise Ratio (PSNR).

It is evident from the graph plotted above that the better values lie between 0.4 and 0.9. At points less than 0.4 the oversampling causes the weights to unbalance and remove non-redundant data as well.

Whereas, for points greater than 0.9 the under sampling affect comes into action. Therefore, the quality also drops there. To see a better quality measure of the images, we have plotted the SSIM graph on the same criterion. The graph is shown in Figure 41.

Figure Structure Similarity Index Measurement (SSIM)

Here in the graph we can clearly see that points between 0.4 and 0.6 have better SSIM than those on the rest of the graph.

Conclusion and Future Work

Conclusion

We've seen that the loss of data increases as we move to later generations. The reason is that the raw data retrieved from the CT-machine is very complex. In order to minimize, the loss of data we make use of more complex algorithms.

At first, we just used the inverse radon transform using BP. Although that reconstructed the image, we had a lot of blurring in the image. So in order to remove the effect of blurring, we used filtering techniques. Filters are 1-D multipliers to the sinograms of the image and are applied across the x' for each angle of projection.

Since point-source fan and cone beam machines are used now and with the advent of short-scan techniques (, there are certain parts of image that exist more in the sinogram than the others. To remove that, we make use of weighting factors. We assign more weight to less redundant data and less weight to more redundant data. This way the reconstructed image doesn't have imbalance of intensity or details.

The earlier methods were time consuming and more than 50% of the time was taken in setting the scanner and detector ring rather than image acquisition. So, a new geometry was proposed with helical orbits rather than axial. So, instead of getting a simple raw data we got data of tilted images to be reconstructed. There was less redundancy but more loss of uncorrelated data. The data retrieved in the projections was overlapping, especially in the case of CBCT where we use multi-detector arrays. The use of techniques such as 'Rebinning' helps us reduce the projections to simple sinograms which can then be reconstructed by using techniques for fan-beam geometry such as FBP and SART.

Future Work

The future of this project is the implementation of more and better reconstruction techniques. There is even a possibility to reconstruct an image with multiple techniques and apply the process of super-resolution to enhance the spatial and intensity resolution of the image.

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