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This research investigates the use of charge coupled device (abbreviated as CCD) linear image sensors in an optical tomographic instrumentation system used for sizing particles. The measurement system, consisting of four CCD linear image sensors are configured around an octagonal shaped flow pipe for a four projections system is explained. The four linear image sensors provide 2,048 pixel imaging with a pixel size of 14 micron × 14 micron, hence constituting a high-resolution system. Image reconstruction for a four-projection optical tomography system is also discussed, where a simple optical model is used to relate attenuation due to variations in optical density, [^{−1}[

Industrial processes are often controlled using process measurements at one or more points. The amount of information contained in such measurements is often minimal, and in some cases (multiphase flow) there are no adequate sensors [

There are several modalities used in process tomography such as electrical (impedance, capacitance, inductance), radiation (optical, x-ray, positron electromagnetic (PET), magnetic resonance) and acoustic (ultrasonic) [

Optical techniques are desirable because of their inherent safety (the transducer does not require direct physical contact with the measurand), high efficiency [

Particle sizing is very important for many industrial processes and has led to much research. Typical problems relate to pulverised coal for combustion and liquid fuels, spray characterisations, analysis and control of particulate emissions, industrial process control, manufacture of metallic powders and the production of pharmaceuticals [

The majority of these techniques are off-line, with direct optical providing the only truly on-line measurement [

Horbury

Mathematical modelling is an important tool in simulating a system. It can predict the output of a system with known conditions. In addition, the model enables the user to understand the output trends or behaviour. Three types of mathematical model are developed in this project for investigating the effects due to particles, the effects due to light sources and the effects due to diffraction on the optical tomography system. The complete optical tomography system consists of a lighting system, the measurement section, the sensor system, the data acquisition system, the PIC microcontroller system and image information.

A well-collimated beam of light is passed through the measurement section, past the object to be measured. For test and calibration purposes, aground bar of known diameter is used. The detection system is the sensing component in the optical tomography system. The standard object (ground bar of known diameter) is sensed using a charge coupled device (CCD) linear image sensor. The output of the CCD linear image sensor is then acquired by the data acquisition system.

Image reconstruction is a process of generating an image from raw data, or a set of unprocessed measurements, made by the imaging system. In general, there is a well-defined mathematical relationship between the distribution of physical properties in an object and the measurements made by the imaging system. Image reconstruction is the process which inverts this mathematical process to generate an image from the set of measurements [

In an optical imaging system, the object density along the optical path (according to Beer-Lambert Law) exponentially attenuates the light intensity:

The optical tomography system consists of four projection systems where each projection is generated by a ray-box which has a laser diode, objective lens, and an spherical lens in it and a CCD linear image sensor at the other side of the pipe. The CCD linear image sensor used in the system has 2,048 effective pixels with a pixel size of 14 micron by 14 micron.

As a result, the tomographic image consists of four projections, with each providing 2,048 measurements. The complete forward problem requires 2,048 × 2,048 values of α (attenuation coefficient) to model the system. The image reconstruction process for the system is modelled using smaller

To explain the forward problem simply the object space is considered to consist of a 7 × 7 array of cells projected onto a circular measurement cross-section, later this is extended to the 2,048 × 2,048 array. The analysis is for four projections (

A simple optical attenuation model is used to model the measurement system. The model uses an array of octagonal shaped cells with square cells to ensure that the light traverses through all cells normally (

The dimension of each side of the octagonal cell is calculated as follows (refer to

The actual measurement of the octagonal and square cells is shown in

The optical attenuation model (based on the combination of octagonal-square shaped cells) is used in the forward problem. These cells are used in the formation of the sensitivity matrix for the image reconstruction process. However, the reconstructed image from the inverse problem has square image pixels due to the standard format of a matrix. The pixel size is 0.014 mm by 0.014 mm for the actual reconstructed image system (2,048 × 2,048 array). The arrangement of cells is shown in

Several techniques have been used in optical tomography to produce images from measured data, such as layergram back-projection (LYGBP) by Ibrahim [

Xie [

In this project, the particle may be translucent or opaque and is a solid surrounded by water. The particle used in the modelling analysis is considered to have a linear attenuation coefficient of 10 mm^{−1}. To explain the forward problem process, the area to be imaged in the pipe is divided into 49 cells (a 7 × 7 array of cells) for simplicity. Later in this section the model is extended to a 2,048 × 2,048 array of cells. The reconstructed image of the particle in water is based on four projections. The arrangement of cells is shown in

From

The linear attenuation of the light is modelled by assuming that each cell has an attenuation coefficient of α_{ij} where i and j represent the row and column, respectively. The length of the octagon pixel is 0.014 mm whilst the length of the square pixel is 0.006 mm. The change in light intensity measured by the sensors may be written as _{out})_{n}_{n}

Based on the Beer-Lambert Law of absorption (

Similarly, for the vertical equations (projection 2) for each sensor (

Projection 3 (

Projection 4 (

These equations may be written in matrix form. The matrices shown above can be represented in a form of matrix [

A similar process is performed on the actual tomography system with 2,048 horizontal and 2,048 vertical cell (2,048 × 2,048 array). A simplified diagram of the cell attenuation coefficients is shown in

The equations for each of the projections for the 2,048 × 2,048 array (

_{0,0} a + α_{0,1} b + α_{0,2} a + ……………..α_{0,2045} a + α_{0,2046} b + α_{0,2047} a = ln (I/I_{out,0})

_{1,0} a + α_{1,1} b + α_{1,2} a + ……………..α_{1,2045} a + α_{1,2046} b + α_{1,2047} a = ln (I/I_{out,1})

and so forth until:

_{2047,0} a + α_{2047,1} b + α_{2047,2} a + …….α_{2047,2045} a + α_{2047,2046} b + α_{2047,2047} a = ln (I/I_{out,0})

Similarly for the vertical equations (

_{0,0} a + α_{1,0} b + α_{2,0} a + ……………..α_{2045,0} a + α_{2046,0} b + α_{2047,0} a = ln (I/I_{out,2048})

_{0,1} a + α_{1,1} b + α_{2,1} a + ……………..α_{2045,1} a + α_{2046,1} b + α_{2047,1} a = ln (I/I_{out,2049})

and so forth until:

_{0,2047} a + α_{1,2047} b + α_{2,2047} a + …….α_{2045,2047} a + α_{2046,2047} b + α_{2047,2047} a = ln (I/I_{out,4095})

_{0,2047} a = ln (I/I_{out,4096})

(α_{0,2045} + α_{1,2046} + α_{2,2047})_{out,4097})

and continue until:

α_{2047,0} _{out,6143})

_{2047,2047} a = ln (I/I_{out,6144})

_{2047,2045} + α_{2046,2046} + α_{2045,2047})a = ln (I/I_{out,6145})

and the same process is repeated until _{0,0}_{out,8191})

The above expressions can be expressed in matrix form,

In practice, measurements are made and then used to estimate values of the linear attenuation coefficients. Equation (5.34) needs to be re-arranged in order to reconstruct the tomographic image:

The main problems associated with the above equation are that the matrix [

The values of matrix [

The following example shows the 2 × 2 array of numbers used to investigate the relationship between inverse, transpose, and pseudo-inverse of a matrix:

The arguments show that pseudo-inverse in this case is equivalent to inversion of the matrix. In this case of matrix-inversion, the pseudo-inverse correctly represents the inverse matrix. However, the transpose does not give the correct answer. The transpose and pseudo inverse transforms are now used to generate images of several different models. The transpose is widely used in many tomographic researches due to its popularity, fast operation and simplicity (in terms of having the required dimensional form) [

_{1} to M_{9}) as the test cells (nine cells of octagonal and square shapes). However, there is no inverse matrix of S because the matrix is too sparse.

^{−1}, whilst for water it is 0.00287 mm^{−1}.

The reconstructed images using the pseudo-inverse matrix are found to be better in representing the actual value of the linear attenuations of water and the particle, compared to the images using the transpose matrix. This is expected, as the pseudo-inverse matrix is closer to the matrix inversion when compared to the transpose matrix. Hence, the contrast is better in the pseudo-inverse image compared to the transpose image. The pseudo inverse image shows no indication of

In this section, three different projections are discussed—two, three and four projections. It is seen that the reconstructed image quality improves as the number of projections is increased.

The image with four projections is less smeared, especially when the image is reconstructed using the pseudo-inverse matrix, compared to the one with two projections. Moreover, the effect of

In the actual system, the effective cells to represent an object vary between approximately 7 cells (for a 100 micron particle) up to 71 cells (for a 1 mm particle). Since the optical tomography system is intended for small diameter particles, only a few of the whole 2,048 pixels are used. Hence, the reconstruction may be simplified by only imaging a small area of the measurement section.

A 21 × 21 array of pixels actually represents 0.294 × 0.294 mm^{2} image area with an image pixel size of 0.014 mm by 0.014 mm. A 100 micron object with a round shape in water is modelled using the forward problem. The object is positioned in the middle of the measurement section, where the attenuation coefficients of water fill image pixels beyond this 0.294 × 0.294 mm^{2} image area.

In the actual system, the effective number of sensors per projection used to generate the image ranges from approximately 7 pixels (for a 100 micron particle) up to 71 pixels (for a 1 mm particle). Since the optical tomography system is intended for measuring small diameter particles, only about 5% (101 pixel/2,048 pixel × 100) of the whole 2,048 pixels per projection are used. Hence, a smaller array of pixels is used so that the reconstruction process is less complex (due to the smaller size of the matrices involved) and faster. In the actual system a 101 × 101 array of pixels is used for the reconstruction of the tomographic images ^{2} area of image.

The full reconstruction operation requires a 2,048 × 2,048 array of pixels, which requires a lot of memory space (approximately 275 Gigabytes). The actual size of the matrix (four projections) = [4n by n^{2}] where n is the number of sensors in each CCD linear image sensor. Therefore, the required memory size is the multiplication of the actual matrix size with 8 bytes ^{3}. This result is only based on the amount of memory needed for the sensitivity matrix (S matrix) with four projections. The actual calculation would need more memory and hence the whole operation is just not possible on a PC.

The limitation of the computer memory limits the reconstructed image area to a 101 × 101 array of pixels (1,414 × 1,414 mm^{2}). In order to reconstruct a larger object, a scaling process is done where several pixels are grouped together before averaging them. For example, averaging five CCD pixels to give a single measurement value which represents a spatial length of 0.07-mm. If the 101 × 101 array of pixels is used combined with this averaging, the image area is increased to 7.07 × 7.07 mm^{2}.

A novel arrangement of cells with a combination of square-shaped cells and octagon-shaped cells makes the image reconstruction process possible for the proposed CCD-based optical tomography system. In this case, the square-shaped cell has two projections whereas the octagon-shaped cell has four projections of measurements. The tomographic images reconstructed using the transpose matrix is found to be producing qualitative images whilst the image reconstructed using the pseudo inverse matrix contains the quantitative information of the image. The main reason is that the pseudo inverse matrix represents the closest generalisation of matrix inversion. Moreover, the pseudo inverse minimises the error between the calculated values and the real values. On the other hand, the transpose calculates a value, which is often used as a basis for iterative image improvement. The transpose is much quicker to calculate than the pseudo inverse. Comparative times for a 101 × 101 pixel image are: 20 seconds (transpose) and 1 hour (pseudo inverse).

A block diagram of particle sizing techniques.

A block diagram of the optical tomography system.

Four projections on 3 × 3 cells.

An octagonal shaped cell.

Actual measurements of octagonal-square shaped cells.

Transformation of octagonal-square cells into square image pixels.

7 × 7 array of pixels with four projections.

2,048 by 2,048 array of pixels with four projections.

A 3 × 3 array of cells with three projections.

One particle in water.

Two-particles in water.

Two-particles image with two projections.

Two-particles image with four projections.

A 100 micron reconstructed image using pseudo inverse.

A 100 micron reconstructed image using transpose.

A graph of