Abstract:
A method for developing a secondary collimator is described. The method includes orienting a plurality of collimator elements in a plane such that a gap is defined between a first collimator element and a second collimator element. The first collimator element has a first curved end, and the first curved end faces the second collimator element across the gap.

Description:
BACKGROUND OF THE INVENTION 
       [0001]    This invention relates generally to imaging systems and more particularly to systems and methods for developing a secondary collimator. 
         [0002]    The events of Sep. 11, 2001 instigated an urgency for more effective and stringent screening of airport baggage. The urgency for security expanded from an inspection of carry-on bags for knives and guns to a complete inspection of checked bags for a range of hazards with particular emphasis upon concealed explosives. X-ray imaging is a widespread technology currently employed for screening. However, existing x-ray baggage scanners, including computed tomography (CT) systems, designed for detection of explosive and illegal substances are unable to discriminate between harmless materials in certain ranges of density and threat materials like plastic explosive. 
         [0003]    A plurality of identification systems based on a plurality of x-ray diffraction (XRD) techniques provide an improved discrimination of materials compared to that provided by the x-ray baggage scanners. The XRD identification systems measure a plurality of d-spacings between a plurality of lattice planes of micro-crystals in materials. However, a signal-to-noise ratio provided by the XRD identification systems is difficult to improve. 
       BRIEF DESCRIPTION OF THE INVENTION 
       [0004]    In one aspect, a method for developing a secondary collimator is described. The method includes developing the secondary collimator having a first collimator element. The first collimator element has a first curved end. 
         [0005]    In another aspect, a processor configured to develop a secondary collimator having a collimator element is described. The collimator element has a curved end. 
         [0006]    In yet another aspect, an imaging system is described. The imaging system includes a source configured to generate energy, a detector configured to detect the energy, and a collimator placed between an object and the detector. The collimator includes a collimator element having a curved end. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0007]      FIG. 1  is an isometric view of an embodiment of a gantry implementing a secondary collimator. 
           [0008]      FIG. 2  is a block diagram of an embodiment of a system for generating a diffraction profile of a substance. 
           [0009]      FIG. 3  is an isometric view of an embodiment of a virtual secondary collimator used to develop the secondary collimator of  FIG. 1 . 
           [0010]      FIG. 4  is an isometric view of an embodiment of the secondary collimator of  FIG. 1 . 
           [0011]      FIG. 5  is a top view of an embodiment of the secondary collimator of  FIG. 4 . 
           [0012]      FIG. 6  is a side view of an embodiment of a system for implementing the secondary collimator of  FIG. 4 . 
           [0013]      FIG. 7  is an isometric view of an embodiment of a system including a plurality of collimator elements that can be implemented within the system of  FIG. 6 . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0014]      FIG. 1  is an isometric view of a block diagram of an embodiment of a gantry  12 . Gantry  12  includes a primary collimator  14 , a transmission detector  17 , a scatter detector  18 , and a secondary collimator  76 . Scatter detector  18  is a segmented semiconductor detector. Gantry  12  has a side wall  19  and another side wall  23  facing side wall  19 . 
         [0015]    Transmission detector  17  includes a plurality of detector elements, such as detector elements  20  and  21 . Scatter detector  18  includes a plurality of detector cells or detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  for detecting coherent scatter. Scatter detector  18  includes any number, such as, ranging from and including 5 to 1200, of detector elements. For example, scatter detector  18  includes a number, such as ranging from and including 5 to 40, of detector elements in a z-direction parallel to a z-axis, and a number, such as ranging from and including 1 to 30 detector elements in a y-direction parallel to a y-axis. An x-axis, the y-axis, and the z-axis are located within an xyz co-ordinate system. The x-axis is perpendicular to the y-axis and the z-axis, and the y-axis is perpendicular to the z-axis, and the x-axis is parallel to an x-direction. X-ray sources, of gantry  12 , including x-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72 , and transmission detector  17  form an inverse single-pass multi-focus imaging system. X-ray sources, of gantry  12 , including x-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 ,  72 , and  74 , have an inverse fan-beam geometry that includes a symmetric location of the x-ray sources relative to the z-axis. 
         [0016]    Scatter detector  18  and transmission detector  17  are located in the same yz plane. The yz plane is formed by the y-axis and the z-axis. Scatter detector  18  is separate from transmission detector  17  by a shortest distance ranging from and including 30 mm to 60 mm in the z-direction. 
         [0017]    Gantry  12  further includes x-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72 . X-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72  are located parallel to and coincident with an arc  75 . It is noted that in an alternative embodiment, gantry  12  includes a higher number, such as 10 or 20, or alternatively a lower number, such as 4 or 6, of x-ray sources than that shown in  FIG. 1 . A centroid of transmission detector  17  is located at a center of circle having arc  75 . Each x-ray source  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72  is an x-ray source that includes a cathode and an anode. Alternatively, each x-ray source  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72  is an x-ray source that includes a cathode and all x-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72  share a common anode. 
         [0018]    A container  79  is placed on a support  80  between x-ray sources  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72 , and scatter detectors  16  and  18 . Container  79  and support  80  are located within an opening  65  of gantry  12 . Examples of container  79  include a bag, a box, and an air cargo container. Examples of each x-ray source  60 ,  62 ,  64 ,  66 ,  68 ,  70 , and  72  include a polychromatic x-ray source. Container  79  includes a substance  82 . Examples of substance  82  include an organic explosive, an amorphous substance having a crystallinity of less than twenty five percent, a quasi-amorphous substance having a crystallinity at least equal to twenty-five percent and less than fifty percent, and a partially crystalline substance having a crystallinity at least equal to fifty percent and less than one-hundred percent. Examples of the amorphous, quasi-amorphous, and partially crystalline substances include a gel explosive, a slurry explosive, an explosive including ammonium nitrate, and a special nuclear material. Examples of the special nuclear material include plutonium and uranium. Examples of support  80  include a table and a conveyor belt. An example of scatter detector  18  includes a segmented detector fabricated from Germanium. 
         [0019]    X-ray source  72  emits an x-ray beam  84  in an energy range, which is dependent on a voltage applied by a power source to x-ray source  72 . Primary collimator  14  outputs a primary beam  86 , such as a pencil beam, upon collimating x-ray beam  84  from x-ray source  72 . Primary beam  86  is incident on a point  85  of substance  82  within container  79  arranged on support  80  to generate scattered radiation including a scattered beam  88 . Scattered beam  88  forms a scatter angle value  89  with respect to primary beam  86 . Secondary collimator  76  is located between support  80  and scatter detector  18 . 
         [0020]    Secondary collimator  76  includes a number of collimator elements, such as sheets, plates, or laminations. The collimator elements of scatter detector  18  are made of a secondary collimator material, which is a radiation-absorbing material, such as, steel, copper, silver, or tungsten. Secondary collimator  76  collimates a portion of the scattered radiation to output the remaining portion of the scattered radiation and the remaining portion includes scattered beam  88 . 
         [0021]    Underneath support  80 , there is arranged transmission detector  17 , which measures an intensity of primary beam  83 . Moreover, underneath support  80 , there is arranged scatter detector  18  that measures photon energies of the remaining portion of the scattered radiation received by scatter detector  18 . Scatter detector  18  measures the x-ray photons within the remaining portion of the scattered radiation in an energy-sensitive manner by outputting a plurality of electrical output signals linearly dependent on a plurality of energies of the x-ray photons detected from within the remaining portion of the scattered radiation. Scatter detector  18  detects the remaining portion, including scattered beam  88 , of the scattered radiation output from secondary collimator  76  and scatter detector  18  detects the remaining portion to output a plurality of electrical signals. 
         [0022]    In an alternative embodiment, gantry  12  includes a second scatter detector other than scatter detector  18 . The second scatter detector is placed in the same yz plane as that of scatter detector  18 . The second scatter detector is placed on a side, with respect to the y-axis, of transmission detector  17  that is the same as a side, with respect to the y-axis, of placement of scatter detector  18 . Moreover, the second scatter detector is the same as scatter detector  18  and a distance of the second scatter detector from a center axis  101  is the same as a distance of scatter detector  108  from center axis  101 . For example, the second scatter detector has the same number of detector elements as that of scatter detector  18 . Additionally, the second scatter detector is placed on a side, with respect to center axis  101 , opposite to a side, with respect to center axis  101 , of placement of scatter detector  18 . In an alternative embodiment, gantry  12  includes additional scatter detectors other than scatter detector  18  and other than the second scatter detector. The additional scatter detectors are located on a side, with respect to the y-axis and transmission detector  17 , opposite to a side, with respect to the y-axis and transmission detector  17 , of location of scatter detector  18  and the second scatter detector. The additional scatter detectors are placed in the same yz plane as that of scatter detector  18 . Each of the additional scatter detectors have the same number of detector elements as that of scatter detector  18 . In yet another alternative embodiment, gantry  12  includes any number of scatter detectors that are placed in the same yz plane as that of scatter detector  18 . 
         [0023]      FIG. 2  is diagram of an embodiment of a system  100  for generating a diffraction profile of a substance. System  100  includes detector element  20  of transmission detector  17 , detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  of scatter detector  18 , a plurality of pulse-height shaper amplifiers (PHSA)  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118 , a plurality of analog-to-digital (A-to-D) converters  120 ,  122 ,  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136 , a plurality of spectrum memory circuits (SMCs)  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  allowing pulse height spectra to be acquired, a plurality of correction devices (CDs)  156 ,  158 ,  160 ,  162 ,  164 ,  166 ,  168 , and  170 , a processor  190 , an input device  192 , a display device  194 , and a memory device  195 . As used herein, the term processor is not limited to just those integrated circuits referred to in the art as a processor, but broadly refers to a computer, a microcontroller, a microcomputer, a programmable logic controller, an application specific integrated circuit, and any other programmable circuit. The computer may include a device, such as, a floppy disk drive or CD-ROM drive, for reading data including the methods for developing a secondary collimator from a computer-readable medium, such as a floppy disk, a compact disc-read only memory (CD-ROM), a magneto-optical disk (MOD), or a digital versatile disc (DVD). In another embodiment, processor  190  executes instructions stored in firmware. Examples of display device  194  include a liquid crystal display (LCD) and a cathode ray tube (CRT). Examples of input device  192  include a mouse and a keyboard. Examples of memory device  195  include a random access memory (RAM) and a read-only memory (ROM). An example of each of correction devices  156 ,  158 ,  160 ,  162 ,  164 ,  166 ,  168 , and  170  include a divider circuit. Each of spectrum memory circuits  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  include an adder and a memory device, such as a RAM or a ROM. 
         [0024]    Detector element  20  is coupled to analog-to-digital converter  120 , and detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  are coupled to pulse-height shaper amplifiers  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118 , respectively. Detector element  20  generates an electrical output signal  196  by detecting primary beam  86  and detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  generate a plurality of electrical output signals  198 ,  200 ,  202 ,  204 ,  206 ,  208 ,  210 , and  212  by detecting scattered radiation within the remaining portion. For example, detector element  22  generates electrical output signal  198  for each scattered x-ray photon incident on detector element  22 . Each pulse-height shaper amplifier amplifies an electrical output signal received from a detector element. For example, pulse-height shaper amplifier  104  amplifies electrical output signal  198  and pulse-height shaper amplifier  106  amplifies electrical output signal  200 . Pulse-height shaper amplifiers  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118  have a gain factor determined by processor  190 . 
         [0025]    An amplitude of an electrical output signal output from a detector element is proportional to an energy of an x-ray quantum that is detected by the detector element to generate the electrical output signal. For example, an amplitude of electrical output signal  196  is proportional to an energy of an x-ray quantum in primary beam  86  detected by detector element  20 . As another example, an amplitude of electrical output signal  198  is proportional to an energy of an x-ray quantum within scattered radiation that is detected by detector element  22 . 
         [0026]    A pulse-height shaper amplifier generates an amplified output signal by amplifying an electrical output signal generated from a detector element. For example, pulse-height shaper amplifier  104  generates an amplified output signal  216  by amplifying electrical output signal  198  and pulse-height shaper amplifier  106  generates an amplified output signal  218  by amplifying electrical output signal  200 . Similarly, a plurality of amplified output signals  220 ,  222 ,  224 ,  226 ,  228 , and  230  are generated. An analog-to-digital converter converts an output signal from an analog form to a digital form to generate a digital output signal. For example, analog-to-digital converter  120  converts electrical output signal  196  from an analog form to a digital format to generate a digital output signal  232  and analog-to-digital converter  122  converts amplified output signal  216  from an analog form to a digital format to generate a digital output signal  234 . Similarly, a plurality of digital output signals  236 ,  238 ,  240 ,  242 ,  244 ,  246 , and  248  are generated by analog-to-digital converters  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136 , respectively. A digital value of a digital output signal generated by an analog-to-digital converter represents an amplitude of energy of a pulse of an amplified output signal. For example, a digital value of digital output signal  234  output by analog-to-digital converter  122  is a value of an amplitude of a pulse of amplified output signal  216 . Each pulse is generated by an x-ray quantum, such as an x-ray photon. 
         [0027]    An adder of a spectrum memory circuit adds a number of pulses in a digital output signal. For example, when analog-to-digital converter  122  converts a pulse of amplified output signal  216  into digital output signal  234  to determine an amplitude of the pulse of amplified output signal  216 , an adder within spectrum memory circuit  140  increments, by one, a value within a memory device of spectrum memory circuit  140 . Accordingly, at an end of an x-ray examination of substance  82 , a memory device within a spectrum memory circuit stores a number of x-ray quanta detected by a detector element. For example, a memory device within spectrum memory circuit  142  stores a number of x-ray photons detected by detector element  24  and each of the x-ray photons has an amplitude of energy or alternatively an amplitude of intensity that is determined by analog-to-digital converter  124 . 
         [0028]    A correction device receives a number of x-ray quanta that have a range of energies and are stored within a memory device of one of spectrum memory circuits  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154 , and divides the number by a number of x-ray quanta having the range of energies received from a memory device of spectrum memory circuit  138 . For example, correction device  156  receives a number of x-ray photons having a range of energies from a memory device of spectrum memory circuit  140 , and divides the number by a number of x-ray photons having the range received from a memory device of spectrum memory circuit  138 . Each correction device outputs a correction output signal that represents a range of energies within x-ray quanta received by a detector element. For example, correction device  156  outputs a correction output signal  280  representing an energy spectrum or alternatively an intensity spectrum within x-ray quanta detected by detector element  22 . As another example, correction device  158  outputs correction output signal  282  representing an energy spectrum within x-ray quanta detector element  24 . Similarly, a plurality of correction output signals  284 ,  286 ,  288 ,  290 ,  292 , and  294  are generated by correction devices  160 ,  162 ,  164 ,  166 ,  168 , and  170 , respectively. 
         [0029]    It is noted that a number of pulse-height shaper amplifiers  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118  changes with a number of detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  of scatter detector  18 . For example, five pulse-height shaper amplifiers are used for amplifying signals received from five detector elements of scatter detector  18 . As another example, four pulse-height shaper amplifiers are used for amplifying signals received from four detector elements of scatter detector  18 . Similarly, a number of analog-to-digital converters  120 ,  122 ,  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136  changes with a number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  and a number of spectrum memory circuits  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  changes with the number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36 . 
         [0030]    Processor  190  receives a plurality of correction output signals, including correction output signals  280 ,  282 ,  284 ,  286 ,  288 ,  290 ,  292 , and  294 , to generate a momentum transfer x, measured in inverse nanometers (nm −1 ), from an energy spectrum r(E) of energy E of x-ray quanta within the remaining portion, including scattered beam  88 , detected by scatter detector  18  ( FIG. 1 ). Processor  190  generates the momentum transfer x by applying 
         [0000]        x =( E/hc )sin(θ/2)   (1) 
         [0031]    where c is a speed of light, h is Planck&#39;s constant, θ represents a scatter angle variable of x-ray quanta of the remaining portion detected by scatter detector  18  ( FIG. 1 ). Scatter angle value  89  is an example of the scatter angle variable θ. Processor  190  relates the energy E to the momentum transfer x by equation (1). Processor  190  receives the scatter angle variable θ from a user, such as a human being, via input device  192 . Processor  190  generates a diffraction profile of substance  82  ( FIG. 1 ) by calculating a number of scatter x-ray photons that are detected by scatter detector  18  and by plotting the number versus the momentum transfer x. 
         [0032]      FIG. 3  is an isometric view of an embodiment of a virtual secondary collimator  300 . Virtual secondary collimator  300  includes a plurality of virtual collimator elements  302 ,  304 , and  306 . Processor  190  generates a virtual primary beam  308 , which is a virtual representation of primary beam  86 . Processor  86  generates a point  310  which is a virtual representation of point  85 . Processor  86  further generates a plurality of virtual scattered beams including virtual scattered beams  312 ,  314 , and  316 . In an alternative embodiment, processor  190  generates any number, such as 2, 4, 10, 20, or 30, of virtual scattered beams. Processor  190  generates a virtual scattered beam forming a virtual scatter angle value with respect to virtual primary beam  308 . For example, processor generates virtual scattered beams  312  forming a virtual scatter angle value  318  with respect to virtual primary beam  308 , generates virtual scattered beam  314  forming a virtual scatter angle value  320  with respect to virtual primary beam  308 , and generates virtual scattered beam  316  forming a virtual scatter angle value  322  with respect to virtual primary beam  308 . 
         [0033]    Processor  190  generates a plurality of virtual scattered beams so that a modulus of a ratio of a first term including a difference between a first virtual scatter angle value formed by a first one of the virtual scattered beams with respect to virtual primary beam  308  and a second virtual scatter angle value formed by a second one of the virtual scattered beams with respect to virtual primary beam  308  and a second term including the first virtual scatter angle value is constant. For example, processor  190  generates virtual scattered beams  312 ,  314 , and  316  so that a modulus of a ratio of a term including a difference between virtual scatter angle value  320  and virtual scatter angle value  318  and another term including virtual scatter angle value  318  is equal to a modulus of a ratio of a term including a difference between virtual scatter angle value  322  and virtual scatter angle value  320  and another term including virtual scatter angle value  320 . The example is represented mathematically as 
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         [0034]    Processor  190  determines a plurality of intersection points of intersection between virtual scattered beams  312 ,  314 , and  316  and a y v z v  plane formed between a y v  axis and a z v  axis perpendicular to y v  axis. For example, processor  190  determines that virtual scattered beam  312  intersects the y v z v  plane at an intersection point  324 , virtual scattered beam  314  intersects the y v z v  plane at an intersection point  326 , and virtual scattered beam  316  intersects the y v z v  plane at an intersection point  328 . The z v  axis is perpendicular to an x v  axis that is perpendicular to the y v  axis. The intersection points  324 ,  326 , and  328  lie within the same y v z v  plane. For example, intersection point  324  lies in the same y v z v  plane as that of location of intersection point  326 . 
         [0035]    Processor  190  generates a number of the intersection points between a first edge intersection point  330  and a second edge intersection point  332  and a shortest distance between first edge intersection point  330  and second edge intersection point  332  is proportional, by a first factor, such as one-half or one-third, to a distance between side walls  19  and  23 . The shortest distance between first edge intersection point  330  and second edge intersection point  332  is a distance that is the shortest among a plurality of distances between first edge intersection point  330  and second edge intersection point  332 . The user provides the number of intersection points to be generated to processor  190  via input device  192 . The user also inputs, via input device  192 , the distance between side walls  19  and  23  to processor  190 . 
         [0036]    Processor  190  generates a curve  334  connecting the intersection points and curve  334  extends from first edge intersection point  330  to second edge intersection point  332 . Processor  190  generates a collimator element axis  336 , generates a curve  338  that is a mirror image of curve  334  and that lies in the same y v z v  plane as curve  334 , and curves  334  and  338  are symmetrical with respect to collimator element axis  336 . For example, processor  190  generates curve  338  and a distance from a point  340  on curve  334  to collimator element axis  336  is equal to a distance from a point  342  on curve  338  to collimator element axis  336 . Processor  190  generates curve  338  so that a distance, along or parallel to the z v  axis, between an edge  344  of curve  338  and first edge intersection point  330  is equal to a distance provided to processor  190  by the user via input device  192 . The distance, along the z v  axis, between edge  344  of curve  338  and first edge intersection point  330  is shortest among a plurality of distances, along the z v  axis, between curves  334  and  338 . 
         [0037]    Processor  190  generates a surface  346 , in the y v z v  plane of curves  334  and  338 , between curves  334  and  338 . Processor  190  generates surface  346  to be symmetrical with respect to collimator element axis  336 . Processor  190  generates virtual collimator element  302  having a thickness, along or parallel to the x v  axis, and virtual collimator element  302  is symmetrical with respect to a centroid of virtual collimator element  302 . For example, processor  190  generates virtual collimator element  302  and a distance between the centroid of virtual collimator element  302  and a point  350  is equal to a distance between the centroid and a point  352 , and points  350  and  352  are located on the same x v  axis. Moreover, point  350  is located on curve  338  and point  352  is located on a curve  359 . Curves  338  and  359  are edges of virtual collimator element  302 . The thickness of virtual collimator element  302  is provided by the user to processor  190  via input device  192 . Processor  190  generates virtual collimator element  302  to have an end  358  located parallel to the z v  axis and another end  360  located parallel to the z v  axis. Virtual collimator element  302  also has a curved end  362  having a thickness along the x v  axis and another curved end  364  having the thickness along the x v  axis. Curves  338  and  359  are edges of curved end  362 . Curved end  364  is a mirror image of curved end  362 . 
         [0038]    Processor  190  generates any number, such as 2, 4, 5, 6, 10, 20, or 30, of virtual collimator elements that lie in the same y v z v  plane as that of virtual collimator element  302  and each of the virtual collimator elements has the same dimensions as that of virtual collimator element  302 . For example, processor  190  generates virtual collimator element  304  that lies in the same y v z v  plane as that of virtual collimator element  302  and generates virtual collimator element  306  that lies in the same y v z v  plane as that of virtual collimator element  302 , and each virtual collimator element  304  and  306  is of the same size as virtual collimator element  302 . Processor  190  generates virtual collimator element  304  having a curved end  366  having a thickness along the x v  axis and another curved end  368  having the thickness along the x v  axis. Each virtual collimator element  302 ,  304 , and  306  has the same uniform thickness, along the x v  axis, and the thickness is input to processor  190  by the user via input device  194 . 
         [0039]    Processor  190  generates a virtual collimator element having an end that lies in the same x v z v  plane, formed between the x v  axis and the z v  axis, as that of end  358  of virtual collimator element  302 . For example, processor  190  generates virtual collimator element  304  having an end  370  lying in the same x v z v  plane as that of end  358  and generates virtual collimator element  306  having an end  372  lying in the same x v z v  plane as that of end  358 . Moreover, processor  190  generates a virtual collimator element having an end that lies in the same x v z v  plane as that of end  360  of virtual collimator element  302 . For example, processor  190  generates virtual collimator element  304  having an end  374  lying in the same x v z v  plane as that of end  360  and generates virtual collimator element  306  having an end  376  lying in the same x v z v  plane as that of end  360 . End  358  does not face end  360 , end  370  does not face end  374 , and end  372  does not face end  376 . 
         [0040]    Processor  190  generates virtual collimator element  304  including a point  354 , which is at a shortest distance, along the z v  axis, from a point  356  of virtual collimator element  302  and generates virtual collimator element  306  including a point  378  at the shortest distance, along the z v  axis, from a point  380  of virtual collimator element  304 . The shortest distance, along the z v  axis, between points of any two adjacent virtual collimator elements is provided by the user to processor  190  via input device  192 . For example, the user operates a keyboard to provide the shortest distance, along the z v  axis, between points  354  and  356  of virtual collimator elements  302  and  304 . The shortest distance, along the z v  axis, between points of two adjacent virtual collimator elements is a distance that is shortest among all distances, along the z v  axis, between the adjacent virtual collimator elements. For example, the shortest distance, along the z v  axis, between virtual collimator elements  302  and  304  is a distance, along the z v  axis, between point  354  of virtual collimator element  304  and point  356  of virtual collimator element  302 . Points  354 ,  356 ,  378 , and  380  are located on an axis  382  that passes through a plurality of centers of a plurality of surfaces  346 ,  384 , and  386  of virtual collimator elements  302 ,  304 , and  306 . Points  356  is located on curve  334 , point  354  is located on an edge of surface  384  and the edge of surface  384  faces an edge of location of point  356  on surface  346 , and point  378  is located on an edge of surface  386  and the edge of surface  386  faces an edge of location of point  380  on surface  384 . Surfaces  346 ,  384 , and  386  are located in the same y v z v  plane. For example, surface  346  is located in the same y v z v  plane as that of surface  384 . 
         [0041]    Processor  190  generates a first virtual collimator element separated by a first virtual opening or spacing or slit from a second virtual collimator element and a size of the first virtual opening is the same as a size of a second virtual opening formed between the second virtual collimator element and a third virtual collimator element. For example, a virtual opening  388  formed between virtual collimator elements  302  and  304  has the same size as that of a virtual opening  390  between virtual collimator elements  304  and  306 . As another example, a distance, along the z v  axis, between ends  358  and  370  is equal to a distance, along the z v  axis, between ends  370  and  374 . Each virtual opening  388  and  390  does not have a constant width as viewed along or parallel to each of the y v  and z v  axes. A width of each virtual opening  388  and  390  is measured along the z v  axis. 
         [0042]      FIG. 4  is an isometric view of an embodiment of a secondary collimator  400 . Secondary collimator  400  includes a plurality of collimator elements  402 ,  404 , and  406  that are located in the same yz plane. Secondary collimator  400  is an example of secondary collimator  76 . Virtual secondary collimator  300  is a virtual representation of secondary collimator  400 . Moreover, virtual collimator element  302  is a ritual representation of collimator  402 , virtual collimator  304  is a virtual representation of collimator element  404 , and virtual collimator element  306  is a ritual representation of collimator element  406 . In an alternative embodiment, secondary collimator  400  includes any number, such as, 2, 4, 5, 6, or 10, of collimator elements lying in the same yz plane. 
         [0043]    The user fabricates secondary collimator  400  to be proportional, by a second factor, such as 2 or 3, to virtual secondary collimator  300 . For example, the user fabricates collimator  402  to have a size that is twice a size of virtual collimator element  302 . As another example, the user fabricates collimator element  404  to have a size that is twice a size of virtual collimator element  304 . As yet another example, the user fabricates collimator element  406  to have a size that is twice a size of virtual collimator element  306 . As still another example, the user fabricates collimator element  402  having a curved end  408  and collimator element  404  having a curved end  410  and a shortest distance between curved ends  408  and  410  is proportional, by the second factor, to a shortest distance between curved ends  364  and  366  of virtual collimator elements  302  and  304 . The shortest distance between curved ends  408  and  410  is a distance that is the shortest among a plurality of distances between curved ends  408  and  410 . Similarly, the shortest distance between curved ends  364  and  366  is a distance that is the shortest among a plurality of distances between curved ends  364  and  366 . Curved end  364  is a virtual representation of curved end  408  and curved end  366  is a virtual representation of curved end  410 . Moreover, collimator element  402  includes another curved end  412  that is a mirror image of curved end  408 . Curved end  362  is a virtual representation of curved end  412 . 
         [0044]    The user fabricates each collimator element of secondary collimator  400  from the secondary collimator material. The user fabricates a collimator element of secondary collimator  400  by using a machining device, such as, a molding machine or a circular rotating diamond saw. For example, the user obtains a block of the secondary collimator material, and cuts, by using the circular rotating diamond saw, each of collimator elements  402 ,  404 , and  406  of a size proportional, by the second factor, to a size of respective virtual collimator elements  302 ,  304 , and  306 . For example, the user obtains a block of the secondary collimator material, and cuts, by using the circular rotating diamond saw, collimator element  402  of a size proportional, by the second factor, to a size of virtual collimator element  302 . As another example, the user pours a liquid form of the secondary collimator material in the molding machine having a size proportional, by the second factor, to a size of virtual collimator element  304 , and cools the secondary collimator material to fabricate collimator element  404 . As yet another example, the user uses the circular rotating diamond saw to fabricate collimator element  406  having a size proportional, by the second factor, to a size of virtual collimator element  306 . The user can measure dimensions of a collimator element by using a measuring tape and determine whether the dimensions are proportional, by the second factor, to the dimensions of a virtual collimator element. Each collimator element, lying in the same yz plane, has the same dimensions. For example, collimator element  402  lies in the yz plane of location of collimator elements  404  and  406  and is of the same size as that of collimator elements  404  and  406 . 
         [0045]    The user attaches, such as glues, welds, or bolts, an end of collimator element to side wall  23 . For example, the user glues an end  414  of collimator element  402  to side wall  23 . As another example, the user welds an end  416  of collimator element  404  to side wall  23  and the user bolts an end  418  of collimator element  406  to side wall  23 . In an alternative embodiment, the user attaches an end of a collimator element to side wall  19  instead of side wall  23 . For example, the user welds an end  420  of collimator element  402  to side wall  19 . As another example, the user glues an end  422  of collimator element  404  to side wall  19  and bolts and end  424  of collimator element  406  to side wall  19 . In yet another alternative embodiment, the user attaches a collimator element to side walls  19  and  23 . For example, the user welds collimator element  402  to side walls  19  and  23 . End  358  is a virtual representation of end  414 , end  370  is a virtual representation of end  416 , and end  372  is a virtual representation of end  418 . Moreover, end  360  is a virtual representation of end  420 , end  374  is a virtual representation of end  422 , and end  376  is a virtual representation of end  424 . Collimator elements  402 ,  404 , and  406  are located in the same yz plane. 
         [0046]    When primary beam  86  is incident on point  85 , a plurality of scattered beams  426 ,  428 , and  430  are output. Scattered beam  426  forms a scatter angle value  432  with respect to primary beam  86 , scattered beam  428  forms a scatter angle value  434  with respect to primary beam  86 , and scattered  430  beam forms a scatter angle value  436  with respect to primary beam  86 . Each of scatter angle values  432 ,  434 , and  436  are values of the scatter angle variable θ. Scattered beam  426  is incident on a point  438  located on collimator element  402 , scattered  428  beam is incident on a point  440  located on collimator element  402 , and scattered  430  beam is incident on a point  442  located on collimator element  402 . Intersection point  324  is a virtual representation of point  438 , intersection point  326  is a virtual representation of point  440 , and intersection point  328  is a virtual representation of point  442 . 
         [0047]    Collimator element  402  is fabricated so that a modulus of a ratio of a third term including a difference between a first scatter angle value formed by a first scattered beam, incident on a curved end of collimator element  402 , with respect to primary beam  86  and a second scatter angle value formed by a second scattered beam, incident on the curved end, with respect to primary beam  86  and a fourth term including the first scatter angle value is constant. For example, collimator element  402  is fabricated so that a modulus of a ratio of a difference between a term including scatter angle value  434  and scatter angle value  432  and another term including scatter angle value  432  is equal to a modulus of a ratio of a difference between a term including scatter angle value  436  and scatter angle value  434  and another term including scatter angle value  434 . The example is represented mathematically as 
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         [0048]    Virtual scatter angle value  318  is a virtual representation of scatter angle value  432 , virtual scatter angle value  320  is a virtual representation of scatter angle value  434 , and virtual scatter angle value  322  is a virtual representation of scatter angle value  436 . Ends  414 ,  416 , and  418  of collimator elements  402 ,  404 , and  406  lie in the same xz plane formed by the x and z axes. For example, end  414  lies in the same xz plane as that of end  416 . Moreover, ends  420 ,  422 , and  424  of collimator elements  402 ,  404 , and  406  lie in the same xz plane formed by the x and y axes. For example, end  420  lies in the same xz plane as that of end  422 . 
         [0049]    A first collimator element, adjacent to a second collimator element, is spaced apart from the second collimator element via a first opening or spacing or slit that has the same dimensions as that of a second opening between the second collimator element and a third collimator element adjacent to the second collimator element. For example, an opening  444  between collimator elements  402  and  404  has the same size as that of an opening  446  between collimator elements  404  and  406 . As another example, a distance, along or parallel to the z-axis, between ends  414  and  416  is equal to a distance, along the z-axis between ends  416  and  418 . As another example, a distance, along the z-axis, between ends  420  and  422  is equal to a distance, along the z-axis between ends  422  and  424 . Each of openings  444  and  446  does not have a constant width as viewed along or parallel to the y-axis. For example, a width, measured along the z-axis and between centroids of collimator elements  402  and  404 , of opening  444  is different than a width, measured along the z-axis, between ends  414  and  416 . A width of each opening  444  and  446  is measured along the z-axis. Virtual opening  388  is a virtual representation of opening  444  and a size of opening  444  is proportional, by the second factor, to a size of virtual opening  388 . For example, a distance between ends  414  and  416  is proportional, by the second factor, to a distance between ends  358  and  370 . As another example, a distance between ends  420  and  422  is proportional, by the second factor, to a distance between ends  360  and  374 . Moreover, virtual opening  390  is a virtual representation of opening  446  and a size of opening  446  is proportional, by the second factor, to a size of virtual opening  390 . For example, a distance between ends  416  and  418  is proportional, by the second factor, to a distance between ends  370  and  372 . As another example, a distance between ends  422  and  424  is proportional, by the second factor, to a distance between ends  374  and  376 . Each collimator element  402 ,  404 , and  406  has the same uniform thickness, that is proportional by the second factor, to a thickness of any of virtual collimator elements  302 ,  304 , and  306 . 
         [0050]      FIG. 5  is a top view of an embodiment of secondary collimator  400  including collimator elements  402  and  404 . As an example, a length, along the y-axis, of each collimator element  402 ,  404 , and  406  ranges from and including 90 millimeters (mm) to 110 mm, a longest width, along the z-axis, of each collimator element  402 ,  404 , and  406  ranges from and including 6 mm to 8 mm, and a thickness, along or parallel to the x-axis, of each collimator element  402 ,  404 , and  406  ranges from and including 0.5 mm to 3 mm. The longest width, along the z-axis, of a collimator element is a width, along the z-axis, that is longest among a plurality of widths, along the z-axis, of the collimator element. Opening  444  lies between collimator elements  402  and  404 . An example of a length of opening  444 , along the y-axis, is a length that is the same as a length of each collimator element  402  and  404 . An example of a shortest width, along the z-axis, of opening  444  ranges from and including 0.5 mm to 1 mm, a thickness, along the x-axis, of opening  444  is the same as a thickness of each collimator element  402  and  404 . The shortest width, along the z-axis, of opening  444  is a width that is the shortest among a plurality of widths of opening  444 . 
         [0051]      FIG. 6  is a side view of an embodiment of system  600  for implementing a secondary collimator. System  600  includes including a gantry  602 , which is an example of gantry  12 . Gantry  602  includes a secondary collimator  604 , which is an example of secondary collimator  76 . Secondary collimator  604  includes a collimator layer  606  of collimator elements  402 ,  404 , and  406 , a collimator element  608 , and a collimator element  610 . Secondary collimator  604  further includes a plurality of collimator layers  612  and  614  of collimator elements including collimator elements  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628 . In an alternative embodiment, secondary collimator  604  includes any number, such as 1, 2, 4, 5, or 10, of collimator layers. Collimator elements within each collimator layer lie within the same yz plane. For example, collimator elements  616 ,  618 , and  626  within collimator layer  612  lies within a first yz plane. As another example, collimator elements  620 ,  622 ,  624 , and  628  within collimator layer  614  lies within a second yz plane parallel to the first yz plane. Collimator elements within a collimator layer have the same size. For example, each of collimator elements  608  and  610  have the same size as any of collimator elements  402 ,  404 , and  406  within collimator layer  606 . Moreover, collimator elements of collimator layers other than collimator layer  606  closest to scatter detector  18  have the same size. For example, each of collimator elements  616 ,  618 , and  626  of collimator layer  612  has the same size as any of collimator elements  620 ,  622 ,  624 , and  628  of collimator layer  614 . Collimator layer  606  is closest, along the x-axis, to scatter detector  18  than collimator layers  612  and  614 . A distance between collimator layer  606  and scatter detector  18  is shortest among any other distances from scatter detector  18  to any other collimator layer of secondary collimator  604 . For example, a distance measured, parallel to an xz plane, between scatter detector  18  and collimator layer  606  is shorter than a distance, measured parallel to the xz plane, between scatter detector  18  and collimator layer  612 . 
         [0052]    Collimator layers  606 ,  612 , and  614  are parallel to each other. For example, collimator layer  606  is parallel to collimator layer  612 . The user places collimator layers  606 ,  608 , and  610  parallel to each other by using a laser pointer. In an alternative embodiment, secondary collimator  604  does not include collimator layers other than collimator layer  606 . For example, secondary collimator  604  does not include collimator layers  612  and  614 . Each collimator element of collimator layers  606 ,  612 , and  614  of secondary collimator  604  is attached to side wall  19  and/or side wall  23 . For example, collimator elements  402 ,  404 ,  406 ,  608 ,  610 ,  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628  are welded to side wall  19 . As another example, collimator elements  402 ,  404 ,  406 ,  608 ,  610 ,  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628  are welded to side wall  23 . As yet another example, collimator elements  402 ,  404 ,  406 ,  608 ,  610 ,  616 ,  618 ,  620 ,  622 ,  624 ,  626 , and  628  are glued to side walls  19  and  23 . Collimator layer  612  is displaced, along the z-axis, by a distance, such as ranging from and including 3 mm to 4 mm, relative to collimator layer  606 , and collimator layer  614  is displaced, along the z-axis, by a distance, such as ranging from and including 3 mm to 4 mm, relative to collimator layer  612 . For example, collimator element  616  is displaced, along the z-axis, by 3.5 mm relative to collimator element  402 . As another example, collimator element  622  is displaced, along the z-axis, by 3.7 mm relative to collimator element  616 . 
         [0053]    A distance, along the x-axis, between a first set of two adjacent collimator layers is the same as a distance, along the x-axis, between a second set of two adjacent collimator layers. For example, a distance, along the x-axis, between collimator layers  606  and  612  is equal to a distance, along the x-axis, between collimator layers  612  and  614 . As another example, a distance, along the x-axis, between collimator layers  606  and  612  ranges from and including 10 mm to 15 mm, and the distance is the same as a distance, along the x-axis, between collimator layers  612  and  614 . In an alternative embodiment, a distance, along the x-axis, between the first set of two adjacent collimator layers is not the same as a distance, along the x-axis, between the second set of two adjacent collimator layers. 
         [0054]    Each collimator layer  606 ,  612 , and  614  includes any number, such as 3, 4, 6, or 10, greater than two collimator elements lying in the same yz plane. For example, collimator layer  606  includes five collimator elements lying in the same yz plane. An opening is formed between any two adjacent collimator elements of a collimator layer. For example, an opening  630  is formed between collimator elements  406  and  608 , which are adjacent to each other and an opening  632  is formed between collimator elements  608  and  610 , which are adjacent to each other. Each of openings  630  and  632  has the same size as that of any of openings  444  and  446 . Moreover, an opening is formed between any two adjacent collimator elements of any of collimator layers  612  and  614 . For example, an opening  634  is formed between collimator elements  622  and  624 , adjacent to each other and opening  634  is formed between collimator elements  618  and  626  adjacent to each other. Openings formed between adjacent collimator elements of a collimator layer have the same size. For example, opening  634  between collimator elements  622  and  624  have the same size as opening  634  between collimator elements  624  and  628 . Moreover, openings formed between adjacent collimator elements of collimator layer  606  have the same size. For example, opening  632  between collimator elements  608  and  610  have the same size as opening  446  between collimator elements  404  and  406 . 
         [0055]    When primary beam  86  intersects container  79 , scattered beam  88 , a scattered beam  636 , and a scattered beam  638 , and a scattered beam  640  are output from container  79 . A portion, extending between collimator layers  606  and  614 , of each of scattered beams  88 ,  636 ,  638 , and  640  form the same scatter angle value  89  with respect to primary beam  86 . For example, portions, extending between collimator layers  606  and  614 , of scattered beams  88 ,  636 ,  638 , and  640  are parallel to each other. A remaining portion, shown as a dark portion, between collimator layer  606  and scatter detector  18  does not form a constant scatter angle with respect to primary beam  86 . Each scattered beam  88 ,  636 ,  638 , and  640  passes through at least one opening of secondary collimator  604 . For example, scattered beam  88  passes through openings  634  of collimator layers  612  and  614 , and through opening  446  of collimator layer  606 . 
         [0056]      FIG. 7  is an isometric view of an embodiment of a system  700  including a plurality of collimator elements  702 ,  704 , and  706 . Collimator elements  702 ,  704 , and  706  are collimator elements of any of collimator layers  612  and  614 . For example, collimator element  702  is an example of collimator element  622  of collimator layer  614 , collimator element  704  is an example of collimator element  624 , and collimator element  706  is an example of collimator element  628 . As another example, collimator element  702  is an example of collimator element  616  of collimator layer  612 . Collimator elements  702 ,  704 , and  706  lie in the same yz plane. Collimator elements  702 ,  704 , and  706  are fabricated from the secondary collimator material and are fabricated by the user by using any of the machining devices. Each collimator element  702 ,  704 , and  706  has the same size. For example, collimator element  702  has a length, along the y-axis, ranging from and including 0.7 metres (m) to 1.3 m, has a width, along the z-axis, ranging from and including 20 mm to 30 mm, and has a thickness, along the x-axis, ranging from and including 3 mm to 5 mm, which is the same as the size of collimator element  704 . Each collimator element  702 ,  704 , and  706  has a uniform width, along the z-axis and does not include a curved end. Moreover, each collimator  702 ,  704 , and  706  has a uniform thickness, along the x-axis. 
         [0057]    Opening  634  is formed between two adjacent collimator elements of system. For example, opening  634  is formed between collimator elements  702  and  704 . As another example, opening  634  is formed between collimator elements  704  and  706 . As an example, each of openings  634  have a length, along the y-axis, that is the same as a length of any of collimator elements  702  and  704 . As yet another example, each of openings  634  have a width, along the z-axis, ranging from and including 0.2 mm to 0.6 mm. As yet another example, each of openings have a thickness, along the x-axis, that is the same as that of a thickness of any of collimator elements  702 ,  704 , and  706 . In an alternative embodiment, system includes any number, such as 2, 4, 6, or 7, of collimator elements lying in the same yz plane. 
         [0058]    Technical effects of the herein described systems and methods for developing a secondary collimator include developing secondary collimator  400  that outputs scattered radiation having a plurality of scatter angles with respect to primary beam  86 . Other technical effects include generating and maintaining separate peaks of the diffraction profile regardless of a value of the scatter angle variable θ by keeping the modulus of the ratio of the first and second terms constant and by keeping the modulus of the ratio of the third and fourth terms constant. The scatter angle variable θ represents a scaling factor between the momentum transfer x and the energy E. Secondary collimator  400  permits a variation, up to a factor of three, in the scatter angle variable θ in a single scan of container  79 . Yet other technical effects include a lower value of the scatter angle variable θ is used for analyzing a dense container and a higher value of the scatter angle variable θ is advantageous for analyzing a light bag because a useful range of the momentum transfer x is increased with an increase in a range of the scatter angle variable θ. Still other technical effects include an increase in passage of scattered radiation through opening  444  compared to an opening of constant width. As a result of the increase in passage, a signal-to-noise ratio is improved and an optimum detection of substance  82  is provided while minimizing a probability of a false alarm. 
         [0059]    While the invention has been described in terms of various specific embodiments, those skilled in the art will recognize that the invention can be practiced with modification within the spirit and scope of the claims.