2 A Primer on Imaging Anatomy and Physiology ground state; in doing so, luminescence radiation with intensity proportional to the which detects v hergy is liberated. This visible light is then amplified by a PMT absorbed x-ray photons and generates a current signal proportional to the num- ber of x-ray photons. The laser scans the PSl plate in a raster format, and de- tection of the PMT current signal is accordingly synchronized. The analog PMt signal is then transformed into a digital value, thereby allowing a digital matrix representation of the visible image to be computed 3. Image processing. The digital image is then optimized, usually via tonal conver ion and edge enhancement. The goal of the tonal conversion is to optimize the image contrast based on: 1)display transfer characteristics; 2)anatomical region of interest; and 3)review medium. In the first stage, a transformation curve is employed based on the capture of the x-rays to the PSl plate is employed. Photo- stimulable phosphors have a much wider dynamic range than x-ray film: the former's exposure range is about 10,000: 1, meaning that the linear response for PSL intensity can go from 5 microroentgens to 50 milliroentgens. In the second stage, the PMT signal digitization to a value(usually a range of 2 values is selected) is transformed dependent upon algorithms that attempt to optimize image contrast based on the anatomical region under examination. Preferre lookup tables and the minimum, maximum, and average PSL values obtained during the pre-read operation are used to compute a transform curve. In the third stage, the transformation goes from a digital pixel value to an analog value(e.g, optical density or luminance), depending upon the characteristic curve of the film or monitor on which the final image is viewed. Edge enhancement is often per- formed on CR images to present an image to the radiologist that is"sharper. This operation is often useful for bone imaging to accentuate fine lines and boundaries between structures. The algorithm used in most CR systems to create an edge enhanced image is called unsharp masking Digital radiography. Not to be confused with CR, digital radiography (DR; also referred to as direct radiography)forgoes the use of the cartridge containing the PSL and separate reader to process the latent image. Instead, digital x-ray sensors are used and the latent image data are transferred directly from a detector to a review monitor 17, 88 provide earlier descriptions of DR systems; and a more recent general discus sion is found in 50]. The different technologies that have been developed as digital detectors can be grouped twofold 1. Indirect conversion. In this first category, a scintillator is used as an intermediate between the detection of x-ray photons and the generation of visible light, much like an intensifying screen. A scintillator is a material that exhibits the property of luminescence when excited by ionizing radiation. The visible light is then detected in a number of ways. Charge-coupled device(CCD) cameras are one method. A CCD camera is a relatively small image sensing device(3-5 cm)that contains a photoactive region. One or more of CCDs are combined in a digital detector, with the image from the scintillator downscaled using an optical lens or fiber optics to project onto the smaller area of the CCD's light-sensitive area From the CCD, the image can then be read. Alternatively, an amorphous silicon There remains some ambiguity in terminology, as digital radiography is sometimes used to refer to an umbrella term for both computed radiography and direct radio- graphy. Here, we refer to digital radiography as a separate entity from computed
2 A Primer on Imaging Anatomy and Physiology 21 ground state; in doing so, luminescence radiation with intensity proportional to the absorbed x-ray energy is liberated. This visible light is then amplified by a PMT, which detects the photons and generates a current signal proportional to the number of x-ray photons. The laser scans the PSL plate in a raster format, and detection of the PMT current signal is accordingly synchronized. The analog PMT signal is then transformed into a digital value, thereby allowing a digital matrix representation of the visible image to be computed. 3. Image processing. The digital image is then optimized, usually via tonal conversion and edge enhancement. The goal of the tonal conversion is to optimize the image contrast based on: 1) display transfer characteristics; 2) anatomical region of interest; and 3) review medium. In the first stage, a transformation curve is employed based on the capture of the x-rays to the PSL plate is employed. Photostimulable phosphors have a much wider dynamic range than x-ray film: the former’s exposure range is about 10,000:1, meaning that the linear response for PSL intensity can go from 5 microroentgens to 50 milliroentgens. In the second stage, the PMT signal digitization to a value (usually a range of 210 values is selected) is transformed dependent upon algorithms that attempt to optimize image contrast based on the anatomical region under examination. Preferred lookup tables and the minimum, maximum, and average PSL values obtained during the pre-read operation are used to compute a transform curve. In the third stage, the transformation goes from a digital pixel value to an analog value (e.g., optical density or luminance), depending upon the characteristic curve of the film or monitor on which the final image is viewed. Edge enhancement is often performed on CR images to present an image to the radiologist that is “sharper.” This operation is often useful for bone imaging to accentuate fine lines and boundaries between structures. The algorithm used in most CR systems to create an edge enhanced image is called unsharp masking. Digital radiography. Not to be confused with CR, digital radiography (DR; also referred to as direct radiography1 ) forgoes the use of the cartridge containing the PSL and separate reader to process the latent image. Instead, digital x-ray sensors are used and the latent image data are transferred directly from a detector to a review monitor. [17, 88] provide earlier descriptions of DR systems; and a more recent general discussion is found in [50]. The different technologies that have been developed as digital detectors can be grouped twofold: 1. Indirect conversion. In this first category, a scintillator is used as an intermediate between the detection of x-ray photons and the generation of visible light, much like an intensifying screen. A scintillator is a material that exhibits the property of luminescence when excited by ionizing radiation. The visible light is then detected in a number of ways. Charge-coupled device (CCD) cameras are one method. A CCD camera is a relatively small image sensing device (~3-5 cm2 ) that contains a photoactive region. One or more of CCDs are combined in a digital detector, with the image from the scintillator downscaled using an optical lens or fiber optics to project onto the smaller area of the CCD’s light-sensitive area. From the CCD, the image can then be read. Alternatively, an amorphous silicon 1 There remains some ambiguity in terminology, as digital radiography is sometimes used to refer to an umbrella term for both computed radiography and direct radiography. Here, we refer to digital radiography as a separate entity from computed radiography
D. Aberle et al photodiode circuitry layer integrated with a thin-film transistor (TFT) array can be used. Using this method, a large flat-panel sensor is constructed by the deposition of silicon with an array of photodiodes, subsequently covered with a scintillator When this scintillator emits light, the photodiodes are activated and generate electric charge; the TFT array then records this information to generate an image, with this array correlating directly to the image's pixels 2. Direct conversion. A range of solid state materials can be used to detect x-rays For instance, lithium-doped germanium or silicon can detect x-rays. Photons hit- ting these materials cause the formation of an electron hole pair that can be detected. Current direct conversion methods use a photoconductor, such as amor- phous selenium, to convert x-ray photons directly into electrical charges. As in xeroradiography, the charge pattern on the selenium is proportional to the incider x-ray exposure; however, a TFt array is instead used to record the electric charge I to create a digital rep Flat-panel detectors(both direct and indirect) have resolution only dependent on the recording device(ie, the TFT array). Present pixel sizes are on the order of 200 um for use in thoracic imaging, and 50-100 um for mammography. a key advantage of direct radiography over CR is the almost immediate generation of the visual image while still preserving the ability to perform digital image processing to improve image quality and reduce noise/artifacts Fluoroscopy. The fundamental idea behind fluoroscopy is to use x-rays to provide real-time display of anatomy/physiology, allowing visualization of movement. The original design entailed the use of a fluorescent screen as the detector, allowing an observer to view images directly. Unfortunately, such images are often dim: significant improvement was brought about when x-ray image intensifier tubes were introduced, providing much brighter images. The simplest image intensifier tubes are composed of an input phosphor layer coupled with a photocathode, a vacuum tube, and a smaller output phosphor layer integrated with an output window for display. X-rays photons hit the input phosphor (usually cesium iodide, doped with sodium onto an aluminum substrate), which in turn scintillates and emits light photons picked up by the photo- cathode. The photocathode, often made of an antimony-cesium alloy, produces electrons that are accelerated through a vacuum and focused onto the output phosphor layer This final phosphor layer luminesces brightly in response to the concentrated elec trons, showing the latent image. The image intensifier thus results in a brighter image rough two effects: 1)Aux gain, where the electrons accelerated through the vacuum produce more light as they strike the output phosphor, and 2)minification, as the lumber of light photons is concentrated in a smaller display. The input image's brightness is enhanced by a factor of approximately 10 times. Frequently used with image intensifiers are video cameras to record the images More recently, digital fluoroscopy has come about with the introduction of flat-panel detectors; additionally, the video camera has been replaced with a CCD-based camera This change introduces new capabilities, such as road mapping, which allows the viewer to temporarily store and display the most recent fluoroscopic image on a screen (e.g, for interventional procedures, such as the placement of guide wires). Digital subtraction angiography is another ability, wherein pre- and post-contrast images are combined together to examine the distribution of contrast. And like computed radiography an advantage of digital fluoroscopy is the ability to perform further image processin
22 D. Aberle et al. photodiode circuitry layer integrated with a thin-film transistor (TFT) array can be used. Using this method, a large flat-panel sensor is constructed by the deposition of silicon with an array of photodiodes, subsequently covered with a scintillator. When this scintillator emits light, the photodiodes are activated and generate an electric charge; the TFT array then records this information to generate an image, with this array correlating directly to the image’s pixels. 2. Direct conversion. A range of solid state materials can be used to detect x-rays. For instance, lithium-doped germanium or silicon can detect x-rays. Photons hitting these materials cause the formation of an electron hole pair that can be detected. Current direct conversion methods use a photoconductor, such as amorphous selenium, to convert x-ray photons directly into electrical charges. As in xeroradiography, the charge pattern on the selenium is proportional to the incident x-ray exposure; however, a TFT array is instead used to record the electric charge and to create a digital representation. Flat-panel detectors (both direct and indirect) have resolution only dependent on the recording device (i.e., the TFT array). Present pixel sizes are on the order of 200 μm for use in thoracic imaging, and 50-100 μm for mammography. A key advantage of direct radiography over CR is the almost immediate generation of the visual image while still preserving the ability to perform digital image processing to improve image quality and reduce noise/artifacts. Fluoroscopy. The fundamental idea behind fluoroscopy is to use x-rays to provide real-time display of anatomy/physiology, allowing visualization of movement. The original design entailed the use of a fluorescent screen as the detector, allowing an observer to view images directly. Unfortunately, such images are often dim: significant improvement was brought about when x-ray image intensifier tubes were introduced, providing much brighter images. The simplest image intensifier tubes are composed of an input phosphor layer coupled with a photocathode, a vacuum tube, and a smaller output phosphor layer integrated with an output window for display. X-rays photons hit the input phosphor (usually cesium iodide, doped with sodium onto an aluminum substrate), which in turn scintillates and emits light photons picked up by the photocathode. The photocathode, often made of an antimony-cesium alloy, produces electrons that are accelerated through a vacuum and focused onto the output phosphor layer. This final phosphor layer luminesces brightly in response to the concentrated electrons, showing the latent image. The image intensifier thus results in a brighter image through two effects: 1) flux gain, where the electrons accelerated through the vacuum produce more light as they strike the output phosphor; and 2) minification, as the number of light photons is concentrated in a smaller display. The input image’s brightness is enhanced by a factor of approximately 105 times. Frequently used with image intensifiers are video cameras to record the images. More recently, digital fluoroscopy has come about with the introduction of flat-panel detectors; additionally, the video camera has been replaced with a CCD-based camera. This change introduces new capabilities, such as road mapping, which allows the viewer to temporarily store and display the most recent fluoroscopic image on a screen (e.g., for interventional procedures, such as the placement of guide wires). Digital subtraction angiography is another ability, wherein pre- and post-contrast images are combined together to examine the distribution of contrast. And like computed radiography, an advantage of digital fluoroscopy is the ability to perform further image processing
2 A Primer on Imaging Anatomy and Physiology optimize image visualization. For example, frame averaging and edge enhancement be used to improve image presentation. Projectional image artifacts. In medical imaging, one often talks of spatial resolution that is, the ability to visually resolve fine details. Higher spatial resolution implies better discrimination of smaller objects In projectional imaging, there are four sources of unsharpness that decrease spatial resolution Motion blur. Although radiologic image exposure times are relatively short, they are not instantaneous. During this time, a patient may move and/or physiologic processes occur(e.g, normal cardiac motion), therefore causing a blurring artifact as the boundaries of an object are spread over the detector's field. 2. Geometric blur. In reality, an x-ray source is not an idealized point source of pho tons. Thus, geometric unsharpness occurs due to the physical geometry of image acquisition and image formation, and is influenced by factors such as the size of le x-ray source, the distance between the source and the patient, and the distance from the patient to the detector. Regions at the edge of an object will be formed such that x-ray intensity will gradually increase/decrease, causing unsharpness. These regions are called penumbra. 3. Absorption blur. X-rays are not uniformly absorbed by an object; rather, there is a in x-ray absorption across its boundary. Consider, for instance. the difference between an object who' s edges are parallel to the cone of an x-ray beam, versus a perfect sphere: the former will have sharply defined edges as absorption will be uniform, whereas the different points of the sphere will encounter varying amounts of x-ray photons (the center will see maximal amounts, the outer regions the minimum 4. Detector blur. Lastly, the detector itself can introduce certain phenomena that will create image blur. For instance, the use of an intensifying screen will result in a finite amount of diffusion given the distance between the screen and the film. [19, 36] provides further details on the geometry of the radiographic image and the reasons for unsharpness Computed Tomography ey work during the mid-20 century in x-ray reconstruction and the theory behind xial tomography led to the development of the first commercial computed tomo- graphy (CT) scanner in 1971 for clinical purposes [37]. Relative to conventional projectional x-ray imaging where subtle differences in attenuation (less than 5%)are often lost, CT provides much improved subject contrast with discrimination less than 1% and the current generation of multi-slice CT scanners provide sub-millimeter reso- lution. We note here that the core physical concept behind CT, x-ray attenuation, is described prior; below, we focus on the principles that enable image formation. The eader is referred to[15] for a complete handling of CT imaging The projection of an x-ray through an object can be defined through a set of line inte grals, representing the total attenuation of the beam as it travels through the different materials composing the object. Recall from the discussion of projectional imaging that x-ray attenuation through one material is described by the equation, I= loe".The attenuation effect is cumulative so that the transmission of an x-ray through multiple substances is given by the formula
2 A Primer on Imaging Anatomy and Physiology 23 to optimize image visualization. For example, frame averaging and edge enhancement can be used to improve image presentation. Projectional image artifacts. In medical imaging, one often talks of spatial resolution – that is, the ability to visually resolve fine details. Higher spatial resolution implies better discrimination of smaller objects. In projectional imaging, there are four sources of unsharpness that decrease spatial resolution: 1. Motion blur. Although radiologic image exposure times are relatively short, they are not instantaneous. During this time, a patient may move and/or physiologic processes occur (e.g., normal cardiac motion), therefore causing a blurring artifact as the boundaries of an object are spread over the detector’s field. 2. Geometric blur. In reality, an x-ray source is not an idealized point source of photons. Thus, geometric unsharpness occurs due to the physical geometry of image acquisition and image formation, and is influenced by factors such as the size of the x-ray source, the distance between the source and the patient, and the distance from the patient to the detector. Regions at the edge of an object will be formed such that x-ray intensity will gradually increase/decrease, causing unsharpness. These regions are called penumbra. 3. Absorption blur. X-rays are not uniformly absorbed by an object; rather, there is a gradated change in x-ray absorption across its boundary. Consider, for instance, the difference between an object who’s edges are parallel to the cone of an x-ray beam, versus a perfect sphere: the former will have sharply defined edges as absorption will be uniform, whereas the different points of the sphere will encounter varying amounts of x-ray photons (the center will see maximal amounts, the outer regions the minimum). 4. Detector blur. Lastly, the detector itself can introduce certain phenomena that will create image blur. For instance, the use of an intensifying screen will result in a finite amount of diffusion given the distance between the screen and the film. [19, 36] provides further details on the geometry of the radiographic image and the reasons for unsharpness. Computed Tomography Key work during the mid-20th century in x-ray reconstruction and the theory behind axial tomography led to the development of the first commercial computed tomography (CT) scanner in 1971 for clinical purposes [37]. Relative to conventional projectional x-ray imaging where subtle differences in attenuation (less than 5%) are often lost, CT provides much improved subject contrast with discrimination less than 1% and the current generation of multi-slice CT scanners provide sub-millimeter resolution. We note here that the core physical concept behind CT, x-ray attenuation, is described prior; below, we focus on the principles that enable image formation. The reader is referred to [15] for a complete handling of CT imaging. Imaging The projection of an x-ray through an object can be defined through a set of line integrals, representing the total attenuation of the beam as it travels through the different materials composing the object. Recall from the discussion of projectional imaging that x-ray attenuation through one material is described by the equation, I = I0e -μt . The attenuation effect is cumulative so that the transmission of an x-ray through multiple substances is given by the formula:
D. Aberle et al where u(r, y is the attenuation coefficient at point (x, y) along the beams path. Given an xy-plane through the object, let represent the path of an x-ray beam in this plane Then the above equation can be rewritten in terms of the total attenuation, p. where 0 is the angle formed between r and the x-axis(ie, r= xcose+ ysine). The function, P(r, 0), is referred to as the Radon transform. By recovering u(x, y) via an inverse Radon transform, a cross-sectional image of the object in the xy-plane is possible this process is the premise behind tomographic reconstruction. In theory, given an infinite number of measurements, one can reconstruct u(x, y perfectly: CT thus uses multiple narrow x-ray beams through the same point in order to collect enough data to sufficiently approximate u(x, y) and reconstruct an image. A sinogram is the raw data obtained from a tomographic reconstruction, a visual representation of the Radon transform. Each row of the sinogram represents a different projection through the object(Fig. 2. 4b) Reconstruction algorithms. Central to CT imaging is a means to efficiently perform he inverse Radon transform. Several algorithms exist for this purpose, and can be 1. Simple back-proiection. The most straightforward method to reconstruct a 2D image starts by assuming an empty, equally-spaced image matrix. As each x-ray beam contributes to the estimation of u, the algorithm aims to sum the attenuation from each beam for point (x, y) in the image matrix. The relative contribution of each x-ray path(ray) through the object can be determined knowing the angle at which the ray is transmitted. This procedure is known as simple back-projection The back-projection is created by"smearing"a ray back through the image in the direction it was originally acquired. Conceptually, one can think of this algorithm as adding the value of u to each pixel based on the rays going through the pixel While simple to implement, simple back-projection tends to blur image features (Fig. 2. 4c) as the point spread function of a back-projection is circularly symmetric. decreasing as an inverse function of the radius(1/r) 2. Filtered back-projection. To overcome the blurring in simple back-projection, each ray can be filtered or convolved with a kernel prior to the back-projection. In filtered back-projection, the filter has the effect of weighting the center of a ray while underweighting the periphery, thus counteracting the blur. Mathematically, the convolution operation in the spatial domain is represented by pIx)=p(x@ k(x) where p(x is the original projection data, kx) is the kernel, p(x is the resultant filtered data, and represents the integral convolution operation. Alternatively, this same operation can be considered in the frequency domain using a Fourier transform(FT), PYx= FT (FT(p(x))x K(), where K(=FT((x). This trans formation is often called the Fourier slice theorem. The advantage of considering this process in the frequency domain is that the convolution operation is transformed into a multiplication. various kernels exist dependent on the imaging application
24 D. Aberle et al. ∫− = dsyx eII ),( 0 μ where μ(x,y) is the attenuation coefficient at point (x, y) along the beam’s path. Given an xy-plane through the object, let r represent the path of an x-ray beam in this plane. Then the above equation can be rewritten in terms of the total attenuation, p: ∫ ∫∫ −= = −+ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = dxdyryxyxdsyx I I rp ln),( )sincos(),(),( 0 θ μ θθδμ where θ is the angle formed between r and the x-axis (i.e., r = xcosθ + ysinθ). The function, p(r, θ), is referred to as the Radon transform. By recovering μ(x, y) via an inverse Radon transform, a cross-sectional image of the object in the xy-plane is possible: this process is the premise behind tomographic reconstruction. In theory, given an infinite number of measurements, one can reconstruct μ(x, y) perfectly; CT thus uses multiple narrow x-ray beams through the same point in order to collect enough data to sufficiently approximate μ(x, y) and reconstruct an image. A sinogram is the raw data obtained from a tomographic reconstruction, a visual representation of the Radon transform. Each row of the sinogram represents a different projection through the object (Fig. 2.4b). Reconstruction algorithms. Central to CT imaging is a means to efficiently perform the inverse Radon transform. Several algorithms exist for this purpose, and can be categorized threefold [36]: 1. Simple back-projection. The most straightforward method to reconstruct a 2D image starts by assuming an empty, equally-spaced image matrix. As each x-ray beam contributes to the estimation of μ, the algorithm aims to sum the attenuation from each beam for point (x,y) in the image matrix. The relative contribution of each x-ray path (ray) through the object can be determined knowing the angle at which the ray is transmitted. This procedure is known as simple back-projection. The back-projection is created by “smearing” a ray back through the image in the direction it was originally acquired. Conceptually, one can think of this algorithm as adding the value of μ to each pixel based on the rays going through the pixel. While simple to implement, simple back-projection tends to blur image features (Fig. 2.4c) as the point spread function of a back-projection is circularly symmetric, decreasing as an inverse function of the radius (1/r). 2. Filtered back-projection. To overcome the blurring in simple back-projection, each ray can be filtered or convolved with a kernel prior to the back-projection. In filtered back-projection, the filter has the effect of weighting the center of a ray while underweighting the periphery, thus counteracting the blur. Mathematically, the convolution operation in the spatial domain is represented by p'(x) = p(x) ⊗ k(x) where p(x) is the original projection data, k(x) is the kernel, p'(x) is the resultant filtered data, and ⊗ represents the integral convolution operation. Alternatively, this same operation can be considered in the frequency domain using a Fourier transform (FT), p'(x) = FT-1(FT(p(x)) x K(f)), where K(f) = FT(p(x)). This transformation is often called the Fourier slice theorem. The advantage of considering into a multiplication. Various kernels exist dependent on the imaging application this process in the frequency domain is that the convolution operation is transformed
2 A Primer on Imaging Anatomy and Physiology Figure 2.4: Demonstration of CT image reconstruction process. (a)Example black and white image source image with two shapes is shown. (b)A sinogram is a visual representation of the Radon transform, where each row/column of the sinogram repre- sents projection information. A sinogram with 180 samples for the image in(a)is shown.(c)Simple back-projection results in a blurring of the image.(d) Different kermels can be applied in filtered back-projection to handle the blurring caused by a point spread function. Here, a Hamming filter is used to improve the reconstruction; although improved, there are still subtle imaging artifacts relative to the original image (e. g, soft tissue or bone visualization), and will affect the in-plane resolution and noise seen in the final image. For instance, in the frequency domain, one apply the Ram-Lak filter to undo the 1/r blurring phenomena; however, this method is highly sensitive to noise, especially at higher frequencies. More robust techniques include the use of a Shepp-Logan, cosine, or Hamming filters that compensate with high-frequency roll-off(Fig 2. 4d; Fig. 2.5). For the most part, filtered back-projection is the method used today in clinical CT'scanners 3. Series expansion. Both simple and filtered back-projection algorithms can be run while raw image data is acquired, allowing for more immediate reconstruction. In comparison, series expansion techniques (also known as iterative techniques and algebraic reconstruction) require that all x-ray attenuation data be available before reconstruction commences. Series expansion techniques involve solving large systems of linear equations based on the observed attenuations from each ray, the linear system represents the target image for reconstruction. Examples of these methods include the algebraic reconstruction technique (ART); iterative least-squares technique (ILST); and simultaneous iterative reconstruction tech- nique(SIRT). [43] provides a thorough discussion of these methods superimposed on a physical. &rid used in reconstruction can be seen as a discretization We note here that the image region; as such, the mapping between a given pixel(voxel)to a single point in space is imperfect. For instance, what if the pixel boundary encom passes two different types of materials(e.g, bone and soft tissue; air-surface boundaries)? The resultant attenuation for that pixel is an intermediate value of the two substances This artifact is referred to as partial voluming Under CT, partial volume averaging often happens when structure boundaries are almost parallel to the ct slice. Hounsfield units. The results of a reconstruction algorithm are a measure of the attenuation for a given pixel location(x, y. These values are normalized to Hounsfield units(HU) prior to generation as final image. Specifically In point of fact, the partial voluming effect is not unique to computed tomography, but also occurs with other imaging modalities, such as magnetic resonance
2 A Primer on Imaging Anatomy and Physiology 25 Figure 2.4: Demonstration of CT image reconstruction process. (a) Example black and white image source image with two shapes is shown. (b) A sinogram is a visual representation of the Radon transform, where each row/column of the sinogram represents projection information. A sinogram with 180 samples for the image in (a) is shown. (c) Simple back-projection results in a blurring of the image. (d) Different kernels can be applied in filtered back-projection to handle the blurring caused by a pointspread function. Here, a Hamming filter is used to improve the reconstruction; although improved, there are still subtle imaging artifacts relative to the original image. (e.g., soft tissue or bone visualization), and will affect the in-plane resolution and noise seen in the final image. For instance, in the frequency domain, one can apply the Ram-Lak filter to undo the 1/r blurring phenomena; however, this method is highly sensitive to noise, especially at higher frequencies. More robust techniques include the use of a Shepp-Logan, cosine, or Hamming filters that compensate with high-frequency roll-off (Fig 2.4d; Fig. 2.5). For the most part, filtered back-projection is the method used today in clinical CT scanners. Series expansion. Both simple and filtered back-projection algorithms can be run while raw image data is acquired, allowing for more immediate reconstruction. In comparison, series expansion techniques (also known as iterative techniques and algebraic reconstruction) require that all x-ray attenuation data be available before reconstruction commences. Series expansion techniques involve solving large systems of linear equations based on the observed attenuations from each ray; the linear system represents the target image for reconstruction. Examples of these methods include the algebraic reconstruction technique (ART); iterative least-squares technique (ILST); and simultaneous iterative reconstruction technique (SIRT). [43] provides a thorough discussion of these methods. We note here that the image grid used in reconstruction can be seen as a discretization superimposed on a physical region; as such, the mapping between a given pixel (voxel) to a single point in space is imperfect. For instance, what if the pixel boundary encompasses two different types of materials (e.g., bone and soft tissue; air-surface boundaries)? The resultant attenuation for that pixel is an intermediate value of the two substances. This artifact is referred to as partial voluming2 . Under CT, partial volume averaging often happens when structure boundaries are almost parallel to the CT slice. Hounsfield units. The results of a reconstruction algorithm are a measure of the attenuation for a given pixel location (x,y). These values are normalized to Hounsfield units (HU) prior to generation as final image. Specifically: 2 In point of fact, the partial voluming effect is not unique to computed tomography, but also occurs with other imaging modalities, such as magnetic resonance. 3