3 Removal of Artifacts Noise is omnipresent! Biomedical images are often a ected and corrupted by various types of noise and artifact Any image, pattern, or signal other than that of interest could be termed as interference, artifact, or simply noise The sources of noise could be physiological, the instrumentation used, or the environment of the experiment The problems caused by artifacts in biomedical images are vast in scope and variety their potential for degrading the performance of the most sophisticated image processing algorithms is high The removal of artifacts without causing any distortion or loss of the desired information in the image of interest is often a signi cant challenge The enormity of the problem of noise removal and its importance are re ected by the placement of this chapter as the rst chapter on image processing techniques in this book This chapter starts with an introduction to the nature of the artifacts that are commonly encountered in biomedical images Several illustrations of images corrupted by various types of artifacts are provided Details of the design of lters spanning a broad range of approaches, from linear space-domain and frequency-domain xed lters, to the optimal Wiener lter, and further on to nonlinear and adaptive lters, are then described The chapter concludes with demonstrations of application of the lters described to a few biomedical images (Note: A good background in signal and system analysis 1, 2, 3, 167] as well as probability, random variables, and stochastic processes 3, 128, 168, 169, 170, 171, 172, 173] is required in order to follow the procedures and analyses described in this chapter.) 3.1 Characterization of Artifacts 3.1.1 Random noise The term random noise refers to an interference that arises from a random process such as thermal noise in electronic devices and the counting of photons A random process is characterized by the PDF representing the probabilities of occurrence of all possible values of a random variable (See Papoulis 128] © 2005 by CRC Press LLC 151 152 Biomedical Image Analysis or Bendat and Piersol 168] for background material on probability, random variables, and stochastic processes.) Consider a random process that is characterized by the PDF p ( ) The process could be a function of time as (t), or of space in 1D, 2D, or 3D as (x), (x y), or (x y z ) it could also be a spatio-temporal function as (x y z t) The argument of the PDF represents the value that the random process can assume, which could be a voltage in the case of a function of time, or a gray level in the case of a 2D or 3D image The use of the same symbol for the function and the value it can assume when dealing with PDFs is useful when dealing with several random processes The mean of the random process is given by the rst-order moment of the PDF, de ned as =E ]= Z1 p ( )d ;1 (3.1) where E ] represents the statistical expectation operator It is common to assume the mean of a random noise process to be zero The mean-squared (MS) value of the random process is given by the second-order moment of the PDF, de ned as E 2] = The variance Z1 ;1 2p ( )d : (3.2) of the process is de ned as the second central moment: =E( ; )2 ] = Z1 ;1 ( ; )2 p ( ) d : (3.3) The square root of the variance gives the standard deviation (SD) of the process Note that = E ] ; If the mean is zero, it follows that = E ], that is, the variance and the MS values are the same Observe the use of the same symbol to represent the random variable, the random process, and the random signal as a function of time or space The subscript of the PDF or the statistical parameter derived indicates the random process of concern The context of the discussion or expression should make the meaning of the symbol clear When the values of a random process form a time series or a function of time, we have a random signal (or a stochastic process) (t) see Figure 3.1 When one such time series is observed, it is important to note that the entity represents but one single realization of the random process An example of a random function of time is the current generated by a CCD detector element due to thermal noise when no light is falling on the detector (known as the dark current ) The statistical measures described above then have physical meaning: the mean represents the DC component, the MS value represents the average power, and the square root of the mean-squared value (the root meansquared or RMS value) gives the average noise magnitude These measures © 2005 by CRC Press LLC Removal of Artifacts 153 are useful in calculating the SNR, which is commonly de ned as the ratio of the peak-to-peak amplitude range of the signal to the RMS value of the noise, or as the ratio of the average power of the desired signal to that of the noise Special-purpose CCD detectors are cooled by circulating cold air, water, or liquid nitrogen to reduce thermal noise and improve the SNR 0.25 0.2 0.15 0.1 noise value 0.05 −0.05 −0.1 −0.15 −0.2 −0.25 50 100 150 sample number 200 250 FIGURE 3.1 A time series composed of random noise samples with a Gaussian PDF having = and = 0:01 MS value = 0:01 RMS = 0:1 See also Figures 3.2 and 3.3 When the values of a random process form a 2D function of space, we have a noise image (x y) see Figures 3.2 and 3.3 Several possibilities arise in this situation: We may have a single random process that generates random gray levels that are then placed at various locations in the (x y) plane in some structured or random sequence We may have an array of detectors with one detector per pixel of a digital image the gray level generated by each detector may then be viewed as a distinct random process that is independent of those of the other detectors A TV image generated by such a camera in the presence of no input image could be considered to be a noise process in (x y t), that is, a function of space and time A biomedical image of interest f (x y) may also, for the sake of generality, be considered to be a realization of a random process f Such a representation © 2005 by CRC Press LLC 154 Biomedical Image Analysis FIGURE 3.2 An image composed of random noise samples with a Gaussian PDF having = and = 0:01 MS value = 0:01 RMS = 0:1 The normalized pixel values in the range ;0:5 0:5] were linearly mapped to the display range 255] See also Figure 3.3 0.016 0.014 Probability of occurrence 0.012 0.01 0.008 0.006 0.004 0.002 −0.5 −0.4 −0.3 FIGURE 3.3 −0.2 −0.1 0.1 Normalized gray level 0.2 0.3 0.4 0.5 Normalized histogram of the image in Figure 3.2 The samples were generated using a Gaussian process with = and = 0:01 MS value = 0:01 RMS = 0:1 See also Figures 3.1 and 3.2 © 2005 by CRC Press LLC Removal of Artifacts 155 allows for the statistical characterization of sample-to-sample or person-toperson variations in a collection of images of the same organ, system, or type For example, although almost all CT images of the brain show the familiar cerebral structure, variations exist from one person to another A brain CT image may be represented as a random process that exhibits certain characteristics on the average Statistical averages representing populations of images of a certain type are useful in designing lters, data compression techniques, and pattern classi cation procedures that are optimal for the speci c type of images However, it should be borne in mind that, in diagnostic applications, it is the deviation from the normal or the average that is present in the image on hand that is of critical importance When an image f (x y) is observed in the presence of random noise , the detected image g(x y) may be treated as a realization of another random process g In most cases, the noise is additive, and the observed image is expressed as g(x y) = f (x y) + (x y): (3.4) Each of the random processes f , , and g is characterized by its own PDF pf (f ), p ( ), and pg (g), respectively In most practical applications, the random processes representing an image of interest and the noise a ecting the image may be assumed to be statistically independent processes Two random processes f and are said to be statistically independent if their joint PDF pf (f ) is equal to the product of their individual PDFs given as pf (f ) p ( ) It then follows that the rst-order moment and second-order central moment of the processes in Equation 3.4 are related as E g] = g = f + = f = E f ] (3.5) 2 2 E (g ; g ) ] = g = f + (3.6) where represents the mean and represents the variance of the random process indicated by the subscript, and it is assumed that = Ensemble averages: When the PDFs of the random processes of concern are not known, it is common to approximate the statistical expectation operation by averages computed using a collection or ensemble of sample observations of the random process Such averages are known as ensemble averages Suppose we have M observations of the random process f as functions of (x y): f1 (x y) f2 (x y) : : : fM (x y) see Figure 3.4 We may estimate the mean of the process at a particular spatial location (x1 y1 ) as M X f (x1 y1 ) = Mlim !1 M fk (x1 y1 ): k=1 (3.7) The autocorrelation function (ACF) f (x1 x1 + y1 y1 + ) of the random process f is de ned as (3.8) f (x1 x1 + y1 y1 + ) = E f (x1 y1 ) f (x1 + y1 + )] © 2005 by CRC Press LLC 156 Biomedical Image Analysis which may be estimated as f (x1 x1 + M X y1 y1 + k) = Mlim !1 M k=1 fk (x1 y1 ) fk (x1 + y1 + ) (3.9) where and are spatial shift parameters If the image f (x y) is complex, one of the versions of f (x y) in the products above should be conjugated most biomedical images that are encountered in practice are real-valued functions, and this distinction is often ignored The ACF indicates how the values of an image at a particular spatial location are statistically related to (or have characteristics in common with) the values of the same image at another shifted location If the process is stationary, the ACF depends only upon the shift parameters, and may be expressed as f ( ) f (x, y) M f (x, y) k spatial average µ (k) f f (x, y) f (x, y) ensemble average µ (x , y ) f 1 FIGURE 3.4 Ensemble and spatial averaging of images The three equations above may be applied to signals that are functions of time by replacing the spatial variables (x y) with the temporal variable t, replacing the shift parameter with to represent temporal delay, and making a few other related changes © 2005 by CRC Press LLC Removal of Artifacts 157 When f (x1 y1 ) is computed for every spatial location or pixel, we get an average image that could be expressed as f (x y) The image f may be used to represent the random process f as a prototype For practical use, such an average should be computed using sample observations that are of the same size, scale, orientation, etc Similarly, the ACF may also be computed for all possible values of its indices to obtain an image Temporal and spatial averages: When we have a sample observation of a random process fk (t) as a function of time, it is possible to compute time averages or temporal statistics by integrating along the time axis 31]: f (k) = Tlim !1 T Z T=2 ;T=2 fk (t) dt: (3.10) The integral would be replaced by a summation in the case of sampled or discrete-time signals The time-averaged ACF f ( k) is given by f ( k) = Tlim !1 T Z T=2 ;T=2 fk (t) fk (t + ) dt: (3.11) Similarly, given an observation of a random process as an image fk (x y), we may compute averages by integrating over the spatial domain, to obtain spatial averages or spatial statistics see Figure 3.4 The spatial mean of the image fk (x y) is given by Z1Z1 f (x y) dx dy (3.12) (k) = A ;1 ;1 k f where A is a normalization factor, such as the actual area of the image Observe that the spatial mean above is a single-valued entity (a scalar) For a stationary process, the spatial ACF is given by f( k) = Z1Z1 ;1 ;1 fk (x y) fk (x + y + ) dx dy: (3.13) A suitable normalization factor, such as the total energy of the image which is equal to f (0 0)] may be included, if necessary The sample index k becomes irrelevant if only one observation is available In practice, the integrals change to summations over the space of the digital image available When we have a 2D image as a function of time, such as TV, video, uoroscopy, and cine-angiography signals, we have a spatio-temporal signal that may be expressed as f (x y t) see Figure 3.5 We may then compute statistics over a single frame f (x y t1 ) at the instant of time t1 , which are known as intraframe statistics We could also compute parameters through multiple frames over a certain period of time, which are called interframe statistics the signal over a speci c period of time may then be treated as a 3D dataset Random functions of time may thus be characterized in terms of ensemble and/or temporal statistics Random functions of space may be represented © 2005 by CRC Press LLC 158 Biomedical Image Analysis f (x, y, t) f (x, y, t ) spatial or intraframe statistics temporal or interframe statistics FIGURE 3.5 Spatial and temporal statistics of a video signal © 2005 by CRC Press LLC Removal of Artifacts 159 by their ensemble and/or spatial statistics Figure 3.4 shows the distinction between ensemble and spatial averaging Figure 3.5 illustrates the combined use of spatial and temporal statistics to analyze a video signal in (x y t) The mean does not play an important role in 1D signal analysis: it is usually assumed to be zero, and often subtracted out if it is not zero However, the mean of an image represents its average intensity or density removal of the mean leads to an image with only the edges and the uctuations about the mean being depicted The ACF plays an important role in the characterization of random processes The Fourier transform of the ACF is the power spectral density (PSD) function, which is useful in frequency-domain analysis Statistical functions as above are useful in the analysis of the behavior of random processes, and in modeling, spectrum analysis, lter design, data compression, and data communication 3.1.2 Examples of noise PDFs As we have already seen, several types of noise sources are encountered in biomedical imaging Depending upon the characteristics of the noise source and the phenomena involved in the generation of the signal and noise values, we encounter a few di erent types of PDFs, some of which are described in the following paragraphs 3, 128, 173] Gaussian: The most commonly encountered and used noise PDF is the Gaussian or normal PDF, expressed as 3, 128] px (x) = p exp ; (x ;2 2x ) : (3.14) x x A Gaussian PDF is completely speci ed by its mean x and variance x2 Figure 3.6 shows three Gaussian PDFs with = = = = and = = See also Figures 3.2 and 3.3 When we have two jointly normal random processes x and y, the bivariate normal PDF is given by px y (x y) = p 2 (1 ; ) x y 2 exp ; 2(1 ;1 ) (x ; x ) ; (x ; x )(y ; y ) + (y ; y ) (3.15) x y x y where is the correlation coe cient given by (3.16) = E (x ; x )(y ; y )] : x y If = 0, the two processes are uncorrelated The bivariate normal PDF then reduces to a product of two univariate Gaussians, which implies that the two processes are statistically independent © 2005 by CRC Press LLC 160 Biomedical Image Analysis 0.35 0.3 Gaussian PDFs 0.25 0.2 0.15 0.1 0.05 −8 −6 FIGURE 3.6 −4 −2 x Three Gaussian PDFs Solid line: = Dotted line: = = = Dashed line: = 10 = The importance of the Gaussian PDF in practice arises from a phenomenon that is expressed as the central limit theorem 3, 128]: The PDF of a random process that is the sum of several statistically independent random processes is equal to the cascaded convolution of their individual PDFs When a large number of functions are convolved in cascade, the result tends toward a Gaussian-shaped function regardless of the forms of the individual functions In practice, an image is typically a ected by a series of independent sources of additive noise the net noise PDF may then be assumed to be a Gaussian Uniform: All possible values of a uniformly distributed random process have equal probability of occurrence The PDF of such a random process over the range (a b) is a rectangle of height (b;1 a) over the range (a b) The mean of the process is (a+2 b) , and the variance is (b;12a) Figure 3.7 shows two uniform PDFs corresponding to random processes with values spread over the ranges (;10 10) and (;5 5) The quantization of gray levels in an image to a nite number of integers leads to an error or noise that is uniformly distributed Poisson: The counting of discrete random events such as the number of photons emitted by a source or detected by a sensor in a given interval of time leads to a random variable with a Poisson PDF The discrete nature of the packets of energy (that is, photons) and the statistical randomness in their emission and detection contribute to uncertainty, which is re ected as © 2005 by CRC Press LLC 270 Biomedical Image Analysis 3.9 Application: Multiframe Averaging in Confocal Microscopy The confocal microscope uses a laser beam to scan and image nely focused planes within uorescent-dye-tagged specimens that could be a few mm in thickness 216, 217, 218] The use of a coherent light source obviates the blur caused in imaging with ordinary white light, where the di erent frequency components of the incident light are re ected and refracted at di erent angles by the specimen Laser excitation causes the dyes to emit light (that is, uoresce) at particular wavelengths The use of multiple dyes to stain di erent tissues and structures within the specimen permits their separate and distinct imaging The confocal microscope uses a pinhole to permit the passage of only the light from the plane of focus light from the other planes of the specimen is blocked Whereas the use of the pinhole permits ne focusing, it also reduces signi cantly the amount of light that is passed for further detection and viewing For this reason, a PMT is used to amplify the light received A scanning mechanism is used to raster-scan the sample in steps that could be as small as 0:1 m The confocal microscope facilitates the imaging of multiple focal planes separated by distances of the order of m several such slices may be acquired and combined to build 3D images of the specimen The use of a laser beam for scanning and imaging carries limitations The use of high-powered laser beams to obtain strong emitted light could damage the specimen by heating On the other hand, low laser power levels result in weak emitted light, which, during ampli cation by the PMT, could su er from high levels of noise Scanning with a low-power laser beam over long periods of time to reduce noise could lead to damage of the specimen by photo-bleaching (the a ected molecules permanently lose their capability of uorescence) Images could, in addition, be contaminated by noise due to auto uorescence of the specimen A technique commonly used to improve image quality in confocal microscopy is to average multiple acquisitions of each scan line or of the full image frame (see Section 3.2) Figure 3.63 shows images of cells from the nucleus pulposus (the central portion of the intervertebral discs, which are cartilaginous tissues lying between the bony vertebral bodies) 216] The specimen was scanned using a laser beam of wavelength 488 nm The red-dye (long-pass cuto at 585 nm) and green-dye (pass band of 505 ; 530 nm) components show distinctly and separately the cell nuclei and the actin lament structure, respectively, of the specimen, in a single focal plane representing a thickness of about m The component and composite images would be viewed in the colors mentioned on the microscope, but are illustrated in gray scale in the gure Images of this nature have been found to be useful in studies of injuries and diseases that a ect the intervertebral discs and the spinal column 216] © 2005 by CRC Press LLC Removal of Artifacts 271 Figures 3.64 (a) and (b) show two single-frame acquisitions of the composite image as in Figure 3.63 (c) Figures 3.64 (c) and (d) show the results of averaging four and eight single-frame acquisitions of the specimen, respectively Multiframe averaging has clearly reduced the noise and improved the quality of the image Al-Kofahi et al 219] used confocal microscopy to study the structure of soma and dendrites via 3D tracing of neuronal topology Comparison of ltering with space-domain and frequency-domain lowpass lters: Figures 3.65 (a) and (b) show the results of 3 mean and median ltering of the single-frame acquisition of the composite image of the nucleus pulposus in Figure 3.64 (a) Figures 3.65 (c) and (d) show the results of 5 mean and median ltering of the same image It is evident that neighborhood ltering, while suppressing noise to some extent, has caused blurring of the sharp details in the images Figure 3.66 shows the results of application of the 5 LLMMSE, re ned LLMMSE, NURW, and adaptive-neighborhood LLMMSE lters to the noisy image in Figure 3.64 (a) The noise was assumed to be multiplicative, with the normalized mean = and normalized standard deviation = 0:2 The optimal, adaptive, and nonlinear lters have performed well in suppressing noise without causing blurring The results of Fourier-domain ideal and Butterworth lowpass ltering of the noisy image are shown in Figures 3.67 (c) and (d) parts (a) and (b) of the same gure show the original image and its Fourier log-magnitude spectrum, respectively The spectrum indicates that most of the energy of the image is concentrated within a small area around the center that is, around (u v) = (0 0) The lters have caused some loss of sharpness while suppressing noise Observe that the ideal lowpass lter has given the noisy areas a mottled appearance Comparing the results in Figures 3.65, 3.66, and 3.67 with the results of multiframe averaging in Figures 3.64 and 3.63, it is seen that multiframe averaging has provided the best results 3.10 Application: Noise Reduction in Nuclear Medicine Imaging Nuclear medicine images are typically acquired under low-photon conditions, which leads to a signi cant presence of Poisson noise in the images Counting the photons emitted over long periods of time reduces the e ect of noise and improves the quality of the image However, imaging over long periods of time may not be feasible due to motion artifacts and various practical limitations Figure 3.68 shows the SPECT images of one section of a resolution phantom acquired over 15 and 40 s Each image has been scaled such that its © 2005 by CRC Press LLC 272 Biomedical Image Analysis (a) (b) (c) FIGURE 3.63 (a) The red-dye (cell nuclei) component of the confocal microscope image of the nucleus pulposus of a dog (b) The green-dye (actin lament structure) component (c) Combination of the images in (a) and (b) into a composite image The images would be viewed in the colors mentioned on the microscope The width of each image corresponds to 145 m Each image was acquired by averaging eight frames Images courtesy of C.J Hunter, J.R Matyas, and N.A Duncan, McCaig Centre for Joint Injury and Arthritis Research, University of Calgary © 2005 by CRC Press LLC Removal of Artifacts 273 (a) (b) (c) (d) FIGURE 3.64 (a) A single-frame acquisition of the composite image of the nucleus pulposus see also Figure 3.63 (b) A second example of a single-frame acquisition as in (a) (c) The result of averaging four frames including the two in (a) and (b) (d) The result of averaging eight frames including the two in (a) and (b) The width of each image corresponds to 145 m Images courtesy of C.J Hunter, J.R Matyas, and N.A Duncan, McCaig Centre for Joint Injury and Arthritis Research, University of Calgary © 2005 by CRC Press LLC 274 Biomedical Image Analysis (a) (b) (c) (d) FIGURE 3.65 Results of ltering the single-frame acquisition of the composite image of the nucleus pulposus in Figure 3.64 (a) with: (a) the 3 mean lter (b) the 3 median lter (c) the 5 mean lter and (d) the 5 median lter © 2005 by CRC Press LLC Removal of Artifacts 275 (a) (b) (c) (d) FIGURE 3.66 Results of ltering the single-frame acquisition of the composite image of the nucleus pulposus in Figure 3.64 (a) with: (a) the 5 LLMMSE lter (b) the re ned LLMMSE lter (c) the NURW lter and (d) the adaptiveneighborhood LLMMSE lter Figure courtesy of M Ciuc, Laboratorul de Analiza si Prelucrarea Imaginilor, Universitatea Politehnica Bucuresti, Bucharest, Romania © 2005 by CRC Press LLC 276 Biomedical Image Analysis (a) (b) (c) (d) FIGURE 3.67 (a) The single-frame acquisition of the composite image of the nucleus pulposus of a dog, as in Figure 3.64 (a) (b) Fourier log-magnitude spectrum of the image in (a) Results of ltering the image in (a) with: (c) the ideal lowpass lter with cuto D0 = 0:4, as in Figure 3.28 (a) and (d) the Butterworth lowpass lter with cuto D0 = 0:4 and order n = 2, as in Figure 3.28 (b) © 2005 by CRC Press LLC Removal of Artifacts 277 minimum and maximum values are mapped to the display range of 255] Also shown is a schematic representation of the section: the circles represent cross-sections of cylindrical holes in a plexiglass block the diameter of the entire phantom is 200 mm the two large circles at the extremes of the two sides are of diameter 39 mm the two inner arrays have circles of diameter 22 17 14 12 and mm The phantom was lled with a radiopharmaceutical such that the cylindrical holes would be lled with the radioactive material It is evident that the image quality improves as the photon counting time is increased The circles of diameter and mm are distinctly visible only in image (c) However, the circles of diameter mm are not visible in any image Figure 3.69 shows six nuclear medicine (planar) images of the chest of a patient in the gated blood-pool mode of imaging the heart using 99m Tc Frame (a) displays the left ventricle in its fully relaxed state (at the end of diastole) The subsequent frames show the left ventricle at various stages of contraction through one cardiac cycle Frame (d) shows the left ventricle at its smallest, being fully contracted in systole Frame (f) completes the cycle at a few milliseconds before the end of diastole Each frame represents the sum of 16 gated frames acquired over 16 cardiac cycles The ECG signal was used to time photon counting such that each frame gets the photon counts acquired during an interval equal to 161 th of the duration of each heart beat, at exactly the same phase of the cardiac cycle see Figure 3.70 This procedure is akin to synchronized averaging, with the di erence that the procedure may be considered to be integration instead of averaging the net result is the same except for a scale factor (Clinical imaging systems not provide the data corresponding to the intervals of an individual cardiac cycle, but provide only the images integrated over 16 or 32 cardiac cycles.) 3.11 Remarks In this chapter, we have studied several types of artifacts that could arise in biomedical images, and developed a number of techniques to characterize, model, and remove them Starting with simple averaging over multiple image frames or over small neighborhoods within an image, we have seen how statistical parameters may be used to lter noise of di erent types We have also examined frequency-domain derivation and application of lters The class of lters based upon mathematical morphology 8, 192, 220, 221, 222] has not been dealt with in this book The analysis of the results of several lters has demonstrated the truth behind the adage prevention is better than cure : attempts to remove one type of artifact could lead to the introduction of others! Regardless, adaptive and © 2005 by CRC Press LLC 278 Biomedical Image Analysis (a) (b) (c) (d) FIGURE 3.68 128 128 SPECT images of a resolution phantom obtained by counting photons over: (a) s (b) 15 s and (c) 40 s (d) Schematic representation of the section Images courtesy of L.J Hahn, Foothills Hospital, Calgary © 2005 by CRC Press LLC Removal of Artifacts 279 (a) (b) (c) (d) (e) (f) FIGURE 3.69 (a) 64 64 gated blood-pool images at six phases of the cardiac cycle, obtained by averaging over 16 cardiac cycles Images courtesy of L.J Hahn, Foothills Hospital, Calgary © 2005 by CRC Press LLC 280 Biomedical Image Analysis R R P T Q P S T Q S PHOTON COUNT DATA phase phase phase phase FIGURE 3.70 phase phase phase phase Use of the ECG signal in synchronized averaging or accumulation of photon counts in gated blood-pool imaging Two cycles of cardiac activity are shown by the ECG signal The ECG waves have the following connotation: P | atrial contraction QRS | ventricular contraction (systole) T | ventricular relaxation (diastole) Eight frames representing the gated images are shown over each cardiac cycle Counts over the same phase of the cardiac cycle are added to the same frame over several cardiac cycles © 2005 by CRC Press LLC Removal of Artifacts 281 nonlinear lters have proven themselves to be useful in removing noise without creating signi cant artifacts Preprocessing of images to remove artifact is an important step before other methods may be applied for further enhancement or analysis of the features in the images Notwithstanding the models that were used in deriving some of the lters presented in this chapter, most of the methods applied in practice for noise removal are considered to be ad hoc approaches: methods that have been shown to work successfully in similar situations encountered by other researchers are tried to solve the problem on hand Di culties arise due to the fact that some of the implicit models and assumptions may not apply well to the image or noise processes of the current problem For this reason, it is common to try several previously established techniques However, the assessment of the results of ltering operations and the selection of the most appropriate lter for a given application remain a challenge Although several measures of image quality are available (see Chapter 2), visual assessment of the results by a specialist in the area of application may remain the most viable approach for comparative analysis and selection 3.12 Study Questions and Problems (Note: Some of the questions may require background preparation with other sources on the basics of signals and systems as well as digital signal and image processing, such as Lathi 1], Oppenheim et al 2], Oppenheim and Schafer 7], Gonzalez and Woods 8], Pratt 10], Jain 12], Hall 9], and Rosenfeld and Kak 11].) Selected data les related to some of the problems and exercises are available at the site www.enel.ucalgary.ca/People/Ranga/enel697 Explain the di erences between linear and circular (or periodic) convolution Given two 1D signals h(n) = 1] and f (n) = 1] for n = 2, compute by hand the results of linear and circular convolution Demonstrate how the result of circular convolution may be made equivalent to that of linear convolution in the above example The two 1D signals f = 4]T and h = ;1]T (given as vectors) are to be convolved Prepare a circulant matrix h such that the matrix product g = h f will provide results equivalent to the linear convolution g = h f of the two signals What are the impulse response and frequency response (transfer function or MTF) of the 3 mean lter? An image is processed by applying the 3 mean lter mask (a) once, (b) twice, and (c) thrice in series What are the impulse responses of the three operations? © 2005 by CRC Press LLC 282 Biomedical Image Analysis Perform linear 2D convolution of the following two images: 3 42 15 (3.174) and 12 66 77 (3.175) 45 5 15 : 12 The image 3 13 66 77 66 5 77 (3.176) 44 15 12 is passed through a linear shift-invariant system having the impulse response 3 40 15 : (3.177) Compute the output of the system The output of a lter at (m n) is de ned as the average of the four immediate neighbors of (m n) the pixel at (m n) is itself not used Derive the MTF of the lter and describe its characteristics P The Fourier transform of a 1D periodic train of impulses, represented as +1 n=;1 (t ; nT ), where T is the period or interval between the impulses, is given by another periodic train of impulses in the frequency domain as !0 P+n=1;1 (! ; n!0 ), where !0 = 2T Using the information provided above, derive the 2D Fourier transform of an image made up of a periodic array of strips parallel to the x axis The thickness of each strip is W , the spacing between the strips is S , and the image is of size A B , with fA B g fW S g Draw a schematic sketch of the spectrum of the image A 3 window of a noisy image contains the following pixel values: 52 59 41 62 74 66 : (3.178) 56 57 59 Compute the outputs of the 3 mean and median lters for the pixel at the center of the window Show all steps in your computation © 2005 by CRC Press LLC Removal of Artifacts 283 10 A digital image contains periodic grid lines in the horizontal and vertical directions with a spacing of cm The sampling interval is mm, and the size of the image is 20 cm 20 cm The spectrum of the image is computed using the FFT with an array size of 256 256, including zero-padding in the area not covered by the original image Sketch a schematic diagram of the spectrum of the image, indicating the nature and exact locations of the frequency components of the grid lines Propose a method to remove the grid lines from the image 11 In deriving the Wiener lter, it is assumed that the processes generating the image f and noise are statistically independent of each other, that the mean of the noise process is zero, and that both the processes are secondorder stationary A degraded image is observed as g = f + : The following expression is encountered for the MSE between the Wiener estimate ~f = Lg and the original image f : h n oi "2 = E Tr (f ; ~f )(f ; ~f )T : (3.179) Reduce the expression above to one containing L and autocorrelation matrices only Give reasons for each step of your derivation 3.13 Laboratory Exercises and Projects Add salt-and-pepper noise to the image shapes.tif Apply the median lter using the neighborhood shapes given in Figure 3.14, and compare the results in terms of MSE values and edge distortion From your collection of test images, select two images: one with strong edges of the objects or features present in the image, and the other with smooth edges and features Prepare several noisy versions of the images by adding (i) Gaussian noise, and (ii) salt-and-pepper noise at various levels Filter the noisy images using (i) the median lter with the neighborhoods given in Figure 3.14 (a), (b), and (k) and (ii) the 3 mean lter with the condition that the lter is applied only if the di erence between the pixel being processed and the average of its 8connected neighbors is less than a threshold Try di erent thresholds and study the e ect Compare the results in terms of noise removal, MSE, and the e ect of the lters on the edges present in the images © 2005 by CRC Press LLC 284 Biomedical Image Analysis Select two of the noisy images from the preceding exercise Apply the ideal lowpass lter and the Butterworth lowpass lter using two di erent cuto frequencies for each lter Study the results in terms of noise removal and the e ect of the lters on the sharpness of the edges present in the images © 2005 by CRC Press LLC [...]... angiography Physiological interference may not be characterized by any speci c waveform, pattern, or spectral content, and is typically dynamic and nonstationary (varying with the level of the activity of relevance and hence with time see Section 3.1.6 for a discussion on stationarity) Thus, simple, linear bandpass lters will usually not be e ective in removing physiological interference © 2005 by CRC... may be extended to the imaging of dynamic systems whose movements follow a rhythm or cycle with phases that can be determined by another signal, such as the cardiac system whose phases of contraction are indicated by the ECG signal Then, several image frames may be acquired at the same phase of the rhythmic movement over successive cycles, and averaged to reduce noise Such a process is known as synchronized... said to be weakly stationary or stationary in the wide sense if its mean is a constant and its ACF depends only upon the di erence (or shift) in time or space Then, we have f (x1 y1 ) = f and f (x1 x1 + y1 y1 + ) = f ( ) The ACF is now a function of the shift parameters and only the PSD of the process does not vary with space © 2005 by CRC Press LLC Removal of Artifacts 167 A stationary process is said... designed for stationary signals may then be extended and applied to nonstationary signals Analysis of signals by this approach is known as shorttime analysis 31, 176] On the same token, the characteristics of the features in an image vary over relatively large scales of space statistical parameters within small regions of space, within an object, or within an organ of a given type may be assumed to remain... image may then be assumed to be block-wise stationary, which permits sectioned or block-by-block processing or moving-window processing using techniques designed for stationary processes 177, 178] Figure 3.11 illustrates the notion of computing statistics within a moving window Certain systems, such as the cardiac system, normally perform rhythmic operations Considering the dynamics of the cardiac system,... events extracted from an observation of the signal over many cycles The cyclical nature of cardiac activity may be exploited for synchronized averaging to reduce noise and improve the SNR of the ECG and PCG 31] The © 2005 by CRC Press LLC 168 Biomedical Image Analysis # @ FIGURE 3.11 Block-by-block processing of an image Statistics computed by using the pixels within the window shown with solid lines... Biomedical Image Analysis Normal anatomical details such as the ribs in chest X-ray images and the skull in brain imaging may also be considered to be artifacts when other details in such images are of primary interest Methods may need to be developed to remove their e ects before the details of interest may be analyzed 3.1.5 Other types of noise and artifact Systematic errors are caused by several factors... acquired.) © 2005 by CRC Press LLC 172 Biomedical Image Analysis Synchronized averaging is a useful technique in the acquisition of several biomedical signals 31] Observe that averaging as above is a form of ensemble averaging Let us represent a single image frame in a situation as above as gi (x y) = f (x y) + i (x y) (3. 33) where gi (x y) is the ith observed frame of the image f (x y) , and i (x y) is the... Then, it follows that 8] E g(x y) ] = f (x y) (3.35) and 1 2 2 (3.36) g(x y) = M (x y) : Thus, the variance at every pixel in the averaged image is reduced by apfactor of M1 from that in a single frame the SNR is improved by the factor M The most important requirement in this procedure is that the frames being averaged be mutually synchronized, aligned, or registered Any motion, change, or displacement... is not commonly applied to spatial functions Signals or processes that do not meet the conditions described above may, in general, be called nonstationary processes A nonstationary process possesses statistics that vary with time or space The statistics of most images vary over space indeed, such variations are the source of pictorial information Most biomedical systems are dynamic systems and produce