Bản chất của hình ảnh y sinh học (Phần 8)

8 Analysis of Oriented Patterns Many images are composed of piecewise linear objects Linear or oriented objects possess directional coherence that can be quanti ed and examined to assess the underlying pattern An area that is closely related to directional image processing is texture identi cation and segmentation For example, given an image of a human face, a method for texture segmentation would attempt to separate the region consisting of hair from the region with skin, as well as other regions such as the eyes that have a texture that is di erent from that of either the skin or hair In texture segmentation, a common approach for identifying the di ering regions is via nding the dominant orientation of the di erent texture elements, and then segmenting the image using this information The subject matter of this chapter is more focused, and concerned with issues of whether there is coherent structure in regions such as the hair or skin To put it simply, the question is whether the hair is combed or not, and if it is not, the degree of disorder is of interest, which we shall attempt to quantify Directional analysis is useful in the e ective identi cation, segmentation, and characterization of oriented (or weakly ordered) texture 432] 8.1 Oriented Patterns in Images In most cases of natural materials, strength is derived from highly coherent, oriented bers an example of such structure is found in ligaments 35, 36] Normal, healthy ligaments are composed of bundles of collagen brils that are coherently oriented along the long axis of the ligament see Figure 1.8 (a) Injured and healing ligaments, on the other hand, contain scabs of scar material that are not aligned Thus, the determination of the relative disorder of collagen brils could provide a direct indicator of the health, strength, and functional integrity (or lack thereof) of a ligament 35, 36, 37, 531] similar patterns exist in other biological tissues such as bones, muscle bers, and blood vessels in ligaments as well 414, 415, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543] Examples of oriented patterns in biomedical images include the following: Fibers in muscles and ligaments see Figure 8.22 © 2005 by CRC Press LLC 639 640 Biomedical Image Analysis Fibroglandular tissue, ligaments, and ducts in the breast see Figures 7.2 and 8.66 Vascular networks in ligaments, lungs, and the heart see Figures 9.20 and 8.27 Bronchial trees in the lungs see Figure 7.1 Several more examples are presented in the sections to follow In man-made materials such as paper and textiles, strength usually relies upon the individual bers uniformly knotting together Thus, the strength of the material is directly related to the organization of the individual bril strands 544, 545, 546, 547, 548, 549] Oriented patterns have been found to bear signi cant information in several other applications of imaging and image processing In geophysics, the accurate interpretation of seismic soundings or \stacks" is dependent upon the elimination of selected linear segments from the stacks, primarily the \ground roll" or low-frequency component of a seismic sounding 550, 551, 552] Thorarinsson et al 553] used directional analysis to discover linear anomalies in magnetic maps that represent tectonic features In robotics and computer vision, the detection of the objects in the vicinity and the determination of their orientation relative to the robot are important in order for the machine to function in a nonstandard environment 554, 555, 556] By using visual cues in images, such as the dominant orientation of a scene, robots may be enabled to identify basic directions such as up and down Information related to orientation has been used in remote sensing to analyze satellite maps for the detection of anomalies in map data 557, 558, 559, 560, 561, 562] Underlying structures of the earth are commonly identi ed by directional patterns in satellite images for example, ancient river beds 557] Identifying directional patterns in remotely sensed images helps geologists to understand the underlying processes in the earth that are in action 553, 562] Because man-made structures also tend to have strong linear segments, directional features can help in the identi cation of buildings, roads, and urban features 561] Images commonly have sharp edges that make them nonstationary Edges render image coding and compression techniques such as LP coding and DPCM (see Chapter 11) less e cient By dividing the frequency space into directional bands that contain the directional image components in each band, and then coding the bands separately, higher rates of compression may be obtained 563, 564, 565, 566, 567, 568, 569] In this manner, directional ltering can be useful in other applications of image processing, such as data compression © 2005 by CRC Press LLC Analysis of Directional Patterns 641 8.2 Measures of Directional Distribution Mardia 570] pointed out that the statistical measures that are commonly used for the analysis of data points in rectangular coordinate systems may lead to improper results if applied to circular or directional data Because we not usually consider directional components in images to be directed elements (or vectors), there should be no need to di erentiate between components that are at angles and 180o therefore, we could limit our analysis to the o semicircular space of 180o ] or ;90o 90o ] 8.2.1 The rose diagram The rose diagram is a graphical representation of directional data Corresponding to each angular interval or bin, a sector (a petal of the rose) is plotted with its apex at the origin In common practice, the radius of the sector is made proportional to the area of the image components directed in the corresponding angle band The area of each sector in a rose diagram as above varies in proportion to the square of the directional data In order to make the areas of the sectors directly proportional to the orientation data, the square roots of the data elements could be related to the radii of the sectors Linear histograms conserve areas and are comparatively simple to construct however, they lack the strong visual association with directionality that is obtained through the use of rose diagrams Several examples of rose diagrams are provided in the sections to follow 8.2.2 The principal axis The spatial moments of an image may be used to determine its principal axis, which could be helpful in nding the dominant angle of directional alignment The moment of inertia of an image f (x y) is at its minimum when the moment is taken about the centroid (x y) of the image The moment of inertia of the image about the line (y ; y ) cos = (x ; x) sin passing through (x y) and having the slope tan is given by m = Z Z x y (x ; x) sin ; (y ; y ) cos ]2 f (x y) dx dy: (8.1) In order to make m independent of the choice of the coordinates, the centroid of the image could be used as the origin Then, x = and y = 0, and Equation 8.1 becomes m = Z Z x y (x sin © 2005 by CRC Press LLC ; y cos )2 f (x y) dx dy 642 Biomedical Image Analysis = m20 sin2 ; m11 sin cos + m02 cos2 where mpq is the (p q)th moment of the image, given by mpq = Z Z x y xp yq f (x y) dx dy: (8.2) (8.3) By de nition, the moment of inertia about the principal axis is at its minimum Di erentiating Equation 8.2 with respect to and equating the result to zero gives (8.4) m20 sin ; m11 cos ; m02 sin = or 11 : (8.5) tan = (m m ;m ) 20 02 By solving this equation, we can nd the slope or the direction of the principal axis of the given image 11] If the input image consists of directional components along an angle only, then If there are a number of directional components at di erent angles, then represents their weighted average direction Evidently, this method cannot detect the existence of components in various angle bands, and is thus inapplicable for the analysis of multiple directional components Also, this method cannot quantify the directional components in various angle bands 8.2.3 Angular moments The angular moment Mk of order k of an angular distribution is de ned as Mk = N X n=1 k (n) p(n) (8.6) where (n) represents the center of the nth angle band in degrees, p(n) represents the normalized weight or probability of the data in the nth band, and N is the number of angle bands If we are interested in determining the dispersion of the angular data about their principal axis, the moments may be taken with respect to the centroidal angle = M1 of the distribution Because the second-order moment is at its minimum when taken about the centroid, we could choose k = for statistical analysis of angular distributions Hence, the second central moment M2 may be de ned as M2 = N X n=1 (n) ; ]2 p(n): (8.7) The use of M2 as a measure of angular dispersion has a drawback: because the moment is calculated using the product of the square of the angular distance and the weight of the distribution, even a small component at a large © 2005 by CRC Press LLC Analysis of Directional Patterns 643 angular distance from the centroidal angle could result in a high value for M2 (See also Section 6.2.2.) 8.2.4 Distance measures The directional distribution obtained by a particular method for an image may be represented by a vector p1 = p1 (1) p1 (2) p1 (N )]T , where p1 (n) th represents the distribution in the n angle band The true distribution of the image, if known, may be represented by another vector p0 Then, the Euclidean distance between the distribution obtained by the directional analysis method p1 and the true distribution of the image p0 is given as v u N uX p1 (n) ; p0 (n)]2: kp1 ; p0 k = t n=1 (8.8) This distance measure may be used to compare the accuracies of di erent methods of directional analysis Another distance measure that is commonly used is the Manhattan distance, de ned as jp1 ; p0 j = N X n=1 jp1 (n) ; p0 (n)j: (8.9) The distance measures de ned above may also be used to compare the directional distribution of one image with that of another 8.2.5 Entropy The concept of entropy from information theory 127] (see Section 2.8) can be e ectively applied to directional data If we take p(n) as the directional PDF of an image in the nth angle band, the entropy H of the distribution is given by H =; N X n=1 p(n) log2 p(n)]: (8.10) Entropy provides a useful measure of the scatter of the directional elements in an image If the image is composed of directional elements with a uniform distribution (maximal scatter), the entropy is at its maximum if, however, the image is composed of directional elements oriented at a single angle or in a narrow angle band, the entropy is (close to) zero Thus, entropy, while not giving the angle band of primary orientation or the principal axis, could give a good indication of the directional spread or scatter of an image 35, 36, 414, 415] (See Figure 8.24.) Other approaches that have been followed by researchers for the characterization of directional distributions are: numerical and statistical characterization of directional strength 535], morphological operations using a rotating © 2005 by CRC Press LLC 644 Biomedical Image Analysis structural element 541], laser small-angle light scattering 538, 539, 549], and optical di raction and Fourier analysis 532, 548, 558, 560] 8.3 Directional Filtering Methods based upon the Fourier transform have dominated the area of directional image processing 36, 532, 550, 551, 552] The Fourier transform of an oriented linear segment is a sinc function oriented in the direction orthogonal to that of the original segment in the spatial domain see Figure 8.1 Based upon this property, we can design lters to select linear components at speci c angles However, a di culty in using the Fourier domain for directional ltering lies in the development of high-quality lters that are able to select linear components without the undesirable e ects of ringing in the spatial domain Schiller et al 571] showed that the human eye contains orientation-selective structures This motivated research on human vision by Marr 282], who showed that the orientation of linear segments, primarily edges, is important in forming the primal sketch Several researchers, including Kass and Witkin 572], Zucker 573], and Low and Coggins 574] used oriented bandpass lters in an e ort to simulate the human visual system's ability to identify oriented structures in images Allen et al 575] developed a very-large-scale integrated (VLSI) circuit implementation of an orientation-speci c \retina" Several researchers 36, 572, 573, 574] have used many types of simple lters with wide passbands at various angles to obtain a redundant decomposition or representation of the given image Such representations were used to derive directional properties of the image For example, Kass and Witkin 572] formed a map of ow lines in the given image, and under conformal mapping, obtained a transformation to regularize the ow lines onto a grid The resulting transformation was used as a parameter representing the texture of the image In this manner, various types of texture could be recognized or generated by using the conformal map speci c to the texture Chaudhuri et al 36] used a set of bandpass lters to obtain directional components in SEM images of ligaments however, the lter used was relatively simple (see Sections 8.3.1 and 8.7.1) Generating highly selective lters in 2D is not trivial, and considerable research has been directed toward nding general rules for the formation of 2D lters Bigun et al 576] developed rules for the generation of least-squares optimal beam lters in multiple dimensions Bruton et al 577] developed a method for designing high-quality fan lters using methods from circuit theory This method results in 2D recursive lters that have high directional selectivity and good roll-o characteristics, and is described in Section 8.3.3 © 2005 by CRC Press LLC Analysis of Directional Patterns 645 (a) (b) (c) (d) FIGURE 8.1 (a) A test image with a linear feature (b) Log-magnitude Fourier spectrum of the test image in (a) (c) Another test image with a linear feature at a di erent angle (d) Log-magnitude Fourier spectrum of the test image in (b) See also Figure 2.30 © 2005 by CRC Press LLC 646 Biomedical Image Analysis 8.3.1 Sector ltering in the Fourier domain Fourier-domain techniques are popular methods for directional quanti cation of images 36, 532, 547, 550, 551, 552, 553, 557, 558, 559, 560, 562, 564, 565, 566, 567, 568, 569, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589] The results of research on biological visual systems provide a biological base for directional analysis of images using lter-based methods 389, 563, 571, 572, 573, 575] The Fourier transform is the most straightforward method for identifying linear components The Fourier transform of a line segment is a sinc function oriented at =2 radians with respect to the direction of the line segment in the spatial domain see Figure 8.1 This fact allows the selective ltering of line segments at a speci c orientation by ltering the transformed image with a bandpass lter Consider a line segment of orientation (slope) a and y-axis intercept b in the (x y) plane, with the spatial limits ;X X ] and ;Y Y ] In order to obtain the Fourier transform of the image, we could evaluate a line integral in 2D along the line y = ax + b To simplify the procedure, let us assume that the integration occurs over a square region with X = Y Because the function f (x y) is a constant along the line, the term f (x y) in the Fourier integral can be normalized to unity, giving the equation f (x y) = along the line y = ax + b Making the substitution x = (y ; b)=a, we have the Fourier transform of the line image given by F (u v) = ja1j ZY ZY ;Y ;Y exp ;j u (y ;a b) + v y h u i sinc + v Y : = 2jaYj exp j bu a a dy dy (8.11) From the result above, we can see that, for the image of a line, the Fourier transform is a sinc function with an argument that is a linear combination of the two frequency variables (u v), and with a slope that is the negative reciprocal of the slope of the original line The intercept is translated into a phase shift of b=a in the u variable Thus, the Fourier transform of the line is a sinc function oriented at 90o to the original line, centered about the origin in the frequency domain regardless of the intercept of the original line This allows us to form lters to select lines solely on the basis of orientation and regardless of the location in the space domain Spatial components in a certain angle band may thus be obtained by applying a bandpass lter in an angle band perpendicular to the band of interest and applying the inverse transform If we include a spatial o set in the above calculation, it would only result in a phase shift the magnitude spectrum would remain the same Figure 8.2 illustrates the ideal form of the \fan" lter that may used to select oriented segments in the Fourier domain © 2005 by CRC Press LLC Analysis of Directional Patterns 647 FIGURE 8.2 Ideal fan lter in the Fourier domain to select linear components oriented between +10o and ;10o in the image plane Black represents the stopband and white represents the passband The origin (u v) = (0 0) is at the center of the gure Prior to the availability of high-speed digital processing systems, attempts at directional ltering used optical processing in the Fourier domain Arsenault et al 558] used optical bandpass lters to selectively lter contour lines in aeromagnetic maps Using optical technology, Duvernoy and Chalasinska-Macukow 560] developed a directional sampling method to analyze images the method involved integrating along an angle band of the Fourier-transformed image to obtain the directional content This method was used by Dziedzic-Goclawska et al 532] to identify directional content in bone tissue images The need for specialized equipment and precise instrumentation limits the applicability of optical processing The essential idea of ltering selected angle bands, however, remains valid as a processing tool, and is the basis of Fourier-domain techniques The main problem with Fourier-domain techniques is that the lters not behave well with occluded components or at junctions of linear components smearing of the line segments occurs, leading to inaccurate results when inverse transformed to the space domain Another problem lies in the truncation artifacts and spectral leakage that can exist when ltering digitally, which leads to ringing in the inverse-transformed image Ringing artifacts may be avoided by e ective lter design, but this, in turn, could limit the spatial angle band to be ltered Considerable research has been reported in the eld of multidimensional signal processing to optimize direction-selective lters 569, 576, 577, 580, 583, 585, 590] Bigun et al 576] addressed the problem of detection of orientation in a least-squares sense with FIR bandpass lters This method has the added bene t of being easily implementable in the space domain Bruton © 2005 by CRC Press LLC 648 Biomedical Image Analysis et al 577] proposed guidelines for the design of stable IIR fan lters Hou and Vogel 569] developed a novel method of using the DCT for directional ltering: this method uses the fact that the DCT divides the spectrum into an upper band and a lower band In 2D, such band-splitting divides the frequency plane into directional lter bands By selecting coe cients of the DCT, the desired spectral components can be obtained Because the DCT has excellent spectral reconstruction qualities, this results in high-quality, directionally selective lters A limitation of this technique is that the method only detects the bounding edges of the directional components, because the band-splitting in the DCT domain does not include the DC component of the directional elements In the method developed by Chaudhuri et al 36], a simple decomposition of the spectral domain into 12 equal angle bands was employed, at 15o per angle band Each sector lter in this design is a combination of an ideal fan lter, a Butterworth bandpass lter, a ramp-shaped lowpass lter, and a raised cosine window as follows: H (fr ) = (1 ; fr ) 2p + ffLr 2q + ffHr 1=2 cos ; o B (8.12) where = 0p:7 = u2 + v2 =6 =4 = 0:5 = 0:02 = angle of the Fourier transform sample = atan(v=u) o = central angle of the desired angle band B = angular bandwidth, and = weighting factor = 0:5: = slope of the weighting function fr = normalized radial frequency p = order of the highpass lter q = order of the lowpass lter fH = upper cuto frequency (normalized) fL = lower cuto frequency (normalized) The combined lter with = 135o and B = 15o is illustrated in Figure 8.3 Filtering an image with sector lters as above results in 12 component images Each component image contains the linear components of the original image in the corresponding angle band Although the directional lter was designed to minimize spectral leakage, some ringing artifacts were observed in the results To minimize the artifacts, a thresholding method was applied to accentuate the linear features in the image Otsu's thresholding algorithm 591] (see Section 8.3.2) was applied in the study of collagen ber images by Chaudhuri et al 36] © 2005 by CRC Press LLC 782 Biomedical Image Analysis The cosine term has a period of therefore, fo = 1= The value of y was de ned as y = l x , where l determines the elongation of the Gabor lter in the orientation direction, with respect to its thickness The values = pixels (corresponding to a thickness of 0:8 mm at a pixel size of 200 m) and l = were determined empirically, by observing the typical spicule width and length in mammograms with architectural distortion in the Mini-MIAS database 376] The e ects of the di erent design parameters are shown in Figure 8.68, and are as follows: Figures 8.68 (a) and (e) show the impulse response of a Gabor lter and its Fourier transform magnitude, respectively In Figure 8.68 (b), the Gabor lter of Figure 8.68 (a) is stretched in the x direction, by increasing the elongation factor l Observe that the Fourier spectrum of the new Gabor lter, shown in Figure 8.68 (f), is compressed in the horizontal direction The Gabor lter shown in Figure 8.68 (c) was obtained by increasing the parameter of the original Gabor lter, thus enlarging the lter in both the x and y directions Correspondingly, the Fourier spectrum of the enlarged lter, shown in Figure 8.68 (g), has been shrunk in both the vertical and horizontal directions The e ect of rotating the Gabor lter by 30o counterclockwise is displayed in Figures 8.68 (d) and (h), that show the rotated Gabor lter's impulse response and the corresponding Fourier spectrum The texture orientation at a pixel was estimated as the orientation of the Gabor lter that yielded the highest magnitude response at that pixel The orientation at every pixel was used to compose the orientation eld The magnitude of the corresponding lter response was used to form the magnitude image The magnitude image was not used in the estimation of the phase portrait, but was found to be useful for illustrative purposes Let (x y) be the texture orientation at (x y), and gk (x y), k = 179, be the Gabor lter oriented at k = ; =2 + k=180 Let f (x y) be the ROI of the mammogram being processed, and fk (x y) = (f gk )(x y), where the asterisk denotes linear 2D convolution Then, the orientation eld of f (x y) is given by (x y) = kmax where kmax = argfmax jf (x y)j]g : (8.94) k k 8.10.4 Characterizing orientation elds with phase portraits In the work of Ayres and Rangayyan 595, 679, 680], the analysis of oriented texture patterns was performed in a two-step process First, the orientation © 2005 by CRC Press LLC Analysis of Directional Patterns 783 (a) (b) (c) (d) (e) (f) (g) (h) FIGURE 8.68 E ects of the di erent parameters of the Gabor lter (a) Example of the impulse response of a Gabor lter (b) The parameter l is increased: the Gabor lter is elongated in the x direction (c) The parameter is increased: the Gabor lter is enlarged in the x and y directions (d) The angle of the Gabor lter is modi ed Figures (e) { (h) correspond to the magnitude of the Fourier transforms of the Gabor lters in (a) { (d), respectively The (0 0) frequency component is at the center of the spectra displayed Reproduced with permission from F.J Ayres and R.M Rangayyan, \Characterization of architectural distortion in mammograms via analysis of oriented texture", IEEE Engineering in Medicine and Biology Magazine, January 2005 c IEEE © 2005 by CRC Press LLC 784 Biomedical Image Analysis eld (x y) of the ROI was computed in a small analysis window The sliding analysis window was centered at pixels within the ROI, avoiding window positions with incomplete data at the edges of the ROI for the estimation of A and b Second, the matrix A and the vector b in Equation 8.90 were estimated such that the best match was achieved between (x y) and (x yjA b) The eigenvalues of A determine the type of the phase portrait present in (x y) the xed point of the phase portrait is given by Equation 8.91 The estimates of A and b were obtained as follows For every point (x y), let (x y) = sin (x y) ; (x yjA b)] represent the error between the orientation of the texture given by Equation 8.94 and the orientation of the model given by Equation 8.92 Then, the estimation problem is that of nding A and b that minimize the sum of the squared error = XX x y (x y) = XX x y fsin (x y) ; (x yjA b)]g2 (8.95) which may be solved using a nonlinear least-squares algorithm 737] The ROI was investigated by sliding the analysis window through the orientation eld of the ROI, and accumulating the information obtained, that is, the type of the phase portrait and the location of the xed point, for each window position, as follows: Create three maps, one for each type of phase portrait (hereafter called the phase portrait maps ), that will be used to accumulate information from the sliding analysis window The maps are initialized to zero, and are of the same size as the ROI or the image being processed Slide the analysis window through the orientation eld of the ROI At each position of the sliding window, determine the type of the phase portrait and compute the xed point of the orientation eld Increment the value at the location of the xed point in the corresponding phase portrait map The size of the sliding analysis window was set at 44 44 pixels (8:8 8:8 mm) The three maps obtained as above provide the results of a voting procedure, and indicate the possible locations of xed points corresponding to texture patterns that (approximately) match the node, saddle, and spiral phase portraits It is possible that, for some positions of the sliding analysis window, the location of the xed point falls outside the spatial limits of the ROI or image being processed the votes related to such results were ignored The value at each location (x y) in a phase portrait map provides the degree of dence in determining the existence of the corresponding phase portrait type centered at (x y) The three phase portraits were expected to relate to di erent types of architectural distortion © 2005 by CRC Press LLC Analysis of Directional Patterns 785 8.10.5 Feature extraction for pattern classi cation The estimates of the xed-point location for a given phase portrait pattern can be scattered around the true xed-point position, due to the limited precision of the estimation procedure, the presence of multiple overlapping patterns, the availability of limited data within the sliding analysis window, and the presence of noise A local accumulation of the votes is necessary to diminish the e ect of xed-point location errors Ayres and Rangayyan 595, 679, 680] employed a Gaussian smoothing lter with a standard deviation of 25 pixels (5 mm) for this purpose For the purpose of pattern classi cation, six features were extracted to characterize each ROI: the maximum of each phase portrait map (three features), and the entropy of each phase portrait map (three features) The maximum of each map conveys information about the likelihood of the presence of the corresponding phase portrait type, and the entropy relates to the uncertainty in the location of the xed point in each map The entropy H of a map h(x y) was computed as H h(x y)] = ; where X X h(x x y Sh = XX x y Sh y) ln h(x y) Sh h(x y): (8.96) (8.97) A map with a dense spatial concentration of votes is expected to have a large maximum value and a low entropy On the contrary, a map with a wide scatter of votes may be expected to have a low maximum and a large entropy 8.10.6 Application to segments of mammograms Ayres and Rangayyan 595, 679] analyzed a set of 106 ROIs, each of size 230 230 pixels (46 46 mm, with a resolution of 200 m), selected from the Mini-MIAS database 376] The set included 17 ROIs with architectural distortion (all the cases of architectural distortion available in the MIAS database), 45 ROIs with normal tissue patterns, eight ROIs with spiculated malignant masses, four ROIs with circumscribed malignant masses, 11 ROIs with spiculated benign masses, 19 ROIs with circumscribed benign masses, and two ROIs with malignant calci cations The size of the ROIs was chosen to accommodate the largest area of architectural distortion or mass identied in the Mini-MIAS database ROIs related to all of the masses in the database were included The normal ROIs included examples of overlapping ducts, ligaments, and other parenchymal patterns Only the central portion of 150 150 pixels of each ROI was investigated using a sliding analysis window of size 44 44 pixels the remaining outer ribbon of pixels was not processed in order to discard the e ects of the preceding ltering steps © 2005 by CRC Press LLC 786 Biomedical Image Analysis TABLE 8.5 Results of Linear Discriminant Analysis for ROIs with Architectural Distortion Using the Leave-one-out Method Architectural #ROIs Classi ed as distortion Architectural distortion Other Benign Malignant 2 Total 17 TP = 13 FN = TP = true positives, FN = false negatives The results correspond to the prior probability of belonging to the architectural distortion class being 0:465 Sensitivity = 76:5% Reproduced with permission from F.J Ayres and R.M Rangayyan, \Characterization of architectural distortion in mammograms via analysis of oriented texture", IEEE Engineering in Medicine and Biology Magazine, January 2005 c IEEE Figure 8.69 illustrates the results obtained for an image with architectural distortion (mdb115) The maximum of the node map is larger than the maxima of the other two maps Also, the scattering of votes in the node map is less than that in the saddle and spiral maps These results indicate that the degree of scattering of the votes (quanti ed by the entropy of the corresponding map) and the maximum of each of the three phase portrait maps could be useful features to distinguish between architectural distortion and other patterns Linear discriminant analysis was performed using SPSS 738], with stepwise feature selection Architectural distortion was considered as a positive nding, with all other test patterns (normal tissue, masses, and calci cations) being considered as negative ndings The statistically signi cant features were the entropy of the node map and the entropy of the spiral map: the other features were deemed to be not signi cant by the statistical analysis package, and were discarded in all subsequent analysis With the prior probability of architectural distortion set to 50%, the sensitivity obtained was 82:4%, and the speci city was 71:9% The fraction of cases correctly classi ed was 73:6% Tables 8.5 and 8.6 present the classi cation results with the prior probability of architectural distortion being 46:5% An overall classi cation accuracy of 76:4% was achieved 8.10.7 Detection of sites of architectural distortion Ayres and Rangayyan 595, 679, 680] hypothesized that architectural distortion would appear as an oriented texture pattern that can be locally approximated by a linear phase portrait model furthermore, it was expected that the © 2005 by CRC Press LLC Analysis of Directional Patterns 787 (a) (b) (c) (d) (e) (f) FIGURE 8.69 Analysis of the ROI from the image mdb115, which includes architectural distortion: (a) ROI of size 230 230 pixels (46 46 mm) (b) magnitude image (c) orientation eld superimposed on the original ROI (d) node map, with intensities mapped from 123] to 255] (e) saddle map, 22] mapped to 255] (f) spiral map, 71] mapped to 255] This image was correctly classi ed as belonging to the \architectural distortion" category (Table 8.5) Reproduced with permission from F.J Ayres and R.M Rangayyan, \Characterization of architectural distortion in mammograms via analysis of oriented texture", IEEE Engineering in Medicine and Biology Magazine, January 2005 c IEEE © 2005 by CRC Press LLC 788 Biomedical Image Analysis TABLE 8.6 Results of Linear Discriminant Analysis for ROIs Without Architectural Distortion Using the Leave-one-out Method Type #ROIs Classi ed as Architectural distortion Other CB SB CM SM 19 11 3 15 Calci cations 1 Normal 45 36 Total 89 FP = 21 Masses TN = 68 CB = circumscribed benign mass, CM = circumscribed malignant tumor, SB = spiculated benign mass, SM = spiculated malignant tumor, FP = false positives, TN = true negatives The results correspond to the prior probability of belonging to the architectural distortion class being 0:465 Speci city = 76:4% Reproduced with permission from F.J Ayres and R.M Rangayyan, \Characterization of architectural distortion in mammograms via analysis of oriented texture", IEEE Engineering in Medicine and Biology Magazine, January 2005 c IEEE © 2005 by CRC Press LLC Analysis of Directional Patterns 789 xed-point location of the phase portrait model would fall within the breast area in the mammogram Then, the numbers of votes cast at each position of the three phase portrait maps would indicate the likelihood that the position considered is a xed point of a node, a saddle, or a spiral pattern Before searching the maps for sites of distortion, the orientation eld was ltered and downsampled as follows Let h(x y) be a Gaussian lter of standard deviation h , de ned as 2 h(x y) = exp ; 21 x +2 y : (8.98) h h De ne the images s(x y) = sin (x y)] and c(x y) = cos (x y)], where (x y) is the orientation eld Then, the ltered orientation eld f (x y) is obtained as h(x y) s(x y) (8.99) f (x y ) = arctan h(x y ) c(x y ) where the asterisk denotes 2D convolution The ltered orientation eld was downsampled by a factor of four, thus producing the downsampled orientation eld d as d (x y ) = f (4x 4y ): (8.100) The ltering and downsampling procedures, summarized in Figure 8.70, were applied in order to reduce noise and and also to reduce the computational e ort required for the processing of full mammograms The ltering procedure described above is a variant of Rao's dominant local orientation method 432]: a Gaussian lter has been used instead of a box lter The following procedure was used to detect and locate sites of architectural distortion, using only the node map: The node map is ltered with a Gaussian lter of standard deviation equal to 1:0 pixel (0:8 mm) The ltered node map is thresholded (with the same threshold value for all images) The thresholded image is subjected to the following series of morphological operations to group positive responses that are close to one another, and to reduce each region of positive response to a single point The resulting points indicate the detected locations of architectural distortion (a) A closing operation is performed to group clusters of points that are less than mm apart The structural element is a disk of radius 10 pixels (8 mm) (b) A \close holes" lter is applied to the image The resulting image includes only compact regions © 2005 by CRC Press LLC 790 Biomedical Image Analysis FIGURE 8.70 Filtering and downsampling of the orientation eld Figure courtesy of F.J Ayres (c) The image is subjected to a \shrink" lter, where each compact region is shrunk to a single pixel The threshold value in uences the sensitivity of the method and the number of false positives per image A high threshold value reduces the number of false positives, but also reduces the sensitivity A low threshold value increases the number of false positives The method was applied to 18 mammograms exhibiting architectural distortion, selected from the Mini-MIAS database 376] The mammograms were MLO views, digitized to 024 024 pixels at a resolution of 200 m and b=pixel Figures 8.71 and 8.72 illustrate the steps of the method, as applied to image mdb115 Observe that the ltered orientation eld Figure 8.71 (d)] is smoother and more coherent as compared to the original orientation eld Figure 8.71 (c)]: the pattern of architectural distortion is displayed better in the ltered orientation eld The architectural distortion present in the mammogram mdb115 has a stellate or spiculated appearance As a consequence, a large number of votes have been cast into the node map, at a location close to the center of the distortion, as seen in Figure 8.72 (c) Another point of accumulation of votes in the node map is observed in Figure 8.72 (c), at the location of the nipple This is not unexpected: the breast has a set of ducts that carry milk to the nipple the ducts appear in mammograms as linear structures converging to the nipple Observe that the node map has the strongest response of all maps, © 2005 by CRC Press LLC Analysis of Directional Patterns Figure 8.71 (a) 791 (b) within the site of architectural distortion given by the Mini-MIAS database The technique has resulted in the identi cation of two locations as sites of architectural distortion: one true positive and one false positive, as shown in Figure 8.72 (d) The free-response receiver operating characteristics (FROC) was derived by varying the threshold level in the detection step the result is shown in Figure 8.73 (See Section 12.8.1 for details on ROC analysis.) A sensitivity of 88% was obtained at 15 false positives per image Further work is required in order to reduce the number of false positives and improve the accuracy of detection 8.11 Remarks Preferred orientation and directional distributions relate to the functional integrity of several types of tissues and organs changes in such patterns could indicate structural damage as well as recovery Directional analysis could, © 2005 by CRC Press LLC 792 Biomedical Image Analysis (c) FIGURE 8.71 (d) (a) Image mdb115 from the Mini-MIAS database 376] The circle indicates the location and the extent of architectural distortion, as provided in the MiniMIAS database 376] (b) Magnitude image after Gabor ltering (c) Orientation eld superimposed on the original image Needles have been drawn for every fth pixel (d) Filtered orientation eld superimposed on the original image Reproduced with permission from F.J Ayres and R.M Rangayyan, \Detection of architectural distortion in mammograms using phase portraits", Proceedings of SPIE Medical Imaging 2004: Image Processing, Volume 5370, pp 587 { 597, 2004 c SPIE See also Figure 8.72 © 2005 by CRC Press LLC Analysis of Directional Patterns Figure 8.72 (a) © 2005 by CRC Press LLC 793 (b) 794 Biomedical Image Analysis (c) FIGURE 8.72 (d) Phase portrait maps derived from the orientation eld in Figure 8.71 (d), and the detection of architectural distortion (a) Saddle map: values are scaled from the range 20] to 255] (b) Spiral map: values are scaled from the range 47] to 255] (a) Node map: values are scaled from the range 84] to 255] (d) Detected sites of architectural distortion superimposed on the original image: the solid line indicates the location and spatial extent of architectural distortion as given by the Mini-MIAS database 376] the dashed lines indicate the detected sites of architectural distortion (one true positive and one false positive) Reproduced with permission from F.J Ayres and R.M Rangayyan, \Detection of architectural distortion in mammograms using phase portraits", Proceedings of SPIE Medical Imaging 2004: Image Processing, Volume 5370, pp 587 { 597, 2004 c SPIE © 2005 by CRC Press LLC Analysis of Directional Patterns 795 0.9 0.8 Sensitivity 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIGURE 8.73 10 15 False positives / image 20 Free-response receiver operating characteristics (FROC) curve for the detection of sites of architectural distortion Reproduced with permission from F.J Ayres and R.M Rangayyan, \Detection of architectural distortion in mammograms using phase portraits", Proceedings of SPIE Medical Imaging 2004: Image Processing, Volume 5370, pp 587 { 597, 2004 c SPIE © 2005 by CRC Press LLC 796 Biomedical Image Analysis therefore, be used to study the health and well-being of a tissue or organ, as well as to follow the pathological and physiological processes related to injury, treatment, and healing In this chapter, we explored the directional characteristics of several biomedical images We have seen several examples of the application of fan lters and Gabor wavelets The importance of multiscale or multiresolution analysis in accounting for variations in the size of the pattern elements of interest has been demonstrated In spite of theoretical limitations, the methods for directional analysis presented in this chapter have been shown to lead to practically useful results in important applications 8.12 Study Questions and Problems Selected data les related to some of the problems and exercises are available at the site www.enel.ucalgary.ca/People/Ranga/enel697 Discuss how entropy can characterize a directional distribution Discuss the limitations of fan lters Describe how Gabor functions address these limitations Using an image with line segments of various widths as an example, discuss the need for multiscale or multiresolution analysis 8.13 Laboratory Exercises and Projects Prepare a test image with line segments of di erent directions, lengths, and widths, with overlap Apply Gabor lters at a few di erent scales and angles, as appropriate Evaluate the results in terms of (a) the lengths of the extracted components, and (b) the widths of the extracted components Discuss the limitations of the methods and the artifacts in the results Decompose your test image in the preceding problem using eight sector or fan lters spanning the full range of 0o ; 180o Apply a thresholding technique to binarize the results Compute the area covered by the ltered patterns for each angle band Compare the results with the known areas of the directional patterns Discuss the limitations of the methods and the artifacts in the results © 2005 by CRC Press LLC [...]... an analysis was performed by Daugman 389] The uncertainty relationship in 2D is given by 1 x y u v (8.34) 16 2 where x and y represent the spatial resolution, and u and v represent the frequency resolution The 2D Gabor functions are given as h(x y) = g(x0 y0 ) exp ;j 2 (Ux + V y) ] (x0 y0 ) = (x cos + y sin ;x sin + y cos ) (8.35) where (x0 y0 ) are the (x y) coordinates rotated by an arbitrary angle... we have the discontinuity in the frequency domain at the origin, or DC, to overcome Wavelet analysis is usually applied to identify discontinuities in the spatial domain however, there is a duality in wavelet analysis, provided by the uncertainty principle, that allows discontinuity analysis in the frequency domain as well In order to analyze the discontinuity at DC, large-scale or dilated versions... transform (STFT) for short-time analysis of nonstationary signals 176, 31] The 2D equivalent of the STFT is given by FS (x0 y0 u v) = Z1 Z1 f (x y) w(x ; x0 y ; y0 ) x=;1 y= ;1 exp ;j 2 (ux + vy)] dx dy (8.37) where w is a windowing function and f is the signal (image) to be analyzed The advantage of short-time (or moving-window) analysis is that if the energy of the signal is localized in a particular part... Toeplitz symmetry with reactive elements supplied by inductive elements that satisfy the nonenergic constraint The nonenergic condition ensures that the lter is stable, because it implies that the lter is not adding any energy to the system The maximum amount of energy that is output from the lter is the maximum amount put into the system by the input image The derivation given by Bruton and Bartley 587]... orientation being speci ed by tan;1 (V=U ) see Figure 8.9 These parameters will then completely specify the Gabor lter bank In the directional analysis algorithm proposed by Rolston and Rangayyan 542, 543], only the real component of the Gabor wavelet is used, with = 1=0:6, = 1:0, and the primary orientation given by tan;1 (V=U ) = 0o 45o 90o and 135o A given image is analyzed by convolving band-limited... contracted versions of the wavelet to analyze spatial discontinuities The wavelet basis is given by hx0 y0 1 2 (x y) = p 1 1 2 0 0 h x;x y; y 1 2 (8. 38) where x0 y0 1 and 2 are real numbers, and h is the basic or mother wavelet For large values of 1 and 2 , the basis function becomes a stretched or expanded version of the prototype wavelet or a low-frequency function, whereas for small 1 and 2 , the... to the well-known Heisenberg uncertainty principle of quantum mechanics Gabor showed that complex, sinusoidally modulated, Gaussian basis functions satisfy the lower bound on the fundamental uncertainty principle that governs the resolution in time and frequency, given by t f 41 (8.33) where t and f are time and frequency resolution, respectively The uncertainty principle implies that there is a resolution... high-frequency function The wavelet transform is then de ned as FW (x0 y0 1 2 ) = p Z1 1 1 2 Z1 x=;1 y= ;1 f (x y) 0 0 h x ; x y ; y dx dy: 1 2 (8.39) From this de nition, we can see that wavelet analysis of a signal consists of the contraction, dilation, and translation of the basic mother wavelet, and computing the projections of the resulting wavelets on to the given signal © 2005 by CRC Press LLC... how nely long-term behavior can be identi ed This is re ected in the above-mentioned uncertainty principle, as established by Gabor Gabor originally suggested his kernel function to be used over band-limited, equally spaced areas of the frequency domain, or equivalently, with constant window functions This is commonly referred to as the short-time Fourier transform (STFT) for short-time analysis of... the order of the polynomial P in m and n, respectively The corresponding frequency response function T (u v) is obtained by the substitution of s1 = j 2 u and s2 = j 2 v © 2005 by CRC Press LLC 654 Biomedical Image Analysis The discontinuous requirement in the continuous prototype lter at the origin results in the lter transfer function T (s1 s2 ) having a nonessential singularity of the second kind