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Aristotle University of Thessaloniki
School of Electrical and Computer Engineering
Department of Electronics and Computer Engineering
Appendix A.Textural Features A.1 First-order statistics (FOS) or Statistical Features (SF) A.2 Gray Level Co-occurrence Matrix (GLCM) method or Spatial Gray Level Dependence Matrix (SGLDM) method A.3 Gray level difference statistics (GLDS) method A.4 Neighborhood gray tone difference matrix (NGTDM) method A.5 Statistical feature matrix (SFM) method A.6 Law's texture energy (LTE) method (TEM) A.7 Fractal Dimension Texture Analysis (FDTA) A.8 Gray Level Run Length Matrix (GLRLM) A.9 Fourier power spectrum (FPS) A.10 Shape parameters A.11 High Order Spectra (HOS) on Radon Transform A.12 Local Binary Pattern (LPB) A.13 Gray Level Size Zone Matrix (GLSZM)
Appendix B.Morhpological Features B.1 Multilevel Binary Morhpological Analysis B.2 Gray Scale Morhpological Analysis Appendix C.Histogram Features C.1 Histogram C.2 Multi-region Histogram C.3 Correlogram Appendix D.Multi-scale Features D.1 Discrete Wavelet Transform (DWT) D.2 Stationary Wavelet Transform (SWT) D.3 Wavelet Packets (WP) D.4 Gabor Transform (GT) D.5 Multiresolution Feature Extraction (DWT, SWT, WP, GT) D.6 Amplitude Modulation-Frequency Modulation (AM-FM)
Appendix E.Other Features E.1 Zernikes' Moments E.2 Hu's Moments E.3 Threshold Adjacency Statistis (TAS)
Those features are resolution independent. Let be the grayscale image. The first order histogram is defined as which is the empirical probability density function for single pixels. FOS/SF consists of the following parameters:
- Mean
- Standar Deviation=Variance1/2
- Median
- Mode
- Skewness
- Kurtosis
- Energy
- Entropy
- Minimal Gray Level
- Maximal Grey Level
- Coefficient of Variation
- (10,25,75,90)
- Histogram Width
A2. Gray Level Co-occurrence Matrix (GLCM) method or Spatial Gray Level Dependence Matrix (SGLDM) method
The spatial gray level dependence matrices as proposed by Haralick et al. are based on the estimation of the second-order joint conditional probability density functions. The is the probability that two pixels and with distance in direction specified by the angle have intensities of gray level and gray level . The estimated values for these probability density functions will be denoted by. Haralick et al. proposed the following texture measures that can be extracted from the spatial gray level dependence matrices:
- Angular Second Moment= Energy2
- Contrast
- Correlation
- Sum of Squares: Variance
- Inverse Difference Moment (1st formula)/Homogeneity (2nd formula)
- Sum Average
- Sum Variance
- Sum Entropy
- Entropy
- Difference Variance
- Difference Entropy
- Information Measures of Correlation
For a chosen distance we have four angular SGLDM. Hence, we obtain a set of four values for each of the proceding measures. The mean and range of each of these measures, averaged over the four values, comprise the set of features which are used as input to the classifier.
The GLDS algorithmuses first order statistics of local property values based on absolute differences between pairs of gray levels or of average gray levels in order to extract texture measures.
- Homogeneity
- Contrast
- Energy/ Angular Second Moment
- Entropy
- Mean
NGTDM corresponds to visual properties of texture. L
- Coarseness
- Contrast
- Busyness
- Complexity
- Strength
The statistical feature matrixmeasures the statistical properties of pixel pairs at several distances within an image which are used for statistical analysis. The contrast, covariance, and dissimilarity are defined. A statistical feature matrix, (SFM) , is an matrix whose element is the statistical feature of the image. In a similar way, the contrast matrix , covariance matrix , and dissimilarity matrix can be defined as the matrices whose elements are the contrast, covariance, and dissimilarity, respectively. Based on the SFM, the following texture features can be computed:
- Coarseness
- Contrast
- Periodicity
- Roughness
Law's texture energy measuresare derived from three simple vectors of length 3, , which represent the one-dimensional operations of center-weighted local averaging, symmetric first differencing for edge detection, and second differencing for spot detection. If these vectors are convolved with themselves, new vectors of length 5, are obtained. By further self-convolution, new vectors of length 7, are obtained, where again performs local averaging, acts as edge detector, and acts as spot detector. If the column vectors of length are multiplied by row vectors of the same length, Laws masks are obtained. In order to extract texture features from an image, these masks are convoluted with the image, and the statistics (e.g., energy) of the resulting image are used to describe texture. The following texture features were extracted:
- LL- texture energy from LL kernel
- EE- texture energy from EE kernel
- SS- texture energy from SS kernel
- LE- average texture energy from LE and ELkernels
- ES- average texture energy from ES and SEkernels
- LS- average texture energy from LS and SL kernels
The averaging of matched pairs of energy measures gives rotational invariance.
FDTA describe the roughness of nature surfaces. It considers naturally occurring surfaces as the end result of random walks. Such random walks are basic physical processes in our universe.
A gray level run is a set of consecutive, collinear picture points having the same gray level value. The length of the run is the number of picture points in the run. For a given picture, we can compute a gray level run length matrix for runs having any given direction . The matrix element specifies the number of times that the picture contains a run of length , in the given direction, consisting of points having gray level (or lying in gray level range ). Computation of these matrices is very simple. The number of calculations is directly proportional to the number of points in the picture. Also, the entire picture need not reside in core. Only two rows of picture values are needed at any one time to compute the matrices. To obtain numerical texture measures from the matrices, we can compute functions analogous to those used by Haralick for gray level co-occurrence matrices.
- Short Run Emphasis
- Long Run Emphasis
- Gray Level Non-Uniformity/Gray Level Distribution
- Run Length Non-Uniformity/Run Length Distribution
- Run Percentage
- Low Gray Level Run Emphasis
- High Gray Level Run Emphasis
- Short Low Gray Level Emphasis
- Short Run High Gray Level Emhpasis
- Long Run Low Gray Level Emphasis
- Long Run High Gray Level Emphasis
Shape parameters consists of the following parameters:
- X-coordinate maximum length
- Y-coordinate maximum length
- area
- perimeter
- perimeter2/area
Radon transform and Hough transform have received more attention in image processing recently. They transform two dimensional images with lines into a domain of possible line parameters, where each line in the image will give a peak positioned at the corresponding line parameters. Hence, the lines of the images are transformed into the points in the Radon domain. High Order Spectra are spectral components of higher moments. The bispectrum , of a signal is the Fourier transform (FT) of the third order correlation of the signal (also known as the third order cumulant function). The frequency may be normalized by the Nyquist frequency to be between 0 and 1. The bispectrum, is a complex-valued function of two frequencies. The bispectrum which is the product of three Fourier coefficients, exhibits symmetry and was computed in the non-redundant region. This is termed as Ω, the principal domain or the nonredundant region. The extracted feature is the entropy 1.
LBP, a robust and efficient texture descriptor, was first presented by Ojala et al. (1996, 2002). The LBP feature vector, in its simplest form, is determined using the following method: A circular neighborhood is considered around a pixel. points are chosen on the circumference of the circle with radius such that they are all equidistant from the center pixel. Let be the gray value of the center pixel and , corresponds to the gray values of the points. These points are converted into a circular bit-stream of 0s and 1s according to whether the gray value of the pixel is less than or greater than the gray value of the center pixel. Ojala et al. (2002) introduced the concept of uniformity in texture analysis. The uniform fundamental patterns have a uniform circular structure that contains very few spatial transitions (number of spatial bitwise 0/1 transitions). Multiscale analysis of the image using LBP is done by choosing circles with various radii around the center pixels and, thus, constructing separate LBP image for each scale. Energy and entropy of the LBP image, constructed over different scales ( with corresponding pixel count respectively) were used as feature descriptors.
A Gray Level Size Zone (GLSZM) quantifies gray level zones in an image. A gray level zone is defined as the number of connected voxels that share the same gray level intensity. A voxel is considered connected if the distance is 1 according to the infinity norm (26-connected region in a 3D, 8-connected region in 2D). In a gray level size zone matrix the th element equals the number of zones with gray level and size appear in image. Contrary to GLCM and GLRLM, the GLSZM is rotation independent, with only one matrix calculated for all directions in the ROI.
- Small Zone Emphasis (SZE) or Small Area Emphasis (SAE)
- Large Zone Emphasis (LZE) or Large Area Emphasis (LAE)
- Gray-Level Non-Uniformity (GLN)
- Zone-Size Nonuniformity(ZSN)
- Zone Percentage (ZP)
- Low Gray-Level Zone Emphasis (LGLZE)
- High Gray-Level Zone Emphasis (HGLZE)
- Small Zone Low Gray-Level Emphasis (SZLGLE) or Small Area Low Gray-Level Emphasis (SALGLE)
- Small Zone High Gray-Level Emphasis (SZHGLE) or Small Area High Gray-Level Emphasis (SAHGLE)
- Large Zone Low Gray-Level Empasis (LZLGLE) or Large Area Low Gray-Level Emphasis (LALGLE)
- Large Zone High Gray-Level Emphasis (LZHGLE) or Large Area High Gray Level Emphasis (LAHGLE)
- Gray-Level Variance (GLV)
- Zone-Size Variance (ZSV)
- Zone-Size Etropy (ZSE)
In multilevel binary morphological analysis, the authors are interested in extracting different plaque components and investigating their geometric properties. Here, binary image outputs are represented as sets of image coordinates where image intensity meets the threshold criteria. Overall, this multilevel decomposition is closely related to a three-level quantization of the original image intensity. The two basic operators in the area of mathematical morphology is erosion and dilation. The basic effect of the erosion on a binary image is to erode away the boundaries of regions of foreground (i.e. white pixels, typically). Thus areas of foreground pixels shrink in size, and holes within those areas become larger. The basic effect of the dilation on a binary image is to gradually enlarge the boundaries of regions of foreground. Thus areas of foreground pixels grow in size while holes within those regions become smaller. The define the n-fold expansion of also known as n-fold Minkowski addition of with itself. Opening and closing are two important operators from mathematical morphology. They are both derived from the fundamental operations of erosion and dilation. The basic effect of an opening is somewhat like erosion in that it tends to remove some of the foreground (bright) pixels from the edges of regions of foreground pixels. However it is less destructive than erosion in general. The effect of the opening is to preserve foreground regions that have a similar shape to this structuring element, or that can completely contain the structuring element, while eliminating all other regions of foreground pixels. Opening is defined as an erosion followed by a dilation. Closing is similar in some ways to dilation in that it tends to enlarge the boundaries of foreground (bright) regions in an image (and shrink background color holes in such regions), but it is less destructive of the original boundary shape. The effect of the closing is to preserve background regions that have a similar shape to this structuring element, or that can completely contain the structuring element, while eliminating all other regions of background pixels. Closing is defined as a dilation followed by an erosion. Opening and closing are idempotent, i.e. their successive applications do not change further the previously transformed result. We define as a multiscale opening of by also known as set-processing (SP) opening at scale. A dual multiscale filter is the closing of by or set-processing (SP) closing. The set difference images can be formed. The pattern spectrum is defined. A probability density function (pdf) measure is considered defined. Given the pdf-measure, the cumulative distribution function (cdf) can also be constructed.
Similarly we henceforth represent graytone images by functions; filters whose inputs and outputs are functions (multilevel signals) are called Function-Processing (FP) filters. Let be a finite support graytone image function on , and let be a fixed graytone pattern. In the context of morphology, is called function structuring element. The mathematical definition for grayscale erosion and dilation is identical except in the way in which the set of coordinates associated with the input image is derived. We define the multiscale function-processing (FP) opening of by at scale n.Likewise, we define the multiscale function-processing (FP) closing. We define the pattern spectrum of relative to a discrete graytone pattern the function. A probability density function (pdf) measure is defined. Given the pdf-measure, the cumulative distribution function (cdf) can also be constructed.
The grey level histogram of the ROI of the plaque image is computed for 32 equal width bins and used as an additional feature set. Histogram despite its simplicity provides a good description of the plaque structure.
Three equidistant ROIs were identified by eroding the plaque image outline by a factor based on the plaque size. The histogram was computed for each one of the three regions as described above and the 96 values comprised the new feature vector. This feature was computed in order to investigate whether the distribution of the plaque structure in equidistant ROIs has a diagnostic value and more specifically if the structure of the outer region of the plaque is critical whether the plaque will rupture or not.
Correlograms are histograms, which measure not only statistics about the features of the image, but also consider the spatial distribution of these features. In this work two correlograms were implemented for the ROI of the plaque image:
- based on the distance of the distribution of the pixels' gray level values from the center of the image, and
- based on their angle of distribution.
For each pixel the distance and the angle from the image center was calculated and for all pixels with the same distance or angle their histograms were computed. In order to make the comparison between images of different sizes feasible, the distance correlograms were normalized into 32 possible distances from the center by dividing the calculated distances with . The angle of the correlogram was allowed to vary among 32 possible values starting from the left middle of the image and moving clockwise. The resulting correlograms were matrices (gray level values over were set to be the white area surrounding the region of interest and were not consider for the calculation of the features).
The DWT of a signal x[n] is defined as its inner product with a family of functions which form an orthonormal set of vectors, a combination of which can completely define the signal, and hence, allow its analysis in many resolution levels . For images, i.e., 2-D signals, the 2-D DWT can be used. This consists of a DWT on the rows of the image and a DWT on the columns of the resulting image. The result of each DWT is followed by downsampling on the columns and rows, respectively. The decomposition of the image yields four subimages for every level. Each approximation subimage is decomposed into four subimages according to the previously described scheme. Each detail subimage is the result of a convolution with two half-band filters: a low-pass and a high-pass for , a highpass and a low-pass for , and two high-pass filters for .
The 2-D SWT is similar to the 2-D DWT, but no downsampling is performed. Instead, upsampling of the low-pass and high-pass filters is carried out. The main advantage of SWT over DWT is its shift invariance property.However, it is nonorthogonal and highly redundant, and hence, computationally expensive.
The 2-D WP decomposition is a simple modification of the 2-D DWT, which offers a richer space-frequency representation. The first level of analysis is the same as that of the 2-D DWT. The second, as well as all subsequent levels of analysis consist of decomposing every subimage, rather than only the approximation subimage, into four new subimages.
The GT of an image consists in convolving that image with the Gabor function, i.e., a sinusoidal plane wave of a certain frequency and orientation modulated by a Gaussian envelope. Frequency and orientation representations of Gabor filters are similar to those of the human visual system, rendering them appropriate for texture segmentation and classification.
The detail subimages contain the textural information in horizontal, vertical, and diagonal orientations. The approximation subimages were not used for texture analysis because they are the rough estimate of the original image and capture the intensity variations induced by lighting. The total number of sub-images or three levels of decomposition, including only the detail images, was
- 9 in the case of DWT
- 9 in the case of SWT
- 63 in the case of WP
- 12 in the case of GT
The texture features that were estimated from each detail subimage were the mean and standard deviation of the absolute value of detail subimages, both commonly used as texture descriptors.
We consider multi-scale AM-FM representations, under least-square approximations. Given the input discrete image , we first apply the Hilbert transform to form a extension of the analytic signal. The result is processed through a collection of bandpass filters with the desired scale. Each processing block will produce the instantaneous amplitude, the instantaneous phase, and the instantaneous frequencies in both and directions. As feature vector, the histogram of the low, medium, high and dc reconstructed images is used with 32 bins as a probability density function of the image.
Zernike moments are used to describe the shape of an object; however, since the Zernike polynomials are orthogonal to each other, there is no redundancy of information between the moments. One caveat to look out for when utilizing Zernike moments for shape description is the scaling and translation of the object in the image. Depending on where the image is translated in the image, your Zernike moments will be drastically different. Similarly, depending on how large or small (i.e. how your object is scaled) in the image, your Zernike moments will not be identical. However, the magnitudes of the Zernike moments are independent of the rotation of the object, which is an extremely nice property when working with shape descriptors. We can resize the object to a constant pixels, obtaining scale invariance. From there, it is straightforward to apply Zernike moments to characterize the shape of the object.
Hu Moments are used to describe, characterize, and quantify the shape of an object in an image. They are normally extracted from the silhouette or outline of an object in an image. By describing the silhouette or outline of an object, we are able to extract a shape feature vector (i.e. a list of numbers) to represent the shape of the object.
Threshold adjacency statistics are generated by first applying a threshold to the image to create a binary image with a threshold chosen as follows. The average intensity, , of those pixels with intensity at least 30 is calculated for the image, the cut off 30 chosen as intensities below this value are in general background. The experimental image is then binary thresholded to the range to . The range was selected to maximise the visual difference of threshold images for which the localisation images had distinct localisations but were visually similar, as in Figure 1. The following nine statistics were designed to exploit the dissimilarity seen in the threshold images. For each white pixel, the number of adjacent white pixels is counted. The first threshold statistic is then the number of white pixels with no white neighbours; the second is the number with one white neighbour, and so forth up to the maximum of eight. The nine statistics are normalised by dividing each by the total number of white pixels in the threshold image. Two other sets of threshold adjacency statistics are also calculated as above, but for binary threshold images with pixels in the ranges to and to , giving in total statistics.
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