1 Point Operations (V1)

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1 Digital Image Processing - Summary 6. Februar Page 1/15 1 Point Operations (V1) 1.1 Definition 1.2 Process M rows (x) and N columns (y) Homogeneous operations don t depend on coordinates, but inhomogeneous do. 1.3 Histogram Gonzales p 120 h(i) = number of pixels in I with intensity value i Cumulative Histogram (Integral): H(i) = i Increasing Contrast: f c (a) = 1.5a Increasing Brightness: f b (a) = a + 10 Invert Image: f i (a) = a max a j=0 h(j) für 0 i K Optimize Dynamic Range (saturate s low % and s high pixels): â low = min{i H(i) M N s low } â high = min{i H(i) M N s high } a min for a â low f mac (a) = a min + (a a low ) amax amin a high a low for â low < a < â high a m ax for â high Histogram Equalisation: H(a) = H eq (a ) a = H(a) 255 M N H eq (i) = M N i H(a) = a M N 255

2 Digital Image Processing - Summary 6. Februar Page 2/15 2 Transformations (V2) Gonzales p 104 Transformation is done using the inverse transformation T 1 which means that the loop is done in the output space and not in the input space. Otherwise, some holes or overlaps are possible. 2.1 Affine Transformations Gonzales p 87 [ ] a11 a [x y] = [w z] Interpolation + [b a 21 a 1 b 2 ] Gonzales p Methods: Nearest Neighbour; Bilinear (image); Bicubic 2.3 Image Registration Image registration is the process of aligning (multiple) images or bringing them into one coordinate system geometrically. Feature extraction Find transform to reference image Transform image 3 Filtering in the Spatial Domain (V3) Gonzales p 144 Operations on images working with the pixels in the neighbourhood. This is a convolution of image and filter matrix in the spatial domain. I = I H If H is 3x3 matrix and the origin is the center position: I = 1 i= 1 j= 1 1 I(u + i, v + j) H(i, j) Beware, to reach an image with the same intensity levels, the calculation should be normalized either by dividing every pixel by the sum of the structural element (SE, H) or by normalizing the SE ( j k H jk = 1). 3.1 Filter Types Box lowpass filter (linear): H 3x3 =

3 Digital Image Processing - Summary 6. Februar Page 3/ Gaussian lowpass filter (linear): H 3x3 = H 5x5 = Rank filter (nonlinear): Select kth element of sorted neighbouring pixels Median filter: Special form of a rank filter where the middle element is selected (remove salt n pepper noise): median(p 0, p 1,..., p k,..., p 2 k ) = p k Minimum filter: Special form of a rank filter where the first element is selected (eliminate white points, thickens dark regions) Maximum filter: Special form of a rank filter where the last element is selected (eliminate dark points, thickens bright regions) Find edges in directions [ of zeros ] in structural [ ] element (gradient operators, based on first derivative) Roberts H1 R =, H R = Prewitt Hx P = Hy P = Sobel (better noise suppression/smoothing than Prewitt): Hx S = Hy S = Compass edge filter (search edges in a specified direction, e.g. Kirsch): H0 K = Hx S = H1 K = H2 K = Hy S = H3 K = H4 K = H5 K = H6 K = H7 K = Laplacian (p. 160), edge intensities sharpening: Lap(f(x, y)) = δ2 f δx + δ2 f 2 δy Beware of the double line effect which creates 2 two lines per edge (one negative and one positive due to the definition of the 2nd derivative. Elimination with zerocrossing detection) H 3 3,anisotropic = H 3 3,isotropic = H 5 5,anisotropic = Additional Comments Derivatives Norm Smoothing Filters All filter elements have to sum up to 1 in order to stay in the correct domain of intensity values. This is easily achievable by dividing all pixels by the sum of all filter elements Border Problems It is not defined what happens at the borders. Therefore, computation is only possible at positions completely inside picture. E.g. with a 3x3 filter mask, the image border is 1 and the resolution is reduced by 2 in both directions (horizontally, vertically). Alternatively, the image can be extended by constant values, by replicating pixels or by replicating cyclic. The second derivative is very sensible to noise (due to its highpass behaviour).

4 Digital Image Processing - Summary 6. Februar Page 4/15 4 Filtering in the Frequency Domain (V4) Gonzales p Discrete Fourier Transform (DFT) Dimensional Gonzales p 220 s(h) = N 1 k=0 ĉ k e jhk 2π N N 1 = k=0 [ ( â k cos hk 2π ) ( + N ˆb k sin hk 2π )] N with N as the periodic number and the coefficients: ĉ k = 1 N 1 N s(h)e jhk 2π N = â k jˆb k = R{c k } + ji{c k } â k = 1 N ˆbk = 1 N h=0 N 1 h=0 N 1 h=0 s(h) cos ( hk 2π N s(h) sin ( hk 2π N ) = R{ĉk } ) = I{ĉk } Dimensional Gonzales p 225 N 1 1 F (u, v) = MN x=0 4.2 Border Problems M 1 y=0 ( ( ux f(x, y) exp j2π M + vy )) N Due to the periodicity of the Fourier transform, borders are reproduced when no measures are taken. F (k + N) = N 1 x n e 2πj N (k+n)n = N 1 n=0 n=0 x n e 2πj N kn e } 2πjn {{} 1 = N 1 n=0 x n e 2πj N kn = F (k) Beware, the Fast Fourier Transform is exactly the same as the DFT but faster. The phase contains important information and should not be discarded. To find out amplitude and phase: F (u, v) = R{F (u, v)} 2 + ) I{F (u, v)} 2, ϕ(u, v) = arctan ( I{F (u,v)} R{F (u,v)} Examples of 2D-Fourier transform pairs Gonzales p 243. Fourier transform properties Gonzales p 253f.. In 2D, the DC values are visible at coordinates (0,0), but to better visualize the spectrum they are usually moved to the center (Matlab: fftshift) Padding One way to avoid border problems is padding the original input image. From p. 263: 1. Obtain padding parameters from input image f(x, y) of size MxN. The padding parameters are P = 2M and Q = 2N 2. Append zeros to input image to obtain the padded image f p (x, y) 3. Center its transform: f c (x, y) = f p (x, y) ( 1) x+y 4. Compute DFT: f c (x, y) F (u, v) 5. Filter: F f = F (u, v) H(u, v) 6. Compute IDFT: F (u, v) g p (x, y) 7. Extract original region MxN from g p (x, y)

5 Digital Image Processing - Summary 6. Februar Page 5/ Window Another way is use windowing function which have nearly the size of the image and which smooth at the border to zero. In the spatial domain, they can be multiplied (which leads to convolution in the frequency domain). 4.3 Filter Types Gonzales p 269 Filtering periodic noise can easily be done in the frequency domain. E.g. band-rejection (notch) filter.

6 Digital Image Processing - Summary 6. Februar Page 6/15 5 Morphological Image Processing (V5, 6, 7) Gonzales p Overview Gonzales p 662 Be aware of theˆwhich is the reflection! Morphological approaches are nonlinear operations on binary or grey images. 5.2 Dilation Gonzales p 633 Dilation (dt. wachsen) grows or thickens objects in a binary image. The structure element H is replicated at every foreground pixel of I:

7 Digital Image Processing - Summary 6. Februar Page 7/ Properties I H = H I (I 1 I 2 ) I 3 = I 1 (I 2 I 3 ) I δ = δ I = I (δ is a neutral object) Border conditions It is best, to set the pixels at the border with the minimum value of H. 5.3 Erosion Gonzales p 631 Erosion (dt. schrumpfen) shrinks or thins objects in a binary image. Only the pixels of the original image where the structure element is completely encapseled, are in the resulting image Properties I H H I (I 1 I 2 ) I 3 I 1 (I 2 I 3 ) Border conditions It is best, to set the pixels at the border with the maximum value of H. 5.4 Duality of Erosion and Dilation Gonzales p 635 (A B) c = A c ˆB (A B) c = A c ˆB with A C being the complement of A (binary inversion) 5.5 Opening and Closing Gonzales p 635 Opening: A B = (A B) B Foreground structures which are smaller than H are eliminated by erosion. Dilation lets the structures grow back to its original size. Closing: A B = (A B) B Holes in the foreground structures are eliminated by dilation. Erosion lets the structures shrink back to its original size. Duality: (A B) c = A c ˆB), (A B) c = A c ˆB) Properties: A B, A B are subsets of A (A B) B = A B, (A B) B = A B 5.6 Hit-or-Miss Transform Gonzales p 640 Finds shapes that are bigger than some (small) structure element D and smaller than some (big) second structure element W : A B = (A D) (A c (W D)). 5.8 Hole Filling Gonzales p 643 Fills out any holes (background regions) starting from a starting point: X k = (X k 1 B) A c with X 0 being an image of same size as A with single pixels as starting point. This is an iterative approach. 5.7 Boundary Extraction Gonzales p 642 Find difference between original image A and erosion of A with B: β(a) = A (A B). 5.9 Connected Components Gonzales p 645 Find objects which are connected: X k = (X k 1 B) A. The only difference to hole filling is that here we are looking for foreground pixels instead of background pixels.

8 Digital Image Processing - Summary 6. Februar Page 8/ Other Tools Convex Hull S647, Thinning S649, Thickening S650, Skeletons S651, Pruning S Morphological Reconstruction Gonzales p 656,676 Used after opening to grow back pieces of the original image that are connected to the opening. In addition to the structure element B (block), here a marker image F with starting points and a mask image G are required. F G Geodesic Dilation Gonzales p 656 Init: D (n) G (F ) = F First it. (size 1): D (1) G Recursion (size n): D (n) G (F ) = (F B) G (F ) = D(1) G [D(n 1) G (F )] Geodesic Erosion Gonzales p 657 Init: E (n) G (F ) = F First it. (size 1): E (1) G Recursion (size n): E (n) G (F ) = (F B) G (F ) = E(1) G [E(n 1) G (F )] Morphological Recon. by Dilation Gonzales p 658 The iterative approach reconstructs the whole object out of a starting point. RG D (F ) = D(k) G (F ) Morphological Recon. by Erosion Gonzales p 658 Reconstruction by erosion reconstructs holes. RG E (F ) = E(k) G (F ) Sample Applications Gonzales p 659 Opening by Reconstruction Mask g is the input image. The structure element b can be selected to find meaningful starting points (e.g. line with 50 pixels length). The marker is then calculated using f = g b. Then magic comes into play: The iterative reconstruction uses this equation: (f nb) g where nb is the iteration number with 0B = and is the minimum operator. This is done until the image is stable (does not change anymore). Hole Filling Automatic filling of holes (including starting point seeking) Border Cleaning e.g. OCR: All characters which contact the border can be removed 5.12 Gray-Scale Morphology Gonzales p 665 Uses function theory instead of set theory (Mengenlehre). In particular, the Top-Hat transformation S672 is interesting for shading correction.

9 Digital Image Processing - Summary 6. Februar Page 9/15 6 Segmentation (V8, V9) Segmentation subdivides an image into its constituent (zusammengehörend) regions or object. One of the most difficult tasks in ImPro. Usually, post processing is required to identify and potentially label objects. Problems: Uneven illumination, noise 6.1 Edge Based Segmentation Gonzales p 692 Aim: Finding borders or lines of objects: Contours, similarity of adjacent pixels can be detected. 3 Steps: 1. Preprocessing & smoothing: noise reduction, small object removal, intensity transformations 2. Edge point detection: Edge pixel detection, derivatives in X,Y direction with 1st and/or 2nd order filters (Laplacian) 3. Postprocessing (localize edges): thresholding, thinning, zero crossings of 2nd order derivative Line Detection Gonzales p 697 See spatial filters (Section 3.1) for Laplacian Sobel, Prewitt, Point detection Gonzales p 696 Point detection: Use 2nd order derivation (Laplacian): 2 f(x, y) = δ2 f δx 2 (f(x, y + 1) + f(x, y 1) 2f(x, y)) H 3 3 = δ2 f δy 2 = (f(x + 1, y) + f(x 1, y) 2f(x, y)) Marr-Hildreth Edge Detector Gonzales p 714 Intensity changes are not independent of image scale and so their detection requires the use of operators of different sizes; and that a sudden intensity change will give rise to a peak or trough (Mulde) in the first derivative or equivalently a zero crossing in the second derivative. ( The operator for smoothing and Laplacian is called Laplacian of a Gaussian (LoG) or Mexican hat: 2 G(x, y) = with σ being the standard deviation of the image. The size of the filter (n n) should be an odd integer with n > ceil(6σ). Discussion: Strong smoothing (sharp edges might be lost) Mostly closed edges Sensitive to noise Post-processing (zero-cross detection) required x 2 +y 2 2σ 2 σ 4 ) e x 2 +y 2 2σ Canny Edge Detector Gonzales p 714 Use first order derivative in both directions. 1. Smooth image with Gaussian 2. Compute gradient (Sobel, Prewitt,... ) in both directions 3. Non-maxima suppression to gradient magnitude image (alles was nicht max ist, unterdrücken) 4. Double thresholding: Pixels with f(x, y) > T 1 belong to edge (strong edge pixels) Pixels with T 1 > f(x, y) > T 2 belong to edge when there is a strong edge pixel in the 8 8 neighbourhood (weak edge pixels) 5. If necessary, edge thinning This algo is considerably slower but performs better than Marr-Hildreth.

10 Digital Image Processing - Summary 6. Februar Page 10/ Edge Linking and Boundary Detection Gonzales p 725 Edges are going to be linked together as they are typically not connected. Local Processing [ ] gx 1. Compute gradient f = (Sobel, Prewitt,... ) g y 2. Compute magnitude M(x, y) = g 2 x + g 2 y g x + g y and angle ϕ(x, y) = arctan(g y /g x ) 3. Set neighbour pixel (8-connected neighbours) at (s, t) of pixel (x, y) as edge pixel when M(s, t) M(x, y) m 0 and ϕ(s, t) ϕ(x, y) ϕ 0 4. If necessary: Thinning and remove single edge pixels Global Processing Find lines instead of simple edges Hough Transform See later... Edge Linking 6.2 Thresholding Gonzales p Definitions Gonzales p 738 Global threshold: One threshold depending on intensity T = T (f(x, y)) Local threshold: Threshold may also depend on neighbourhood (e.g. grey level average, variance,... ) T = T (f(x, y), p(x, y)) Dynamic or variable or adaptive threshold: Depending on position: T = T (x, y, f(x, y)) Hysteresis thresholding: 1. Global threshold with T H, 2. Threshold in the k-connected (e.g. k = 4, 8) neighborhood of all pixels detected in (1) with lower threshold T L. Problems of thresholding: Noise, illumination Basic Method Gonzales p Initial T (e.g. middle point between 2 maxima) 2. Segment image with T into G 1 T, G 2 > T 3. Compute average intensity value m 1, m 2 in G 1, G 2 4. Compute new threshold value: T = 1 2 (m 1 + m 2 ) 5. Repeat steps 2..4 until the difference of the T s is smaller than a constant ɛ Otsu s Method Gonzales p 742 Statistical approach to find best threshold. 1. Compute the normalized histogram of the input image. Denote the components of the histogram by p i, i = 0, 1,..., L Compute cumulative sums: P (k) = k i=0 p(i) for k = 0, 1,..., L 1 3. Compute cumulative means: m(k) = k i=0 ip(i) for k = 0, 1,..., L 1 4. Compute global intensity mean: m G = L 1 i=0 ip(i) 5. Compute between-class variance: σ 2 B (k) = (m GP 1(k) m(k)) 2 P 1(k)(1 P 1(k)) for k = 0, 1,..., L 1 6. Obtain Otsu threshold k = max k=0,1,...,l 1 σb 2 (k). If max is not unique, average these ks. Otsu is not always better than basic approach! Smoothing Gonzales p 747 Again, smoothing might improve results when noise is a problem.

11 Digital Image Processing - Summary 6. Februar Page 11/ Edges Gonzales p 749 Improving the shape of histograms by considering only thoe pixels that lie on or near the edges between the objects and the background. less dependency of size of objects and background. 1. Find edges 2. Threshold edge image binary image 3. Mask original image with thresholded edge image 4. Compute the histogram using only the pixels in the origianl image that correspond to the locations of the 1-valued pixels in image from step 3. Evaluate threshold with the basic method, Otsu, etc. 5. Segment original image using this threshold Adaptive Thresholding Gonzales p 756 Subdivide image and compute threshold for every region. Hint: Regions must consist of both object and background, otherwise the thresholding would not work and only distinguish between noise. Therefore, regions must be not too big and not too small Multivariable Thresholds Gonzales p 761 Use RGB color information for thresholding: z = [r, g, b] T. This is a 3D vector which can also be referred as voxel (volume element). 6.3 Region Based Segmentation Gonzales p 763 Find boundaries between regions directly (in contrast to thresholding) according to some criteria (gray level, color, texture, form,... ). This is more efficient in terms of noisy and blurred images Region Growing Gonzales p Find seed points in input image f(x, y) (e.g. with a very high threshold): S(x, y) 2. Morphological erosion of S(x, y) to reduce connected components to 1 pixel 3. Calculate predicate: E.g. similarity in intensity (intensity threshold) on f(x, y). This leads to f Q (x, y) check 4. Append found values to seed image and label every connected area. Starting with seed points, grow regions according to some criteria as mentioned above Splitting and Merging Gonzales p 766 Instead of growing regions from seed points, this strategy grows from arbitrary disjoint regions. 1. Split into four disjoint quadrants any region R i for which Q(R i ) = F ALSE (Q(R i ) is the predicate, pixels belonging to object) 2. Split all regions again into disjoint regions (if possible) 3. End splitting when no more splitting is possible 6.4 Watersheds Gonzales p 769 The image can be viewed as a 3D relief with local minima. When flooding from these minima step by step and the water merges between two regions, a dam (Damm) can be built. These dams (watersheds/wasserscheiden) are the dividers between regions. Use when uniform regions are searched Problems: Oversegmentation due to noise or non-uniform regions. Possible solutions: Smoothing (as always) but with relatively large kernels Apply watershed transformation to gradient image (often still oversegmentation smoothing) Use markers as starting point (e.g. shortest distance between black and white pixels)

12 Digital Image Processing - Summary 6. Februar Page 12/15 7 Description and Representation (V10, 11, 12) Gonzales p Representation Gonzales p 796 A region can be represented in terms of its external characteristics (its boundary - basically for shape properties) or its internal characteristics (pixels comprising the region - basically for regional properties (colors, texture)). Description is the extraction of information out of the representation. For example the length of a boundary. These are some basic methods: Chain Codes S798, Polygonal Approximations S801 and Skeletons S Signatures Gonzales p 808 Signature: 1d representation of a boundary that can be generated in various ways. Example: Distance from the centroid to the boundary as a function of angle: Fourier Descriptors Gonzales p 818 FDs represent the frequency content of a shape (of the border) which can also be viewed as the form of an object. It is possible to make them independent of various transformations: Translation F (0) = 0 Scale F (u) = F (u) F (1) Rotation, starting point, direction F (u) = F (u) FDs are insensitive to noise. However, this might be dangerous as information (phase) is thrown away and may lead to wrong assumptions: This signature generation is translation variant but not rotation or scaling variant. Features that can be used further are e.g. the number of maxima or the distance between maximum and minimum Hough Transform (HT) Gonzales p 735 Aim: Find lines, circles or any freeform shapes in an image after edge detection. HT for Lines The objects are found using the following algorithm: Step Example: Line Define mathematical equation which describes the shape. y = a k x + b k Define parameter space with limits a k [a min, a max ], b k [b min, b max ] For every point x i, y i solve b = y i ax i and increment each visited point: H(a, b) = H(a, b) + 1 check Find maximum value in parameter space Parameters inserted into equation leads to most likely object Marked deep blue in graphics above (with 3 in it) y = ax + b Problem with this is that a horizontal line leads to a. Therefore, the parameter equation x cos(θ) + sin(θ) = r should be used for finding a line. Equations for circles

13 Digital Image Processing - Summary 6. Februar Page 13/15 Cartesian equation: r 2 = (x x 0 ) 2 + (y y 0 ) 2 Parametric representation: x = x 0 + r cos(θ) y = y 0 + r cos(θ) with Θ not being a free parameter Generalized Hough Transform (GHT) For freeform shapes which are rotation and scaling invariant as well as some a priori knowledge of the shape is available. Here, the gradient of the boundary is used instead of a parameter model. The space is now called reference space and not parameter space anymore. Conclusions Quite high computational effort ( Brute Force ), but control by quantization of parameter space Multiple objects can be detected HT tolerates also objects with holes Harris Corner Detection Corners are reference landmarks in images and help in localization. A corner has two large eigenvalues in the structure tensor T s. 1. if necessary (eg. noise, etc.) low-pass filter the image 2. compute gradients G x, G y (sobel) 3. build the structure tensor: 4 (3) values per pixel 4. low-pass filter each component of the structure tensor 5. compute Harris corner measure R(x, y) from structure tensor 6. threshold R(x, y) with T t h > 0 potential corners 7. use non-maxima suppression to find maxima in R(x, y) sort maxima in descending order select largest maximum suppress maxima within radius r around selected maximum select remaining largest maximum... etc

14 Digital Image Processing - Summary 6. Februar Page 14/ Simple Descriptors Gonzales p 822 Simple descriptors include area, perimeter, minimum/maximum diameter, area of bounding rectangle and should be independent of translation, rotation and scale. Area based descriptors, continued 2D central moments of order p + q: µ pq = M 1 x=0 N 1 (x x) p (y ȳ) p f(x, y) y=0 with x = m10 m 00 and ȳ = m01 m 00 which are also the centroid coordinates. Use principal components for normalizing with respect to variations in size, translation and rotation. Moments provide: Centroid coordinates Principal axes of inertia (Trägheit) or direction of maximal and minimal variance (principal components) Minimal bounding box 7.4 Texture Based Descriptors Gonzales p Statistical Approaches Based on normalized histogram h(g) (global): Mean: m = L 1 i=0 g ih(g i ) Variance: µ 2 = σ 2 = L 1 i=0 (g i m) 2 h(g i ) Skewness: µ 3 = L 1 i=0 (g i m) 3 h(g i ) 1 Roughness: R(g) = 1 1+σ 2 /(L 1) 2 R 1: large σ, rough; R 0: σ = 0, smooth Uniformity (uniformity): U(g) = L 1 i=0 h2 (g i ) Average entropy: H(g) = L 1 i=0 h(g i) log 2 (h(g i )) Co-Occurence Matrix The construction of COO uses the following scheme, it is local: 7.3 Area Based Descriptors (Moments) Gonzales p 839 Moments are statistical measures of the shape of a set of points. Due to its statistical nature, they are hardly variant to translation, rotation and scale. 2D moments of order p + q: m pq = M 1 x=0 N 1 y=0 x p y p f(x, y) The row index indicates the value of the center and the columns the number of values in the PO. The COO can also be reduced (instead of 255 grey intensities only 64 or 8 could be evaluated). Energy Contrast Entropy Homogenity E = L 1 i=0 C = L 1 i=0 H = L 1 S = L 1 i=0 L 1 j=0 L 1 L 1 i=0 j=0 L 1 j= Spectral Approach COO(i, j)2 j=0 (i j)2 COO(i, j) COO(i, j) log(coo(i, j)) COO(i,j) 1+ i j Energy, contrast, entropy can also be found out using the scaled spectrum: S(u, v) S(u, v) = F F T (f(x, y)) S n (u, v) = m n S(u, v) u=2 v=2

15 Digital Image Processing - Summary 6. Februar Page 15/15 8 Object Recognition V13, 14 Gonzales p 861 Problems: Which features are to be used? Where are the boundaries? Definition of a pattern or feature vector: x = [x 1,..., x n ] T Classes: x K 1,..., K W 8.1 Decision Theory Minimum Distance Classification Gonzales p 866 Sample will be assigned to the class to which the mean is the shortest. 8.2 Neural Nets Gonzales p 882 No assumptions about statistical model, training via samples. 2-Layer perceptron for linearly separable classes 3-Layer perceptron for non-linearly separable classes with nonlinear perceptron function When the Mahalanobis distance is taken instead of the Euclidean distance, the variance is also included in the min-dist-classifier: D j (x) 2 = (x m j ) T C 1 j (x m j ) 8.3 Structural Methods Gonzales p 903 Classification based on common descriptors (order descriptors) according to degree of similarity: with C j as the covariance matrix for every sample per class: C j = 1 N j x K j xx T m j m T j Matching by Correlation Gonzales p 869 Correlation is not robust in terms of background and freeform objects (only square objects) Bayes Classifier Gonzales p 874 Measure the similarity: Largest number to which k is the same. Distance between two objects a and b: D(a, b) = 1 k Build similarity matrix: Binary loss function: d j (x) = p(x K j )p(k j ) (j = 1,..., W ). p(x K j ) is assumed to be Gaussian distributed: d j (x) = ln p(k j ) 1 2 ln C j 1 2 (x m j) T C 1 j (x m j )) 8.4 Grammar Gonzales p 904