9 length of contour = no. of horizontal and vertical components + ( 2 no. of diagonal components) diameter of boundary B

Transcription

1 8. Boundary Descriptor 8.. Some Simple Descriptors length of contour : simplest descriptor - chain-coded curve 9 length of contour no. of horiontal and vertical components ( no. of diagonal components diameter of boundary B - Diam (B max[d(p,p ] i, j i j where D : distance measure p i, p j : points on boundary - value and orientation of diameter : useful descriptors curvature - the rate of change of slope - approximately difference between the slopes of adjacent boundary segments useful descriptor of curvature - in clockwise direction 9 the change in slope at p : nonnegative vertex point p : part of convex segment 9 the change in slope at p : negative vertex point p : part of concave segment - ranges in the change of slope 9 p : part of a nearby straight segment if the change < : corner point if the change > Shape Numbers Shape number - the first difference of smallest magnitude Order n - the no. of digits in its representation - n : even for a closed boundary 8

2 Eccentricity (9u, ÁÑu of boundary - ratio of major to minor axis - Major axis (straight-line segment joining the two points farthest from each other - Minor axis (perpendicular to the major axis For unique shape number Alignment the chain-code grid with the basic rectangle ex. fig

3 8..3 Fourier Descriptor fig N-point digital boundary in x y-plane - Starting point ( x, y, sequence ( x, y,,( x N, yn counterclockwise - Coordinates x ( k xk, y ( k yk s ( k [ x( k, y( k] for k,,,, N- complex number s ( k x( k jy( k advantage : -D -D problem DFT of s(k N - a( u N k s( kexp[ jπuk / N] : Fourier descriptors of boundary - IDFT of a(u s( k N u - approximation to s(k M u a( uexp[ jπuk / N] sˆ( k a( uexp[ jπuk / N] for k,,,, N- that is, a( u for u > M - low-freq. Components : determine global shape high-freq. Components : account for fine detail

4 the smaller M the more detail : lost on boundary ex. fig. 8.5 A few Fourier descriptor - capture the gross essence of boundary : valuable property - used as the basis for differentiating between distinct boundary shape ( chap. 9 : detail Change of boundary : translation, rotation, scaling simple transformation on the descriptors - rotation 9 multiplying the point by e 9 rotated sequence : s ( k e N Fourier descriptor ar ( u N jθ jθ k s( kexp[ jθ ][ jπuk / N] jθ ( for u,,,, N- a u e 9 rotation : coeffs. X multiplicative constant table 8. jθ e some basic properties of Fourier descriptor

5 8..4 Moments shape of boundary segments - described quantitatively by using moments fig. 8.6 (a segment of boundary (b represented as a -D function g(r of variance r - amplitude histogram of g p ( v i, i,,, k v i : random variable where k : number of discrete amplitude increment - n-th moment of v about its mean ( v n where k i ( v m i m k i n p( v i v i i v p( : mean - : variance m: mean - generally, only the first few moments : required to difference between distinct shapes alternatives - normaliation of g(r to unit area - (r : measures the spread of curve about mean

6 3 (r : measures its symmetry with reference to the mean description task reduce -D function - moments : the most popular method 9 advantages : straight-forward implementation physical interpretation of boundary shape insensitive to rotation sie normaliation : achieved by scaling the range of r 8.3 Regional Descriptors 8.3. Some simple Descriptors - area : no. of pixels contained within its boundary - perimeter : length of boundary applied in the case that sie of object : invariant or in measuring compactness of a region ( perimeter / area : no. dimension 9 min. for a disk-shaped region 9 insensitive to rotation - principal axes of region - eigenvectors of covariance matrix obtained by using the pixels within region - two eigenvectors 9 point in the directions of maximal region spread 9 orthogonal each other - degree of spread : measured by the corresponding eigenvalues - principal spread and direction of region 9 described by the largest eigenvalues and its corresponding eigenvector - insensitive to rotation - depend on scale change (eigenvalue : changed 9 alternative : ratio of the large to the small eigenvalues descriptor (insensitive to scale change 8.3. Topological Descriptors : useful for global description topology - a study of properties of figure that are unaffected by any deformation, as long as there is no 3

7 tearing or joining of figure (called rubber-sheet distortions - ex. fig region with two holes 9 if topological descriptor : no. of holes not affected by a stretching or rotation transformation 9 if region is torn or folded no. of holes : changed another topological property : no. of connected components - connected component of a set 9 a subset of maximal sie such that any two of its points : joined by a connected curve lying entirely within the subset fig connected components Euler number E - E C H where C : no. of connected components H : no. of holes - topological property fig

8 (a E C H (b E C H Polygon network - no. of vertices, edges, and face : W, Q, F - Euler formular W Q F C H E fig vertices, edges, faces connected region, 3 holes 7 3 Topological property : rather general Texture Texture - measure of properties such as smoothness, coarseness, and regularity - three principal approach 9 statistical, structural, spectral Statistical approach 5

9 : smooth, coarse, graining and so on : characterie Moment of histogram : the simplest approach describing texture - p ( i for i,,, L : histogram where i : discrete image intensity - moment of about the mean ( n where L i ( m i m L n p( p( i i i 9 u, u Variance σ ( : the second moment - important factor in texture description - measure of gray level contrast 9 descriptor of relative smoothness - measure R σ ( 9 for area of constant intensity ( σ ( 9 for large values of σ ( Third moment : measure of skewness ( Mºb of histogram Fourth moment : measure of relative flatness i Fifth and higher moments : not so easily related to histogram shape - provide further quantitative discrimination of texture content Problem - no information regarding the relative position of pixels with respect to each other P : position operator A : k k matrix whose element a ij is the no. of times that points with gray level i occur relative to points with gray level j with i, j k ex.,, 3 6

10 an image p : one pixel to the right and one pixel to the left j a ij : x x fá i õ i } i ý e j i M MNm pixel 4 A 3 ex. a 3 : } i ýá E ý Sie of A : determined by the no. of distinct gray level in the input image - intensities : should requantied into a few gray level bands in order to keep the sie of A manageable n : total no. of point pairs in image that satisfy P ( in ex. n6 C A n : called gray level co-occurrence matrix - ij c : estimate of the joint prob. that a pair of points satisfying P will have values i, ( j - presence of texture pattern 9 detected by choosing an appropriate position operator analysis of C matrix - a set of descriptors useful for this purpose max. prob. max( c : an indication of the strongest response to P i, j ij Element difference moment of order k i j ( i j : relative low value when the high values of C are near the main diagonal, k c ij because the differences (i - j are smaller there 3 Inverse element difference moment of order k k cij /( i j, i j : apposite effect of i 4 Entropy j i j c log : measure of randomness, achieving its highest value when all elements of C ij c ij 7

11 are equal 5 Uniformity i j c ij : the lowest value when ij c s are all equal basic idea - characterie the content of C via these descriptors use of these descriptor - teach a system representative descriptor value or a set of different texture - unknown region 9 its descriptor : matched with those stored in the system memory Structural approach Rewritten S as Fig. 8. symbol - a : a circle - a meaning of circles to the right 9 assigned to a string of the form aaa. - rule S as : generation of fig. 8. (b some new rules - S as, S ba, A ca, A c, A bs, S a 9 b : circle down 8

12 9 c : circle to the left - string 9 aaas aaaba aaabca aaabccbs aaabccbaa 3 3 matrix of circles in fig. 8. (c basic idea - simple texture primitive more complex texture patterns by some rules that limit the no. of possible arrangements of primitives(s 3 Spectral approach Three features of the Fourier spectrum that are useful for texture descriptions prominent peaks in the spectrum : principal direction of the texture patterns location of peaks in the spectrum : fundamental spatial period of patterns 3 eliminating any periodic components via filtering : leaves non-periodic image elements Polar form spectrum - S ( r, θ : spectrum function, r, θ : variable - -D function 9 S θ (r : variable r for fixed θ 9 r (θ S : variable θ for fixed r More global description π - S( r Sθ ( r θ R - S( θ S r ( θ : R : radius of a circle r A pair of value : S (r, S(θ - constitute a spectral-energy description of texture for an entire image - descriptor of these functions 9 mean, variance of both the amplitude and axial variations 9 distance between the mean and the highest value of the function ex. fig

13 8.3.4 Moments moment of order (p q for function f (x, y m p q pq x y f x, y ( dxdy for p, q,,, uniqueness theorem - if f (x, y 9 piecewise continuous and 9 nonero values only in a finite part of the xy-plane moments of all orders : exist the moment sequence ( m pq is uniquely determined by f (x, y - conversely, ( m pq uniquely determines f (x, y central moments

14 p q pq ( x x ( y y f ( x, y - where dxdy m m x and - for digital images pq x y m y m p q ( x x ( y y f ( x, y - the central moments of order up to 3,,, 3,,, - in summary, m m ym 3 3 m 3 xm 3 m x m ym xm y m m xm m xm ym x m m ym 3 m 3 3ym y m - normalied central moments pq pq γ p q where γ for p q, 3, a set of seven invariant moments - derived from the second and third moments φ φ ( 4 φ 3 ( 3 3 (3 3 φ 4 ( 3 ( 3 φ ( 3 ( [( (3 ] (3 3( [3( 3 ( 3 ]

15 ( ( 4 ] ( [( ( φ ( (3 ] (3 [( ( 3 ( φ ] ( [3( 3 3 ex.fig. 8.4 table 8.