Robust Image Topological Feature Extraction Krlis Freivalds Paulis
Robust Image Topological Feature Extraction Kārlis Freivalds, Paulis Ķikusts University of Latvia Theory Days at Jõulumäe October 2008
Image Analysis n Image analysis steps q q n Pixels features Features objects Objects knowledge Knowledge decision Features q q Edge Ridge Valley (ravine) Corner
Topological Features n n n n Image is treated as 3 D surface Ridge Valley Junction Peak Pit Saddle (Edge)
Ridges in Images
Ridge Image Examples Text Natural objects Line drawings Industrial applications
Mathematics n Image – smooth function f(x, y) n Derivatives n Directional derivative in direction p = (px, py) n Second derivative in two directions (unit vectors) p and q
Principal Orthogonal Directions n n Two directions play the key role Gradient direction Tangent direction h = (hx, xy) = (–gy, gx). fh = 0 n n Decision is based on fgg, fhh and fhg
Ridge Definition n Point (x, y) is called a ridge point if and only if
Analytical example f(x, y) = y 2 – x 2 K 3: xy = 0 gives x = 0 and y = 0 K 1: K 2: always true Ridge: x = 0
Another example f(x, y) = –y 2 – 10 x 2 K 3: xy = 0 gives x = 0 and y = 0 K 1: always true K 2: Ridge: x = 0 Theorem: Ridge of a quadratic surface is always a line.
Naive implementation n n n fx = (f(k + 1, l) – f(k – 1, l)) / 2, fy = (f(k, l + 1) – f(k, l – 1)) / 2, g= gx = fx / g, gy = fy / g, hx = – gy, hy = gx, fxx = f(k + 1, l) – 2 f(k, l) + f(k – 1, l), fyy = f(k, l + 1) – 2 f(k, l – 1) + f(k, l – 1), fxy = (f(k + 1, l + 1) + f(k – 1, l – 1) – f(k + 1, l – 1) – f(k – 1, l + 1)) / 4, fgg = fxxgxgx + 2 fxygxgy + fyygygy, fhh = fxxhxhx + 2 fxyhxhy + fyyhyhy, fgh = fxxgxhx + fxy(gxhy + gyhx) + fyygyhy. Ridge conditions:
Naive implementation
Equivalent Formulation n Hessian matrix n n Eigenvalues of H Eigenvectors of H n Ridge: R. Haralick, “Ridges and Valleys on Digital Images, ” Computer Vision, Graphics, and Image Processing, vol. 22, no. 10, pp. 28– 38, Apr. 1983.
Algorithm n n Calculate eigenvalues and eigenvectors See if there is a sign change of in the direction of Problems: q q q Unreliable Requires interpolation Relatively Slow
Lee and Kim Algorithm n n 4 directions of discrete grid are analyzed Orthogonal directions of greatest curvature are selected Point is classified as ridge, valley, peak or pit based on sign changes of gradients in these directions. Problems: q q Unreliable No clear distinction between ridges and peaks Lee, S. and Kim, Y. J. 1995. Direct Extraction of Topographic Features for Gray Scale Character Recognition. IEEE Trans. Pattern Anal. Mach. Intell. 17, 7 (Jul. 1995), 724 -729.
New Discrete Algorithm n n n n Consider 4 directions of discrete grid Find direction of minimum second derivative (that will be direction perpendicular to ridge), let fhh be that second derivative. Test if the current point is a local maximum in that direction Let fgg be the second derivative in the perpendicular direction. Test if abs(fhh) >= abs(fgg) If all tests are true return abs(fhh) as ridge strength Else return that this is not a ridge point
Implementation n n n // Calculation of ridge pixel strength at the given point, 0 if the point is not a ridge point // assumes that int *image holds the image values in row by row. int Ridge. Pixel. Strength. Optimized(int x, int y){ int *p = image + (x + y*width); int f = *p; int fpp[4]; int fhh, fhh. I, fhh_tmp, fhh. I_tmp; fpp[0] = 2*(f - p[1] - p[-1]); // calculate magnitudes of directional second derivatives fpp[1] = - p[width+1] - p[-width-1]; fpp[2] = 2*(f - p[width] - p[-width]); fpp[3] = - p[width-1] - p[-width+1]; n n n if (fpp[1]>fpp[0]){fhh = fpp[1]; fhh. I = 1; } else {fhh = fpp[0]; fhh. I = 0; } // find the maximum one if (fpp[2]>fpp[3]){fhh_tmp = fpp[2]; fhh. I_tmp = 2; } else {fhh_tmp = fpp[3]; fhh. I_tmp = 3; } if(fhh_tmp > fhh){fhh = fhh_tmp; fhh. I = fhh. I_tmp; } fhh += 2*f; // diagonal n if(fhh<lower. Threshold) return 0 ; // optimization: early exit n n n int fgg = fpp[(fhh. I+2)&3] + 2*f; // second derivative in the orthogonal direction int f 1 Ofs[4] = {1, width+1, width-1}; int ofs = f 1 Ofs[fhh. I]; if(p[ofs] < f && p[-ofs] < f && abs(fgg) <= fhh) return fhh; n n return 0; n n }
Results
Sensitivity to Noise
Sensitivity to Noise (1)
Sensitivity to Noise (2)
Sensitivity to Noise (3) Note: All images were processed with the same parameters
Results n n n Connected lines Thin lines No artifacts Low noise sensitivity High performance 45 MPix/s = 108 video frames(720 × 576) per second (AMD Athlon 2. 6 GHz)
Compared to Lee and Kim Algorithm n n n Simpler Faster No spurious ridges Original image Lee & Kim Freivalds & Ķikusts
Scale Selection n Gaussian blurring (Geusebroek et. al. 2002) Scale space and automatic scale selection (Lindeberg 1996) Adaptive blurring (Elder, Zucker 1998) Geusebroek, J. , Smeulders, A. W. , and Weijer, J. v. 2002. Fast Anisotropic Gauss Filtering. In Proceedings of the 7 th European Conference on Computer Vision-Part I. LNCS, vol. 2350. pp. 99 -112. T. Lindeberg: "Edge detection and ridge detection with automatic scale selection", International Journal of Computer Vision, vol 30, number 2, pp. 117 --154, 1998. Earlier version presented at IEEE Conference on Pattern Recognition and Computer Vision, CVPR'96, San Francisco, California, pages 465 --470, june 1996 Elder, J. and Zucker, S. 1998. Local scale control for edge detection and blur estimation. IEEE Pattern Anal. Machine Intell. , 20(7): 699– 716
Topological Edge Detection n Edge conditions: q q n fgg = 0 fggg < 0 Algorithm q q Find the discrete direction of maximum gradient See if the second derivative has sign change from positive to negative in that direction
Comparison to Canny Detector n n n Similar quality Simpler Faster Original Canny Proposed
Results – Shooting Target
Conclusion n n New robust ridge, edge detection algorithms High quality High performance; suitable for real-time applications Simplementation Straightforward hardware implementation
- Slides: 30
