Hough transform and geometric transform
Jen-Chang Liu, Spring 20056 Motivation for edge linking
Edge detection: find pixels lying on edges => How to link edge points ? Sobel: horizontal Sobel: vertical original Goal: Global processing Given n edge points in an image
=>Find subsets of the points that lie on straight lines Step 1: for a pair of points, determine their line function Step 2: determine whether there are points that are close to the line Step 3: repeat step 1 until all pair of points are tested Global processing: complexity
n edge points n =>C2 ~ n2 lines n =>(n-2)C2 ~ n3 tests to determine whether other points are near this line Any faster method? Hough transform Hough [1962] How to determine a line ? yi = axi + b
Tow point (xi, yi) and (xj, yj) determine a line (ak, bk) (a, b) yi = axi + b yj = axj + b Line with parameters (a, b) * Many lines pass a point * (xi, yi) and (xj, yj) both have a line with parameters (a, b) passing through Idea: Parameter space of line
A line is determined by (a, b) B (ak, bk) (a, b) B = -Axi + yi (ak, bk) (a, b) yi = Axi + B B = -Axj + yj A Example: find all lines [i.e. parameter (a,b)] pass a point
y a + b = 1 (a,b)=(, ?) (1,1) b (0,1) (a,b)=(1,0) (1,0) (a,b)=(0,1) x a Replace in y = ax+b 1 = a + b Problem: How to define the horizontal lines? Problem with y=ax+b Solution for line function with slope a =
Normal representation of a line Inner product = 0 (x, y) Implementing Hough transform
1. decide on a discrete set of values of and to use 2. For each edge point, calculate the values of = xcos+ysinfor all discrete 3. Create an accumulator array whose size are the number of angles and values 4. Updating the accumulator array as we go Limit and 1. Restrict 0 y -3 3 x Limit and (cont.) 2. -90180 Line in the 2nd quadrant
will not appear in the image y x Example 128 128 128 -90 180 Example: Edge image: Gradient => threshold original Project 2 Topic: Generalize Hough transform to detect circles
1-3 persons a team Prepare PowerPoint slides to explain your methods Write the MATLAB function to perform it Give me team members on 4/29 Present and demo on 5/6 Image rotation Rotation matrix (x, y) (x, y) Real situation when rotating
Boundary of image changes Rotated pixels are not on the integer image grid Real situation when rotating (cont.)
Take nearest integer grid What are the gray values? Rotated back and interpolation !!! Gray-level interpolation
Nearest neighbor Bilinear interpolation Use 4 nearest neighbors Cubic convolution interpolation Fit a surface over a large number of neighbors Interpolation 1D Nearest neighbor Linear interpolation Interpolation 2D Bilinear interpolation Interpolate for (x,y)
and (x+1, y) first 2. Interpolate for (x,y) High order interpolation
Cubic interpolation weighting f(x)=R3(-1-)f(x1)+ R3(-)f(x2)+R3(1-)f(x3)+ R3(2-)f(x4) Example: Anamorphosis Spatial transformations
r(x,y) s(x,y) tiepoints f(x,y) g(x,y) distorted original Spatial transformations (cont.)
Within the quadrilateral region, geometric transformation is modeled by a pair of bilinear equation 4 pairs of tiepoints to solve the equations Restored image: Gray-level interpolation
Non-integer coordinate x or y Tiepoints after original distortion restored distorted
Nearest neighbor restored distorted Bilinear interp. restored distorted original distorted (Using Previous Slide) restored Diff.