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• Edge detec*on schemes can be grouped in three classes: – Gradient operators: Robert, Sobel, Prewi>, and Laplacian (3x3 and 5x5 masks) – Surface fiHng operators: Hueckel, Hartly and Haralick – Based on Gaussian deriva*ves: Canny
• Typically two stages are needed in edge detec*on 1. Detec*on of short linear edge segments (edgels) using the edge detec*on operators 2. Aggrega*on of edgels into extended edges
Edge detectors
Gradient based methods
• Many edge-‐detec*on operators are based upon the 1st deriva*ve of the intensity.
Where the biggest change occurs, the deriva*ve has maximum magnitude. Using this informa*on we can search an image for peaks in the intensity gradient. The gradient of an image f (x,y) is:
• The edge direc.on is given by: • The edge strength is given by the gradient magnitude:
θ
Δy Δx Δy Δx (x-‐1,y+1) (x,y+1) (x+1,y+1)
(x-‐1,y) (x,y) (x+1,y)
(x-‐1,y-‐1) (x,y-‐1) (x+1,y-‐1)
3x3 Sobel operator Sobel window
Sobel operator
• The Sobel operator calculates the gradient of the image intensity at each point, giving the direc*on of the largest possible increase from light to dark and the rate of change in that direc*on.
• Mathema*cally, the operator uses two 3×3 kernels which are convolved with the original image to calculate approxima*ons of the deriva*ves, one for horizontal changes, and one for ver*cal. The Sobel operator represents a rather inaccurate approxima*on of the image gradient:
– it uses intensity values only in a 3×3 region around each image point – it uses only integer values for the coefficients
• But it is s*ll of sufficient quality to be of prac*cal use in many applica*ons.
(a) 2x2 Roberts operator (b) 3x3 Prewi> operator (c) 4x4 Prewi> operator
Other gradient operators
Δy Δx
Δx Δx Δy Δy
Δy Δx
(c)
a) b)
d) c)
Canny edge detector
• The Canny algorithm aims at performing op*mal edge detec*on. It is comprised of several cascade stages:
Original image Norm of the gradient aaer noise reduc*on
Thresholding with hysteresis
Thinning (non-‐maximum suppression)
• Noise reduc*on First deriva*ves are suscep*ble to noise present on raw unprocessed image data. Canny edge
detec*on performs convolu*on of the original image with a Gaussian filter to remove noise. The result is a slightly blurred version of the original image. Consider a single row or column of the image and plot intensity as a func*on of posi*on. If pixels are disturbed with noise and the if deriva*ve of the signal is computed, edges are not any more dis*nguishable.
Smoothing the image by using a low pass Gaussian filter h permits to remove noise. Edges are clearly evidenced in correspondence of peaks of
• According to the deriva*ve theorem of convolu*on one opera*on can be saved:
• Finding the intensity gradient direc*on the Canny algorithm uses four filters to detect horizontal, ver*cal and diagonal edges in the blurred image. For each pixel, the direc*on of the filter which gives the largest response magnitude is determined. This direc*on together with the filter response then gives the complete es*ma*on of the intensity gradient at each point in the image.
• Non-‐maximum suppression a search is then carried out to determine if the gradient magnitude assumes a local maximum in the gradient direc*on. This requires checking interpolated pixels p and r. Non maximum pixels are suppressed. As a result a set of edge points, in the form of a binary image, is obtained ("thin edges”).
• Tracing edges
important edges are along con*nuous curves in the image. Therefore edges are detected if the faint sec*on of the edge line is followed and noisy pixels that do not cons*tute a line but have produced large gradients are discarded. Edges are traced in the image using thresholding with hysteresis.
Hysteresis uses a high threshold to start edge curves and a low threshold to con*nue them. Use of high threshold marks out the edges that are genuine. Star*ng from these, using the direc*onal informa*on, edges can be traced through the image.
For each edge point, the tangent to the edge curve (which is normal to the gradient at that point) is used to predict the next points (here either r or s).
While tracing an edge, we apply the lower threshold, allowing us to trace faint sec*ons of edges as long as we find a star*ng point. We obtain a binary image where each pixel is marked as either an edge pixel or a non-‐edge pixel.
• The Canny algorithm contains a number of adjustable parameters, which can affect the
computa*on *me and effec*veness of the algorithm (h>p://matlabserver.cs.rug.nl):
– The size of the Gaussian filter: • smaller filters cause less blurring, and allow detec*on of small, sharp lines; • larger filters cause more blurring and are more useful for detec*ng larger,
smoother edges.
– Thresholds: • Eliminates noise edges and makes edges smoother. • Removes fine detail. • the use of two thresholds with hysteresis allows more flexibility than in a single-‐
threshold approach;
Canny with large σ Canny with small σ original
• The choice of σ depends on desired behavior: – large detects large scale edges – small detects fine features
Effects of σ (Gaussian kernel size)
Effects of thresholding
Coarse scale low threshold
• A too high threshold can miss important informa*on; a too low threshold will falsely iden*fy irrelevant informa*on as important.
Fine scale high threshold Coarse scale high threshold
Lines
What is a line
• Lines can be detected following different approaches: – searching for changes of the intensity gradient (double edges) at every possible
posi*on/orienta*on – using a vo*ng scheme in the line parameter space
• An edge is not a line... a line is a double edge
• In correspondence of lines there is an intensity gradient on one side of the line, followed immediately by the opposite gradient on the opposite side. Therefore lines exhibit a very high change in intensity gradient. According to this, line detec*on can be based upon local maxima of the 1st deriva*ve of the intensity that gives the rate of change in intensity gradient.
• Assume that the image has been pre-‐smoothed by Gaussian smoothing and a scale-‐space representa*on L(x,y;t) at scale t is obtained and introduce t every image point a local coordinate system (u,v), with the v-‐direc*on parallel to the gradient direc*on. It is therefore required that the first-‐order direc*onal deriva*ve in the v-‐direc*on Lv has its first order direc*onal deriva*ve in the v-‐direc*on equal to zero while the second-‐order direc*onal deriva*ve in the v-‐direc*on should be nega*ve.
• Local maxima of the intensity gradient are hence obtained by detec*ng zero-‐crossings of the second-‐order direc*onal deriva*ve in the gradient direc*on:
that sa*sfies the following sign-‐condi*on on the third-‐order direc*onal deriva*ve in the same
direc*on: where Lx, Ly ... Lyyy, are local par*al deriva*ves.
Zero-‐crossing of Laplacian line detector
Laplacian of Gaussian Gaussian Deriva*ve of Gaussian
• In prac*ce the image is convolved with the 2D Laplacian of Gaussian:
that can be approximated by compu*ng the first image deriva*ves with first order difference operators as with edges and second order deriva*ves as:
• Lines are at zero crossings of the Laplacian of Gaussian:
Hough transform-‐based line detector
– A line in the image corresponds to a point in the Hough space – To go from image space to Hough space: given a set of points (x,y), find all (m,b) such
that y = mx + b
x
y
m
b
m0
b0
image space Hough space
• Hough transform vo*ng scheme is the most commonly used solu*on to find lines in images. In this approach, points vote for a set of parameters describing a line or a curve. The more votes for a par*cular set determine the line parameters. Mul*ple lines or curves can be detected in one shot.
• Typically a different parameteriza*on is used:
d is the perpendicular distance from the line to the origin θ is the angle between d and the x axis
• Basic Hough transform algorithm 1. Ini*alize an accumulator H[d, θ] with a suitable number of bins corresponding
to all possible line orienta*ons so that H[d, θ] = 0 2. for each pixel that is verified is an edge point e[x,y]
a. plot lines at different angles and draw the perpendicular line that intersects the origin (θ = 0 to 180° )
b. calculate parameters d and θ for each line c. increase the corresponding accumulator bin H[d, θ] = + 1
3. Find the value(s) of (d, θ) where H[d, θ] is maximum 4. The detected line in the image is given by
Example
Lines plo>ed at different angles and distances to the origin for three dis*nct points
angle-‐distance plot: the intersec*on point is the H[d, θ] maximum and determines the parameters of the line passing through the three points