Edge pixel with coordinates (s,t) in S xy has an angle similar to pixel at α(s,t) α(x,y) <A

Size: px

Start display at page:

Download "Edge pixel with coordinates (s,t) in S xy has an angle similar to pixel at α(s,t) α(x,y) <A"

Dorthy Rice
5 years ago
Views:

1 Image segmentation(10.2.7) SLIDE 1/ Edge linking and boundary detection Edge detection is always followed by edge linking Local processing AnalyzepixelsinsmallneighbourhoodS xy ofeachedgepoint Pixelsthataresimilararelinked Principal properties used for establishing similarity: (1)M(x,y)= f(x,y) : Magnitudeofgradientvector (2) α(x, y): Direction of gradient vector Edgepixelwithcoordinates(s,t)inS xy issimilarinmagnitudetopixelat (x,y)if M(s,t) M(x,y) <E Edge pixel with coordinates (s,t) in S xy has an angle similar to pixel at (x,y)if α(s,t) α(x,y) <A Edgepixel(s,t)inS xy islinkedwith(x,y)ifbothcriteriaaresatisfied

2 Image segmentation(10.2.7) SLIDE 2/13 Theabovestrategyisexpensive. Arecordhastobekeptofalllinkedpoints by, for example, assigning a different label to every set of linked points Simplification suitable for real-time applications: (1)ComputeM(x,y)andα(x,y)ofinputimagef(x,y) (2) Form binary image { 1, ifm(x,y)>tm ANDα(x,y) [A T g(x,y)= A,A+T A ] 0, otherwise (3)Scan rows of g and fill (set to 1) all gaps (sets of 0s) in each row that donotexceedaspecifiedlengthk (4)Rotateg byθandapplystep(3). Rotateresultbackby θ. Image rotation is expensive when linking in numerous directions is required, steps(3) and(4) are combined into a single, radial scanning procedure.

3 Image segmentation(10.2.7) SLIDE 3/13 Example 10.10:

4 Image segmentation(10.2.7) SLIDE 4/13 Regional processing (Polygonal approximations) A conceptual understanding of this idea is sufficient Requirements: (1) Two starting points must be specified;(2) All the points must be ordered Large distance between successive points, relative to the distance between otherpoints boundarysegment(opencurve) endpointsusedasstarting points Seperation between points uniform boundary (closed curve) extreme points used as starting points

5 Image segmentation(10.2.7) SLIDE 5/13 Example 10.11: Edge linking using polygonal approximation Example 10.12: (READ)

6 Image segmentation(10.2.7) SLIDE 6/13 Global Processing using the Hough transform Weattempttolinkedgepixelsthatlieonspecifiedcurves Brute force method: When the specified curve is a straight line, the line between each pair of edge pixels in the image is considered. The distance between every other edge pixel and the line in question is then calculated. When the distance is less than a specified threshold, the pixel is considered tobepartoftheline Number of calculations for n edge pixels: Numberofpossiblelines: n 1 k=1 k=n(n 1) 2 n 2 Distances per line: n Totalnumberofdistances: n2 (n 1) 2 n 3 Whenn=256 2 thennumberofcalculationsis !!!

7 Image segmentation(10.2.7) SLIDE 7/13 Hough transform(1962) Whendifferentvaluesforaandbareconsidered,y i =ax i +bgivesallpossible linesthroughthepoint(x i,y i ) Theequationb= x i a+y i givesonelineintheab-planeforaspecific(x i,y i ) When another point (x j,y j ) is considered, b= x j a+y j represents another line in the ab-plane Supposethatthesetwolinesintersectatthepoint(a,b ),theny=a x+b representsthelineinthexy-planeonwhichboth(x i,y i )and(x j,y j )lie Since a computer can only deal with a finite number of straight lines, we subdivide the parameter space ab into a finite number of accumulator cells...

8 Image segmentation(10.2.7) SLIDE 8/13 (FigfromEd2) > Algorithm: (1)Setallcellsequaltozero (2)Forevery(x k,y k ) (2.1)Leta=everysubdivisiononthea-axis (2.2)Calculateb= x k a+y k (2.3)Roundoffbtothenearestallottedvalueontheb-axis (2.4) Increment accumulator cell(a, b) with 1 Note: When there are K subdivisions on the a-axis, we need only nk calculations, which is linear (recall that we needed n 3 calculations for the brute force method)

9 Image segmentation(10.2.7) SLIDE 9/13 Westillhaveaproblemthough,since <a< and <b<! Inordertodealwiththisproblem,wenowrepresentastraightlineasfollows sothat(a,b) (ρ,θ) xcosθ+ysinθ=ρ

10 Image segmentation(10.2.7) SLIDE 10/13 Derivation: y=ax+b b ρ =cscθ= 1 sinθ b=ρcscθ a = 1 tanθ (negativereciprocal) = cosθ sinθ y= cosθ x+ρcscθ ysinθ= xcosθ+ρ xcosθ+ysinθ=ρ sinθ Now we have that ρ [ 2D, 2D] and θ [ 90 o,90 o ], where 2D is the diagonal distance between two opposite corners in the image. Problem solved! Algorithm: (1)Setallcellsequaltozero (2)Forevery(x k,y k ) (2.1)Letθ=everysubdivisionontheθ-axis (2.2)Calculateρ=x k cosθ+y k sinθ (2.3)Roundoffρtothenearestallottedvalueontheρ-axis (2.4) Increment accumulator cell(ρ, θ) with 1

11 Image segmentation(10.2.7) SLIDE 11/13 Example 10.13: Illustration of Hough transform properties

12 Image segmentation(10.2.7) SLIDE 12/13 Algorithm for edge linking: (1) Compute f and isolate edge pixels through thresholding (2) Specify subdivisions in the ρθ-plane (3) Apply Hough transform to edge pixels (4) Identify accumulator cells with highest values (5) Examine continuity of pixels that constitute cell (6) Link these pixels if gaps are smaller than threshold Extension to more general graphs Houghtransformapplicabletoanygraphg(v,c)=0,wherevisvectorof coordinates and c is vector of coefficients Example: Findthepointsthatlieonacircle (x c 1 ) 2 +(y c 2 ) 2 =c 2 3 Thepresenceofthreeparameters(c 1,c 2 andc 3 )resultsina3-dparameter spacewithcubelikecellsandaccumulatorsoftheforma(i,j,k)!

13 Image segmentation(10.2.7) SLIDE 13/13 Example 10.14: Using the Hough transform for edge linking

Edge linking. Two types of approaches. This process needs to be able to bridge gaps in detected edges due to the reason mentioned above

Edge linking. Two types of approaches. This process needs to be able to bridge gaps in detected edges due to the reason mentioned above Edge linking Edge detection rarely finds the entire set of edges in an image. Normally there are breaks due to noise, non-uniform illumination, etc. If we want to obtain region boundaries (for segmentation)