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Measuring perimeter of a discrete object Nataša Sladoje Introduction Main story Area coverage digitization Measuring perimeter of a discrete object Slopes and difference of column sums Optimization Local computations Rotations of the Nataša Sladoje plane Length estimation Complete algorithm Evaluation and examples Summer School of Image Processing, Vienna Conclusions July 11, 2008

Measuring perimeter of a Outline discrete object Nataša Sladoje Introduction 1 Introduction Main story Area coverage 2 Main story digitization Slopes and Area coverage digitization difference of column sums Slopes and difference of column sums Optimization Local computations Optimization Rotations of the plane Length estimation Local computations Complete algorithm Rotations of the plane Evaluation and examples Length estimation from 3 × 3 configurations Conclusions Complete algorithm Evaluation and examples Conclusions 3

Measuring perimeter of a discrete object Nataša Sladoje Introduction Main story Area coverage digitization Slopes and difference of column sums Introduction Optimization Local computations Rotations of the Measuring perimeter of discrete binary objects plane Length estimation Complete algorithm Evaluation and examples Conclusions

Measuring perimeter of a To introduce myself... discrete object Nataša Sladoje Nataša Sladoje Introduction Assistant Professor at the Department of Mathematics Main story Faculty of Engineering Area coverage University of Novi Sad, Serbia digitization Slopes and difference of column sums Optimization Local computations B.Sc. in Mathematics, University of Novi Sad Rotations of the M.SC. in Discrete Mathematics, University of Novi Sad plane Length estimation Ph.D. in Image Analysis, Centre for Image Analysis, Uppsala, Sweden Complete algorithm Evaluation and examples Conclusions Once a student at SSIP (Budapest, 2003) Twice a teacher at SSIP (Szeged, 2006 and 2007) My department is in the CEEPUS network “Medical Imaging & Medical Information Processing” since 2005, when Serbia joined CEEPUS programme.

Measuring perimeter of a To introduce myself... discrete object Nataša Sladoje Nataša Sladoje Introduction Assistant Professor at the Department of Mathematics Main story Faculty of Engineering Area coverage University of Novi Sad, Serbia digitization Slopes and difference of column sums Optimization Local computations B.Sc. in Mathematics, University of Novi Sad Rotations of the M.SC. in Discrete Mathematics, University of Novi Sad plane Length estimation Ph.D. in Image Analysis, Centre for Image Analysis, Uppsala, Sweden Complete algorithm Evaluation and examples Conclusions Once a student at SSIP (Budapest, 2003) Twice a teacher at SSIP (Szeged, 2006 and 2007) My department is in the CEEPUS network “Medical Imaging & Medical Information Processing” since 2005, when Serbia joined CEEPUS programme.

Measuring perimeter of a To introduce myself... discrete object Nataša Sladoje Nataša Sladoje Introduction Assistant Professor at the Department of Mathematics Main story Faculty of Engineering Area coverage University of Novi Sad, Serbia digitization Slopes and difference of column sums Optimization Local computations B.Sc. in Mathematics, University of Novi Sad Rotations of the M.SC. in Discrete Mathematics, University of Novi Sad plane Length estimation Ph.D. in Image Analysis, Centre for Image Analysis, Uppsala, Sweden Complete algorithm Evaluation and examples Conclusions Once a student at SSIP (Budapest, 2003) Twice a teacher at SSIP (Szeged, 2006 and 2007) My department is in the CEEPUS network “Medical Imaging & Medical Information Processing” since 2005, when Serbia joined CEEPUS programme.

Measuring perimeter of a To introduce myself... discrete object Nataša Sladoje Nataša Sladoje Introduction Assistant Professor at the Department of Mathematics Main story Faculty of Engineering Area coverage University of Novi Sad, Serbia digitization Slopes and difference of column sums Optimization Local computations B.Sc. in Mathematics, University of Novi Sad Rotations of the M.SC. in Discrete Mathematics, University of Novi Sad plane Length estimation Ph.D. in Image Analysis, Centre for Image Analysis, Uppsala, Sweden Complete algorithm Evaluation and examples Conclusions Once a student at SSIP (Budapest, 2003) Twice a teacher at SSIP (Szeged, 2006 and 2007) My department is in the CEEPUS network “Medical Imaging & Medical Information Processing” since 2005, when Serbia joined CEEPUS programme.

Measuring perimeter of a ...and to introduce the topic discrete object Nataša Sladoje Introduction • The task of image analysis is to extract relevant information from Main story images. Area coverage digitization • Images contain discrete representations of real continuous objects. Slopes and difference of column sums • Our aim is usually to obtain information about continuous real Optimization Local computations objects, having available their discrete representations. Rotations of the plane • Different numerical descriptors, such as area, perimeter, moments, of Length estimation Complete algorithm the objects are often of interest, for the tasks of shape analysis, Evaluation and classification, etc. examples Conclusions • Accurate and precise perimeter estimates of the real objects, based on their discrete representations, have been of interest for more than forty years, and many papers are published on that topic; the problem still attracts attention.

Measuring perimeter of a Formulation of the problem discrete object Nataša Sladoje Having a discrete representation of a real object, inscribed and digitized in an integer grid, Introduction estimate its perimeter (length of its border) with as small Main story Area coverage digitization error as possible. Slopes and difference of column sums Small error provides not only accurate, but also precise Optimization Local computations estimates. Rotations of the We obtain correct feature values - accuracy . plane Length estimation Complete algorithm Repeated measurements provide very similar results - Evaluation and examples precision . Conclusions

Measuring perimeter of a ... and a way to solve it ... discrete object Nataša Sladoje Introduction Main story Area coverage digitization Slopes and difference of column sums Optimization Local computations Rotations of the plane Length estimation Complete algorithm ...walk along the object boundary and accumulate local step Evaluation and examples lengths . Conclusions A digital boundary of an object (green) consists of horizontal and vertical links corresponding to pixel edges. Its length is obtained by simply counting pixel edges traversed. Such a perimeter estimate leads to a large overestimate. Freeman chain coding can be used instead. Eight directions of steps connecting consecutive pixel centres lead to two step lengths . Accumulating the lengths of all the steps along the (inner) boundary of the object leads to perimeter estimation. Boundary detection is required.

Measuring perimeter of a ... and a way to solve it ... discrete object Nataša Sladoje Introduction Main story Area coverage digitization Slopes and difference of column sums Optimization Local computations Rotations of the plane Length estimation Complete algorithm ...walk along the object boundary and accumulate local step Evaluation and examples lengths . Conclusions A digital boundary of an object (green) consists of horizontal and vertical links corresponding to pixel edges. Its length is obtained by simply counting pixel edges traversed. Such a perimeter estimate leads to a large overestimate. Freeman chain coding can be used instead. Eight directions of steps connecting consecutive pixel centres lead to two step lengths . Accumulating the lengths of all the steps along the (inner) boundary of the object leads to perimeter estimation. Boundary detection is required.

Measuring perimeter of a ... and a way to solve it ... discrete object Nataša Sladoje Introduction Main story Area coverage digitization Slopes and difference of column sums Optimization Local computations Rotations of the plane Length estimation Complete algorithm ...walk along the object boundary and accumulate local step Evaluation and examples lengths . Conclusions A digital boundary of an object (green) consists of horizontal and vertical links corresponding to pixel edges. Its length is obtained by simply counting pixel edges traversed. Such a perimeter estimate leads to a large overestimate. Freeman chain coding can be used instead. Eight directions of steps connecting consecutive pixel centres lead to two step lengths . Accumulating the lengths of all the steps along the (inner) boundary of the object leads to perimeter estimation. Boundary detection is required.

Measuring perimeter of a Marching Squares technique discrete object Nataša Sladoje Introduction Main story Area coverage digitization Slopes and difference of column sums Optimization Local computations b/2 a b b Rotations of the plane 4 2 = 16 configurations reduce to 4 Length estimation Complete algorithm different perimeter contributions. Evaluation and examples Conclusions By checking local ( 2 × 2 ) pixel configurations, local perimeter contributions can be assigned. Boundary detection simultaneously with perimeter estimation!