1. Measurements and Scale Arjan Kuijper arjan@itu.dk

2. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 2 15 Before doing anything Things do not have a shape Like Santa Claus has a suit. Jan Koenderink

3. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 3 15 Measurements How to measure a cloud?

4. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 4 15 Measurements What do we measure?

5. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 5 15 Observations Objects have a size.Objects have a size. Objects consists of objects of various sizes.Objects consists of objects of various sizes.  They contain several scales. Objects are measured by some device.Objects are measured by some device.  Cameras, the eye, … Devices are finite.Devices are finite.  They have a minimum and a maximum detection range: the inner and outer scale. They determine the spatial resolution. The device must allow multi-scale structures.The device must allow multi-scale structures.  It has to respect the various sizes of the object. The inner scale isn’t always the best scale.

6. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 6 15 Founding fathers of scale space

7. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 7 15 The visual system We see multi-scale:We see multi-scale: The images only contain two values (black and white).The images only contain two values (black and white). We regards them as grey level images, or see structure.We regards them as grey level images, or see structure.

8. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 8 15 The eye – to make you interested

9. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 9 15 The retinal device

10. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 10 15 To model a device It is finite.It is finite.  Infinite resolution is impossible. Take uncommitted observationsTake uncommitted observations  There is no bias, no knowledge, no memory. We know nothing.We know nothing.  At least, at the first stage. Refine later on. Allow different scales.Allow different scales.  There’s more than just pixels. View them simultaneously.View them simultaneously.  There is no preferred size. Noise is part of the measurement.Noise is part of the measurement.  Beware!

11. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 11 15 To model (II) Don’tDon’t trust the resolution.  What does a detector of a 3 pixels circular size detect?

12. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 12 15 To model (III) Don’t trust the grid.

13. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 13 15 Axioms When observing the world a device is necessary: When observing the world a device is necessary: Real world -> Device -> image To model a device axioms are necessary. To model a device axioms are necessary. When observing images a device is necessary: When observing images a device is necessary: Image -> Device -> observation What kind of axioms are reasonable? What kind of axioms are reasonable?

14. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 14 15 Axioms of measurements and scale I1I2I3O KYBL1F1APN L2F2 Convolution kernel xxxxxxxxx xx Semigroup property xxxxxxxxx Locality x Regularity xxxxxxxx Infinitesimal generator x Max. loss principle x Causality xxxxx Nonnegativity xxxxxx Tikhonov regularization x Average grey level invar. xxxxxx Flat kernel for t to infinity x Isometry invariance xxxxxxxxxxx Homogeneity & isotropy x Separability xx Scale invariance xxxxxxxx Valid for dimension12221,2 11>1>1N NNN

15. Measurements and scale; PhD course on Scale Space, Cph 1-5 Dec 2003 15 Sources Front-End Vision and Multi-Scale Image Analysis, Bart ter Haar RomenyFront-End Vision and Multi-Scale Image Analysis, Bart ter Haar Romeny Linear Scale-Space has First been Proposed in Japan, Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Journal of Mathematical Imaging and Vision (10), 237-252, 1999.Linear Scale-Space has First been Proposed in Japan, Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Journal of Mathematical Imaging and Vision (10), 237-252, 1999. The structure of images, Jan Koenderink, Biological Cybernetics (50), 363-370, 1984.The structure of images, Jan Koenderink, Biological Cybernetics (50), 363-370, 1984. Solid Shape, Jan KoenderinkSolid Shape, Jan Koenderink On the Gaussian Scale-Space Taizo Iijima IEICE Transactions D (E86-D), 1162-1164, 2003On the Gaussian Scale-Space Taizo Iijima IEICE Transactions D (E86-D), 1162-1164, 2003