measurements and scale arjan kuijper [email protected]
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Before doing anythingBefore doing anything
Things do not have a shapeThings do not have a shape
Like Santa Claus has a suit.Like Santa Claus has a suit.
Jan KoenderinkJan Koenderink
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MeasurementsMeasurements
How to measure a How to measure a cloud?cloud?
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MeasurementsMeasurements
What do we measure?What do we measure?
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ObservationsObservations
•Objects have a Objects have a sizesize..
•Objects consists of objects of various Objects consists of objects of various sizessizes.. They contain several They contain several scalesscales..
•Objects are measured by some Objects are measured by some devicedevice.. Cameras, the eye, …Cameras, the eye, …
•Devices are Devices are finitefinite.. They have a minimum and a maximum detection They have a minimum and a maximum detection
range: the range: the innerinner and and outerouter scale. They determine the scale. They determine the spatial resolutionspatial resolution..
•The device must allow The device must allow multi-scalemulti-scale structures. structures. It has to respect the various sizes of the object. The It has to respect the various sizes of the object. The
inner scale isn’t always the best scale.inner scale isn’t always the best scale.
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Founding fathers of scale spaceFounding fathers of scale space
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The visual systemThe 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.
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The eye – to make you interestedThe eye – to make you interested
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The retinal deviceThe retinal device
fovea
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To model a deviceTo model a device
• It is finite.It is finite. Infinite resolution is impossible.Infinite resolution is impossible.
• Take uncommitted observations Take uncommitted observations There is no bias, no knowledge, no memory.There is no bias, no knowledge, no memory.
• We know nothing. We know nothing. At least, at the first stage. Refine later on.At least, at the first stage. Refine later on.
• Allow different scales. Allow different scales. There’s more than just pixels .There’s more than just pixels .
• View them simultaneously.View them simultaneously. There is no preferred size.There is no preferred size.
• Noise is part of the measurement.Noise is part of the measurement. Beware! Beware!
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To model (II)To model (II)
• Don’t Don’t trust the resolution. What does a detector of a 3 pixels circular size detect?
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To model (III)To model (III)
Don’t trust the grid.Don’t trust the grid.
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AxiomsAxioms• When observing the world a device is When observing the world a device is
necessary:necessary:
Real world -> Device -> imageReal world -> Device -> image
• To model a device axioms are necessary. To model a device axioms are necessary.
• When observing images a device is When observing images a device is necessary:necessary:
Image -> Device -> observationImage -> Device -> observation
• What kind of axioms are reasonable?What kind of axioms are reasonable?
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Axioms of measurements and Axioms of measurements and scalescale
I1 I2 I3 O K Y B L1 F1 A P N L2 F2
Convolution kernel x x x x x x x x x x x
Semigroup property x x x x x x x x x
Locality x
Regularity x x x x x x x x
Infinitesimal generator x
Max. loss principle x
Causality x x x x x
Nonnegativity x x x x x x
Tikhonov regularization x
Average grey level invar. x x x x x x
Flat kernel for t to infinity x
Isometry invariance x x x x x x x x x x x
Homogeneity & isotropy x
Separability x x
Scale invariance x x x x x x x x
Valid for dimension 1 2 2 2 1,2 1,2 1 1 >1 N 1,2 N N N
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SourcesSources
• Front-End Vision and Multi-Scale Image Analysis, Front-End Vision and Multi-Scale Image Analysis, Bart ter Haar Romeny Bart ter Haar Romeny
• Linear Scale-Space has First been Proposed in Japan, Linear Scale-Space has First been Proposed in Japan, Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Joachim Weickert, Seiji Ishikawa, Atsushi Imiya Journal of Mathematical Imaging and Vision (10),Journal of Mathematical Imaging and Vision (10),237-252, 1999. 237-252, 1999.
• The structure of images, The structure of images, Jan Koenderink,Jan Koenderink,Biological Cybernetics (50), 363-370, 1984.Biological Cybernetics (50), 363-370, 1984.
• Solid Shape,Solid Shape,Jan KoenderinkJan Koenderink
• On the Gaussian Scale-Space On the Gaussian Scale-Space Taizo IijimaTaizo IijimaIEICE Transactions D (E86-D), 1162-1164, 2003IEICE Transactions D (E86-D), 1162-1164, 2003