Hierarchical Polynomial-Bases & Sparse Grids 1/21 grid: Gitter сéтка sparse: spärlich, dünn рéдкий.
Sparse Feature Maps in a Scale Hierarchy -...
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Sparse Feature Maps in a Scale Hierarchy Per-Erik Forss´ en and G¨ osta Granlund Computer Vision Laboratory, Link¨oping University, Sweden {perfo,gosta}@isy.liu.se, http://www.isy.liu.se/cvl/ Introduction This paper describes an essential step towards a view centred representation of the low-level structure in an image. The low-level structure (lines and edges) is separated into three sparse feature maps per scale. A value in one of the maps has a descriptive, and at the same time locally linear behaviour, making these maps ideal as input to a learning machinery. phase π dark lines π 2 phase edges 0 phase bright lines Image scale pyramid φ Line filters, and edge filters are quadrature complements. The local phase of a signal can be defined us- ing the ratio of line and edge filter responses. Characteristic phase The three maps in each scale correspond to local signal phases of 0, π, and ±π/2 respectively. The reason that these are chosen is that they are spatially stable over scale, and thus characterize the local image region. Signal Line responses in three scales Edge responses in three scales 10 20 30 40 50 60 70 0 0.5 1 10 20 30 40 50 60 70 -0.2 -0.1 0 0.1 0.2 10 20 30 40 50 60 70 -0.2 -0.1 0 0.1 0.2 By requiring that a line event is present in two neighbouring scales, one octave apart, and requiring that there is no edge response in the second lower octave, we are able to eliminate the ringings in the response, and produce a sparse output. Lines and edges in scale space The resultant representation exhibits a smooth transition between line and edge description of events. Signal Response at 0 Phase Response at ±π/2 Phase 0 50 100 150 200 250 300 0 0.5 1 0 50 100 150 200 250 300 0 0.2 0.4 0 50 100 150 200 250 300 0 0.2 0.4 Since images can be successfully approximated as locally one- dimensional, this line of reasoning can be extended to 2D. Real Imag The locally dominant orientation is represented in the maps as a complex argument. The colour rep- resents this argument, and the intensity represents the magnitude. Image 0 Phase bright lines π Phase dark lines ±π/2 Phase edges No responses Scale hierarchy of a natural image
Transcript of Sparse Feature Maps in a Scale Hierarchy -...
Sparse Feature Maps in a Scale HierarchyPer-Erik Forssen and Gosta Granlund
Computer Vision Laboratory, Linkoping University, Sweden{perfo,gosta}@isy.liu.se, http://www.isy.liu.se/cvl/
Introduction
This paper describes an essential step towards a viewcentred representation of the low-level structure in an image.The low-level structure (lines and edges) is separated into threesparse feature maps per scale. A value in one of the maps hasa descriptive, and at the same time locally linear behaviour,making these maps ideal as input to a learning machinery.
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dark lines
π2 phase
edges
0 phasebright lines
Image scale pyramid
φ
Line filters, and edge filters arequadrature complements. The localphase of a signal can be defined us-ing the ratio of line and edge filterresponses.
Characteristic phase
The three maps in each scale correspond to local signalphases of 0, π, and ±π/2 respectively. The reason that theseare chosen is that they are spatially stable over scale, and thuscharacterize the local image region.
Signal
Line responsesin three scales
Edge responsesin three scales
10 20 30 40 50 60 700
0.5
1
10 20 30 40 50 60 70
−0.2
−0.1
0
0.1
0.2
10 20 30 40 50 60 70
−0.2
−0.1
0
0.1
0.2
By requiring that a line event is present in two neighbouringscales, one octave apart, and requiring that there is no edgeresponse in the second lower octave, we are able to eliminatethe ringings in the response, and produce a sparse output.
Lines and edges in scale space
The resultant representation exhibits a smooth transitionbetween line and edge description of events.
Signal
Response at0 Phase
Response at±π/2 Phase
0 50 100 150 200 250 3000
0.5
1
0 50 100 150 200 250 3000
0.2
0.4
0 50 100 150 200 250 3000
0.2
0.4
Since images can be successfully approximated as locally one-dimensional, this line of reasoning can be extended to 2D.
Real
Imag The locally dominant orientation is represented inthe maps as a complex argument. The colour rep-resents this argument, and the intensity representsthe magnitude.
Image 0 Phasebright lines
π Phasedark lines
±π/2 Phaseedges
No responses
Scale hierarchy of a natural image
