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