the topographic primal sketch slides

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The Topographic Primal Sketch

Robert Haralick

Layne WatsonThomas Laffey

The International Journal Of RoboticsResearch

Vol 2 No 1 Spring 1983

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The Topographic Primal Sketch

Classify the underlying image inten-sity surface patches according to thecategories defined by monotonic graytone invariant functions of directionalderivatives.

Peak

Pit

Ridge

Ravine

Saddle

Flat

Hillside

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The Gradient Vector

The gradient foffis a vector whosemagnitude at a given point (r, c) is themaximum rate of change of f at thatpoint and whose direction is the direc-tion in which the surface has the great-

est rate of change.

f=frfc

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Directional Derivative

The directional derivative of a functionfat the point(r, c)in the direction isdenoted by f(r, c).

f

(r, c) = limh0

f(r+h sin, c+h cos) f(r, c)h

The direction is the clockwise anglefrom the column axis.

f(r, c) = f

r(r, c)sin +

f

c(r, c)cos

= (sin cos)f

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Hessian

H=

2f2r

2frc

2frc

2f2c

We denote the second derivative offat the point (r, c) in the direction by

f(r, c)

f(r, c) = (sin cos)H

sincos

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Notation

fgradient vector of a function f f gradient magnitude off1 unit vector in the direction in

which second directional derivative-

has the greatest magnitude2 unit vector in the direction in

which second directional derivativehas the smallest magnitude

1value of second directional deriva-tive in the direction of1

2value of second directional deriva-tive in the direction of2

f1 value of first directional deriva-tive in the direction of1

f

2 value of first directional deriva-

tive in the direction of2

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Curvature

There is a direct relationship betweenthe eigenvalues 1 and 2 and the cur-vature in the directions 1 and 2.

When the first directional derivativef i=0, then the curvature in thedirection

i

is=

i1 + f f

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Peak

A peak occurs where there is local max-ima in all directions.

f(r, c) =01


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Pit

A pit occurs where there is local min-ima in all directions.

f(r, c) =01 >0

2 >0

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Ridge

A ridge occurs where there is a lo-cal maximum in one direction. As wewalk along a ridge, the points to theright and left of us are lower than theones we are on. The direction across

the ridge is the direction in which thecurvature is extremized.

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Sloped or Curved Ridge

A sloped or curved ridge must satisfyone or the other of the following setsof conditions.

f(r, c) = 01


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Flat Ridge

A flat ridge must satisfy one of thefollowing sets of conditions.

f(r, c) = 01


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Ravine

A ravine (valley) is identical to a ridgeexcept that it is a local minimum (ratherthan maximum) in one direction. Aswe walk along the ravine line, the pointsto the right and left of us are higher

than the point we are on. A ravinemay be sloped or curved or flat.

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Sloped or Curved Ravine

f(r, c) = 01 >0

f(r, c) 1 = 0

f(r, c) = 02 >0

f(r, c) 2 = 0

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Flat Ravine

A flat ravine must satisfy one of thefollowing sets of conditions.

f(r, c) = 01 >0

2= 0

f(r, c) = 01= 0

2 >0

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Saddle Point

f(r, c) = 012


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Flat

A flat (plain) is a simple horizontalsurface. It must have zero gradientand no curvature.

f(r, c) = 0

1= 02= 0

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Hillside

A hillside is anything not covered bythe previous categories. It has nonzerogradient and no strict extrema in thedirections of maximum and minimumsecond directional derivative.

Any one of the four conditions implya hillside.

f 1 = 0 f 2 = 0 f 1 = 02= 0

f 2 = 01= 0

f

= 0

1= 0

2= 0

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Hillside Subcategories

Slope if1=2= 0

Convex if1 >0 and 2 0 or2 >0 and 1 0

Concave if1


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Ridge Ravine Continua

Entire areas may be classified as ridgeor ravine.

Right circular cone: f(r, c) =r2 +c2

Hemisphere: f(r, c) = k2

r2

c2

Any function of the form h(r2 +c2)

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Surface Estimation

If we assume that the observed sur-face is a noisy discretely sampled ob-servation, then there is a problem ofhow to estimate derivatives. Methodsbased strictly on finite differences will

fail miserably in noise. Least squaresestimation or robust least squares es-timation must be used.

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Classification1. Estimate the surface around each

pixel by a local least squares or localrobust least squares.

2. Use the estimated surface to findthe gradient, gradient magnitude, and

the eigenvalues and eigenvectors ofthe Hessian at the center of the pixelsneighborhood (0,0).

3. Search in the direction of the eigen-vectors calculated in step 2 for azero-crossing of the first directional

derivative within the pixels area. Ifthe eigenvalues of the Hessian areequal and nonzero, then search inthe Newton direction.

4. Recompute the gradient, gradientmagnitude, and values of the sec-

ond directional derivative extremaat each zero-corssing. Then applythe labeling scheme.

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