Transcript
CSE486, Penn StateRobert Collins
Lecture 2:Intensity Surfaces
and Gradients
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Intensity pattern
Visualizing ImagesRecall two ways of visualizing an image
2d array of numbers
We “see it” at this level Computer works at this level
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Bridging the Gap
Motivation: we want to visualize images at a level highenough to retain human insight, but low enough to allowus to readily translate our insights into mathematicalnotation and, ultimately, computer algorithms that operate on arrays of numbers.
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Surface heightproportional topixel grey value(dark=low, light=high)
Images as Surfaces
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Examples
Mean = 164 Std = 1.8
Note: see demoImSurf.m in matlab examples directory oncourse web site if you wantto generate plots like these.
CSE486, Penn StateRobert Collins
Examples
CSE486, Penn StateRobert Collins
Examples
CSE486, Penn StateRobert Collins
Examples
CSE486, Penn StateRobert Collins
Examples
Mean = 111 Std = 15.4
CSE486, Penn StateRobert Collins
Examples
CSE486, Penn StateRobert Collins
How does this visualization help us?
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Terrain Concepts
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Terrain Concepts
www.pianoladynancy.com/ wallpapers/1003/wp86.jpg
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Terrain ConceptsBasic notions:
Uphill / downhillContour lines (curves of constant elevation)Steepness of slopePeaks/Valleys (local extrema)
Gradient vectors (vectors of partial derivatives) will help us define/compute all of these.
More mathematical notions:Tangent PlaneNormal vectorsCurvature
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Math Example : 1D GradientConsider function f(x) = 100 - 0.5 * x^2
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Math Example : 1D GradientConsider function f(x) = 100 - 0.5 * x^2
Gradient is df(x)/dx = - 2 * 0.5 * x = - x
Geometric interpretation: gradient at x0 is slope of tangent line to curve at point x0
x0
f(x)
tangent line
Δy
Δx
slope = Δy / Δx
= df(x)/dx
x0
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Math Example : 1D Gradientf(x) = 100 - 0.5 * x^2 df(x)/dx = - x
x0 = -2
grad = slope = 2
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Math Example : 1D Gradientf(x) = 100 - 0.5 * x^2 df(x)/dx = - x
gradients 3
2
10
-3
-2
-1
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Math Example : 1D Gradientf(x) = 100 - 0.5 * x^2 df(x)/dx = - x�
gradients 3
2
10
-3
-2
-1
Gradientson this sideof peak arepositive
Gradientson this sideof peak arenegative
Note: Sign of gradient at point tells you what direction to go to travel “uphill”
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Math Example : 2D Gradientf(x,y) = 100 - 0.5 * x^2 - 0.5 * y^2df(x,y)/dx = - x df(x,y)/dy = - y
Gradient = [df(x,y)/dx , df(x,y)/dy] = [- x , - y]
Gradient is vector of partial derivs wrt x and y axes
CSE486, Penn StateRobert Collins
Math Example : 2D Gradientf(x,y) = 100 - 0.5 * x^2 - 0.5 * y^2
Gradient = [df(x,y)/dx , df(x,y)/dy] = [- x , - y]
Plotted as a vector field,the gradient vector at each pixel points “uphill”
The gradient indicates thedirection of steepest ascent.
The gradient is 0 at the peak(also at any flat spots, and local minima,…butthere are none of those for this function)
CSE486, Penn StateRobert Collins
Math Example : 2D Gradientf(x,y) = 100 - 0.5 * x^2 - 0.5 * y^2
Gradient = [df(x,y)/dx , df(x,y)/dy] = [- x , - y]
Let g=[gx,gy] be the gradient vector at point/pixel (x0,y0)
Vector g points uphill (direction of steepest ascent)
Vector - g points downhill (direction of steepest descent)
Vector [gy, -gx] is perpendicular,and denotes direction of constantelevation. i.e. normal to contourline passing through point (x0,y0)
CSE486, Penn StateRobert Collins
Math Example : 2D Gradientf(x,y) = 100 - 0.5 * x^2 - 0.5 * y^2
Gradient = [df(x,y)/dx , df(x,y)/dy] = [- x , - y]
And so on for all points
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Image Gradient
The same is true of 2D image gradients.
The underlying function is numerical(tabulated) rather than algebraic. So need numerical derivatives.
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Numerical DerivativesSee also T&V, Appendix A.2
Taylor Series expansion
Finite forward difference
Manipulate:
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Numerical DerivativesSee also T&V, Appendix A.2
Taylor Series expansion
Finite backward difference
Manipulate:
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Numerical Derivatives
Finite central difference
See also T&V, Appendix A.2
Taylor Series expansion
subtract
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Numerical DerivativesSee also T&V, Appendix A.2
Finite backward difference
Finite forward difference
Finite central differenceMoreaccurate
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Example: Temporal Gradient
derivative
df(x)/dx
A video is a sequence of image frames I(x,y,t).
Each frame has two spatial indices x, y andone temporal (time) index t.
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Example: Temporal Gradient
Consider the sequence ofintensity values observedat a single pixel over time.
derivative
df(x)/dx
1D function f(x) over time 1D gradient (deriv) of f(x) over time
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Temporal Gradient (cont)What does the temporal intensity gradient at each pixel look like over time?
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Example: Spatial Image Gradients
I(x+1,y) - I(x-1,y)
2I(x,y)
I(x,y+1) - I(x,y-1)
2
Ix=dI(x,y)/dx
Iy=dI(x,y)/dy
Partial derivative wrt x
Partial derivative wrt y
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Functions of GradientsI(x,y) Ix Iy
Magnitude of gradient sqrt(Ix.^2 + Iy.^2)
Measures steepness ofslope at each pixel
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Functions of GradientsI(x,y) Ix Iy
Angle of gradient atan2(Iy, Ix)
Denotes similarityof orientation of slope
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Functions of Gradients
What else dowe observe inthis image?
atan2(Iy, Ix)
Enhanced detailin low contrastareas (e.g. foldsin coat; imagingartifacts in sky)
I(x,y)
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Next Time: Linear Operators
Gradients are an example of linear operators,i.e. value at a pixel is computed as a linearcombination of values of neighboring pixels.
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