linear algebra simplifiedcis580/spring2015/... · homogeneous representation line in is represented...
TRANSCRIPT
![Page 1: Linear Algebra Simplifiedcis580/Spring2015/... · Homogeneous representation Line in Is represented by a point in : But correspondence of line to point is not unique We define set](https://reader033.vdocuments.site/reader033/viewer/2022042611/5f3b07a10697137fb86d42bb/html5/thumbnails/1.jpg)
Linear Algebra Simplified
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Readings
http://szeliski.org/Book/drafts/SzeliskiBook_20100903_draft.pdf -2.1.5 for camera geometry, -2.1.3, 2.1.4 for rotation representation
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Inner (dot) Product
v w
α
332211
3
2
1
321 ..),,( yxyxyxyyy
xxxwvT ++=!!!
"
#
$$$
%
&
=
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Inner (dot) Product
v w
α 332211
3
2
1
321 ..),,( yxyxyxyyy
xxxwvT ++=!!!
"
#
$$$
%
&
=
wvwvT ⊥⇔= 0
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Cross product
X is normal to both v and w
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Cross Product
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Cross Product
+ -
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Cross Product
+ -
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Cross Product
+ -
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Example
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Cross product as a matrix operator
v ⇥ u = v̂ · u
v̂ =
2
40 �x3 x2
x3 0 �x1
�x2 x1 0
3
5
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Lines and Points
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Line Representation
• a line is
• is the distance from the origin to the line • is the norm direction of the line
• It can also be written as
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Example of Line
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Example of Line (2)
0.42*pi
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Line Homogeneous Representation
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Homogeneous representation
Line in
Is represented by a point in :
But correspondence of line to point is not unique
We define set of equivalence class of vectors in R^3 - (0,0,0) As projective space
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Point
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Homogenous representation of point
A point lies on a line:
A point in P^2 is defined by the equivalence class of k(x,y,1)
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Homogeneous Coordinates Homogeneous coordinates
• represent coordinates in 2 dimensions with a 3-vector
!!!
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''''' →'!"
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1
coords shomogeneou yx
yx
A point:
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Homogeneous -> Real Coordinates divide the third number out:
• (x, y, w) represents a point at location (x/w, y/w) • (x, y, 0) represents a point at infinity (in direction x,y)
• (0, 0, 0) is not allowed
Convenient coordinate system to represent many useful transformations
1 2
1
2 (2,1,1) or (4,2,2) or (6,3,3)
x
y
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Example of Point
>> 0.6*42+0.8* 141 ans = 138
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Line passing two points
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Line passing through two points
l is the line passing two points
Two points:
Define a line
Proof:
x
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Line passing through two points
• More specifically,
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Matlab codes
• function l = get_line_by_two_points(x, y)
• x1 = [x(1), y(1), 1]'; • x2 = [x(2), y(2), 1]'; • l = cross(x1, x2); • l = l / sqrt(l(1)*l(1)+l(2)*l(2));
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Example of Line
>> 218*0.6+13*0.8 ans = 141.2000
Verify p2 lies on the line P2
P1 (close)
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Example of Line P2
P1
>> p1 = [42,142,1]; >> p2=[218,13,1]; >> l = cross(p1,p2) l = 129 176 -30410
Test line:
>> l = l/sqrt(l(1)^2+l(2)^2) l = 0.5912 0.8066 -139.3591
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>> x = [221;23]; >> y = [218;268]; >> l = get_line_by_two_points(x,y); >> l l = -0.2448 -0.9696 265.4744
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Point passing two lines
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Intersection of lines
Define a point
Given two lines: ,
X is the intersection of the two lines
l
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Intersection of lines
• More specifically,
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Matlab codes
• function x0 = get_point_by_two_line(l, l1)
• x0 = cross(l, l1); • x0 = [x0(1)/x0(3); x0(2)/x0(3)];
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Example of Lines Intersection
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Point and Line at infinity
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Point at infinity
Example: Consider two parallel horizontal lines:
Intersection =
Point at infinity in the direction of y
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Point at infinity, Ideal points
Intersection:
Any point (x1,x2,0) is intersection of lines at infinity
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Points at infinity
• Under projective transformation, • All parallel lines intersects at the point at infinity
• One point at infinity ó one parallel line direction
• Where are the points at infinity in the image plane?
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Line at infinity
• A line passing all points at infinity:
• Because :
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Projective Transformation
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From Plane to Plane
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Projective transformation Goal: study geometry of image projection from one plane to another plane(the image plane).
Facts: 1) parallel lines intersect, 2) circle becomes ellipses, 3) straight line is still straight
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Projective transformation
Definition: Line remains a line! Projective transform is an invertible mapping from to itself, such that three points lies on a same line iff do.
0 0.5 1
0
0.2
0.4
0.6
0.8
1
0.6 0.8 1 1.2
2
2.2
2.4
2.6
2.8
3
3.2
3.4
x1
x2
x3
h(x1)
h(x2)
h(x3)
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• Theorem:
Projective transformation
Check what happened to lies on line l?
Line Mapping:
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=
x’ y’
x y
How many independent para? Can we always set h33 = 1?
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Classes of 2D projective transformations
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Special case: Similarity Transformation
Similarity
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
2
2.5
3
3.5
4
4.5
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Special Case: Affine Transformation
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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Special case: Projective transformation
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
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Example
0.6 0.8 1 1.2
2
2.2
2.4
2.6
2.8
3
3.2
3.40 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
10 0.1 0.2 0.3
0
0.1
0.2
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0.4
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0.6
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
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0.25
0.3
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Example
• Projective
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
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Example
Affine
0 0.1 0.2 0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
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Example
0 0.1 0.2 0.3
0
0.1
0.2
0.3
0.4
0.5
0.6
Similarity
0.6 0.8 1 1.2
2
2.2
2.4
2.6
2.8
3
3.2
3.4
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Matlab codes
• function img_now = projective_transform(img, H);
• px = [0 1]; • py = [0 1]; • tform = maketform('projective', H'); • [img_now, xdata, ydata]= imtransform(img, tform,
'udata', ... • px, 'vdata', py, 'size', size(img));
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• Extra slides
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Points, Lines & Projective Transformation
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Lines under Projective Transformation • With homogenous coordinates:
• Point:
• Line:
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Before transformation
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After transformation
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Just checking…
• Verify:
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Vanishing point, revisited
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Points at infinity: Revisited
• Where are the points at infinity in the image plane? • The point at infinity can be in the FINITE region of the image !
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Example
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Seeing vanishing point
• vanishing point of horizontal direction:
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• In the image plane:
Line at infinity: Revisited
• A line passing all points at infinity:
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Line of infinity
• line of infinity is:
Notice: this is the last column of
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The line of infinity