Linear Subspaces - Geometry
No Invariants, so Capture
Variation
• Each image = a pt. in a high-dimensional
space.
– Image: Each pixel a dimension.
– Point set: Each coordinate of each pt. A
dimension.
• Simplest rep. of variation is linear.
– Basis (eigen) images: x1…xk
– Each image, x = a1x1 + … + akxk
• Useful if k << n.
When is this accurate?
• Approximately right when:
– Variation approximately linear. Always true for
small variation.
– Some variations big, some small, can discard
small.
• Exactly right sometimes.
– Point features with scaled-orthographic projection.
– Convex, Lambertian objects and distant lights.
Principal Components Analysis
(PCA)
• All-purpose linear approximation.
• Given images (as vectors)
• Finds low-dimensional linear subspace
that best approximates them.
– Eg., minimizes distance from images to
subspace.
Derivation on whiteboard
• This is all taken from Duda, Hart and
Stork Pattern Classification pp. 114-117.
Excerpt in library.
SVD
• Scatter matrix can be big, so computation
non-trivial.
• Stack data into matrix X, each row an image.
SVD gives X = UDVT
– D is diagonal with non-increasing values.
– U and V have orthonormal rows.
• VT(:,1:k) gives first k principal components.
• matlab
Linear Combinations
u

v
u

v
 .

 .
 .

u
v

1
1
1
1
2
1
2
1
m
1
m
1
u
v
u
v
1
2
.
.
.
1
2
2
2
u
v
u
1
n
1
n
2
n
2
v
.
.
.
u
v
.
.
.
u
v
2
m
2
n
m
2
m
2
n
.
.
I
.
m
n
 s
 
 s
 s
 
 s

 
 
 
 
 s
 s
 
1
1,1
1
2 ,1
2
1,1
2
2 ,1
m
1,1
m
2 ,1
s
s
s
1
1, 2
1
2,2
2
1, 2
s
.
.
.
s
s
2
2,2
m
1, 2
m
2,2
S
s
s
s
1
1, 3
1
2,3
2
1, 3
s
2
2,3




 x
t 
 y
 z

 1

t 
t 
t
t
t
1
x
1
y
2
x
2
1
y
1
1
s
s
m
1, 3
m
2,3
x
y
z
1
2
.
.
.
2
x

y
z 

1
n
n
2
n
m
x
m
y
P
Immediately apparent that u and v coordinates lie in a 4D
linear subspace
We Can Remove Translation (1)
• This is trivial, because we can pick a simple
origin.
– World origin is arbitrary.
– Example: We can assume first point is at origin.
• Rotation then doesn’t effect that point.
• All its motion is translation.
– Better to pick center of mass as origin.
• Average of all points.
• This also averages all noise.
Remove Translation (2)
x 
   0
n  i  
WLOG :   y    0 
i  1 i   0 
z   
 i
n
u 
i
u
k 1
i
k
u

v
u

v
~
I 




u
v

1
v 
v
k 1
2
1
2
1
1
m
1
i
1
1
m
n
n
1
u
v
u
1
v
2
.
.
.
u
v
1
u u
v v
u u
1
v v
2
1
1
2
1
.
.
.
2
2
m
m
2
2
m
2
1
2
2
n
m
m
2
m
n
n
2
.
.
.
u u
v v
1
n
2
2
1
1
2
2
u u 

v v 
u u 

v v 
. 

. 
. 

u u 
v  v 
m
n
m
.
.
.
m
m
n
i
k
n
Notice this is just the first step of
PCA.
First Step: Remove Translation
(3)
 u~
~
v
 u~
~
v
 .

 .
 .

 u~
 v~

1
1
2
u~
v~
u~
v~
.
.
.
u~
v~
.
.
.
u~
v~
1
2
1
1
.
.
.
1
2
2
1
2
m
1
2
2
n
1
2
2
2
m
m
n
.
.
.
n
1
1,1
1
2
1,1
2
2 ,1
n
m
 s
 
 s
 s
 
 s

 
 
 
 
 s
 s
 
2 ,1
n
2
m
1
n
2
1
1
u~
v~
u~
v~
m
m
1,1
m
2 ,1
s
1
1, 2
s
s
s
.
.
1
2,2
2
1, 2
2
2,2
.
s
s
m
1, 2
m
2,2
s 

s 
s 

s  x
 y

 z


s 
s 
1
1, 3
1
2,3
2
1, 3
1
x
y
1
z
2
2,3
m
1, 3
m
2,3
1
2
2
2
.
.
.
x

y
z 
n
n
n
Rank Theorem
 u~
~
v
 u~
~
v
 .

 .
 .

 u~
 v~

1
1
2
u~
v~
u~
v~
.
.
.
u~
v~
.
.
.
u~
v~
1
2
1
1
.
.
.
1
2
2
1
2
m
1
2
2
1
2
2
2
m
m
n
.
.
~
I
.
n
1
1,1
1
2
1,1
2
2 ,1
n
m
 s
 
 s
 s
 
 s

 
 
 
 
 s
 s
 
2 ,1
n
2
m
1
n
n
2
1
1
u~
v~
u~
v~
m
m
1,1
m
2 ,1
s
1
1, 2
s
s
s
.
.
1
2,2
2
1, 2
2
2,2
.
s
s
m
1, 2
m
2,2
s 

s 
s 

s  x
 y

 z


s 
s 
1
1, 3
1
2,3
2
1, 3
1
x
y
1
z
2
2,3
m
1, 3
1
2
.
2
.
.
x

y
z 
n
n
2
n
P
m
2,3
S
So, given any object, u and v coordinates
of any image of it lie in a 3-dimensional
linear subspace.
~
I has rank 3.
This means there
are 3 vectors
such that every
~
row of I is a
linear
combination of
these vectors.
These vectors
are the rows of P.
Lower-Dimensional Subspace
• (a4,b4)  affine invariant coordinates of
point 4 relative to first three.
• Represent image: (a4,b4, a5,b5,… an,bn)
– This representation is complete.
– The a or b coordinates of all images of an
object occupy a 1D linear subspace.
Representation is Complete
s

s
s
s
1,1
1, 2
2 ,1
2,2
s
s
t
t
1, 3
2,3
x
y

a
 P  

a
1,1
2 ,1
a
a
1, 2
2,2
t
t
r

 r
 0

1,1
x
2 ,1
y
r
r
0
1, 2
2,2
r
r
0
1, 3
2,3
0

0P
1 
where
(r , r , r )  (r , r , r )  0
1,1
1, 2
1, 3
2 ,1
2,2
2,3
(r , r , r )  (r , r , r )  1
1,1
1, 2
1, 3
2 ,1
2,2
2,3
• 3D-2D affine transformation is projection in some
direction + 2D-2D affine transformation.
• 2D-2D affine maps first three image points anywhere.
So they’re irrelevant to a complete representation.
• Once we use only affine coordinates, 2D-2D affine
transformation no longer matters.
1D Subspace
Summary
• Projection can be linearized.
• So images produced under projection can be
linear and low-dimensional.
• Are these results relevant to surfaces of real
3D objects projected to 2D?
– Maybe to features; probably not to intensity
images.
• Why would images of a class of objects be
low-dimensional?