Department of Electrical and Computer Engineering
16:332:570 ROBUST COMPUTER VISION
Some Vector Calculus
Let x Rn , x = [x1 xn ] , g (x) a scalar valued function of x, z(x) Rp , z = [z1 zp ] , a
vector valued function of x, and A(x) Rpq a matrix valued fu

Distribution of cells on the retina
The Human Eye
Visual
System
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The Gradient Motivation.
is the arclength paramater along a curve in
We want to compute the gradient of an -dimensional surface with the coordinated
. The surface
is represented with an additional dimension. Please see the gure.
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Department of Electrical and Computer Engineering
16:332:570 ROBUST COMPUTER VISION
Some Matrix Facts
Block Matrix Multiplication
Consider two block matrices, the n p matrix A and the p m matrix B
B11 B1t
A11 A1q
B = .
A = .
Bq1 Bqt
As1 Asq
where Aik is

3D Antisymmetric (Skew-symmetric) Matrices.
In 3D we have
a = [a1 a2 a3 ]
b = [b1 b2 b3 ]
and can compute the vector (cross) product a b, with the result orthogonal to
both a and b, as
a2 b3 a3 b2
0 a3 a2
0 a1 b = a3 b1 a1 b3
a b = Ab = [a] b = a3
a1 b2

Translating & Scaling
= Scaling & Translating ?
1
P ' ' ' T S P 0
0
0
1
0
t x s x
t y 0
1 0
s x
P' ' ' S T P 0
0
s x
0
0
0
sy
0
0
sy
0
0
0 x s x
0 y 0
1 1 0
0
sy
0
t x x s x x t x
t y y s y y t y
1 1
1
0 1 0 t x x
s y 0 0 1 t y y
0 1 0 0 1

Department of Electrical and Computer Engineering
16:332:570 ROBUST COMPUTER VISION
Singular Vector Decomposition
and some things around
LINEAR MAPPING
z = X
Rp
X is an (n, p) matrix of rank r p.
x
1
.
X = . = [x1 xp ]
.
x
n
z Rn
np
the difference is in

The Eye
The human eye is a camera!
Iris - colored annulus with radial muscles
Pupil - the hole (aperture) whose size is controlled by the iris
Lens - changes shape by using ciliary muscles (to focus on objects
at different distances)
diff
di
Retina -

Why the smoothed differentiation lters are so simple?
In one-dimension, we have the discrete data g (i) and a discrete orthogonal polynomial basis l (i) where l = 0, 1, . . . , p. Because of orthogonality, the data and
the basis have to be dened on
i = m,

A few relations in Edge detection with embedded condence.
Relation (A.10). Trace of two matrices.
Given two matrices A of n p dimension and B of p m dimension, we have
a11 . . . a1p
b11 . . . b1m
.
. .
.
= vec[A ] vec[B]
trace[AB] =
.
.
.
.
an1 . .

What Can Be Achieved With Only Bottom-Up Segmentation.
Each image is a 2D projection of the 3D world. Therefore, each distinct image is
a little bit different than the other images showing the same scene. For example,
have slightly different projection an

Epipolar Geometry. Basic Facts.
Essential and fundamental matrices describe the geometric relationship between
corresponding points of two cameras. We assume only inliers in the two images.
Having a point P in the 3D space and given two cameras with image

Algebraic distance.
A surface (or curve) in Rq is dened implicitly as f (yo ) = 0, where yo is a point
on the surface and f () is a scalar. If we take a point y not on the surface, the value
f (y) = 0. The sign can be negative or positive function of the

The least squares supplement.
The Ordinary Least Squares (OLS) estimate.
The traditional least square (OLS) has a variable z pulled out and corrupted by
noise. The input variables yi = yio are correctly measured.
zio = + x
io
i = 1, . . . , n
zi = zio +

The M-estimator.
The simplest M-estimator is homoscedastic and the inlier measurements are corrupted with y GI (0, 2 Ip ), independent and identically distributed (i.i.d.)
symmetric distribution. Only the unknown n1 n inlier residuals are of the form
g (y