Discriminant Functions for the 3 Normal Densitycseweb.ucsd.edu/classes/sp14/cse190-b/lec5.pdfImage Formation Biometrics CSE 190 Lecture 5 CSE190, Spring 2014 Announcements • HW1:

1

CSE190, Spring 2014

Image Formation

Biometrics CSE 190 Lecture 5

CSE190, Spring 2014

Announcements

•  HW1: To be assigned tonight

Pattern Classification, Chapter 2 (Part 3)

3 Discriminant Functions for the Normal Density

• We saw that the minimum error-rate classification can be achieved by the discriminant function gi(x) = ln P(x | ωi) + ln P(ωi)

•  Case of multivariate normal

•  Decision boundary gi(x) = gj(x) €

gi(x) = −12(x − µi)

tΣi−1(x − µi) −

d2ln2π − 1

2lnΣi + lnP(ω i)


4

CASE I

Case Σi = σ2I (I stands for the identity matrix)

€

gi(x) = wit x + wi0 (linear discriminant function)

where :

wi =µiσ 2

; wi0 = −1

2σ 2µitµi + lnP(ω i)

(ω i0 is called the threshold for the ith category!)


5

•  The decision surfaces for a linear machine are pieces of hyperplanes defined by:

gi(x) = gj(x)


6

2


7

The hyperplane separating Ri and Rj are given by

always orthogonal to the line linking the means!

)()()(ln)(

21

0)(

2

2

0

0

jij

i

ji

ji

ji

t

PP

µµωω

µµ

σµµ

µµ

−−

−+=

−=

=−

x

wxxw

)(21)()( 0 jiji PPif µµωω +== x then


8


9 Case II Case Σi = Σ (covariance of all classes are identical but arbitrary!)

Now consider case where P(ωi)=P(ωj),

€

gi(x) = −12(x − µi)

tΣi−1(x − µi) −

d2ln2π − 1

2lnΣi + lnP(ω i)

= −12(x − µi)

tΣ−1(x − µi) −12lnΣ + lnP(ω i)

= −12(x − µi)

tΣ−1(x − µi) + lnP(ω i)

€

gi(x) = −(x − µi)tΣ−1(x − µi)

€

(x − µi)tΣ−1(x − µi) is sometimes called the Mahalanobis distance


10 Case II Case Σi = Σ (covariance of all classes are

identical but arbitrary!)

Hyperplane separating Ri and Rj

Here the hyperplane separating Ri and Rj is generally not orthogonal to the line between the means!

[ ])(

)()()(/)(ln

)(21

)(

0)(

10

10

jiji

tji

jiji

ji

t

PPx µµ

µµµµ

ωωµµ

µµ

−−Σ−

−+=

−Σ=

=−

−

−wxxw


11


12 CASE III

Case Σi = arbitrary The covariance matrices are different for each category

Here the separating surfaces are Hyperquadrics which are: hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoids, hyperparaboloids, hyperhyperboloids)

)(Plnln21

21 w

w21W

:wherewxwxWx)x(g

iii1

iti0i

i1

ii

1ii

0itii

ti

ωΣµΣµ

µΣ

Σ

+−−=

=

−=

=+=

−

−

−

3


13

CSE190, Winter 2011 Biometrics

Image Formation: Outline •  Factors in producing images •  Projection •  Perspective •  Vanishing points •  Orthographic •  Lenses •  Sensors •  Quantization/Resolution •  Illumination •  Reflectance


Earliest Surviving Photograph

•  First photograph on record, “la table service” by Nicephore Niepce in 1822. •  Note: First photograph by Niepce was in 1816.


Images are two-dimensional patterns of brightness values.

They are formed by the projection of 3D objects.

Figure from US Navy Manual of Basic Optics and Optical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc., 1969.


Effect of Lighting: Monet


Change of Viewpoint: Monet

Haystack at Chailly at Sunrise (1865)

4


Pinhole Camera: Perspective projection

• Abstract camera model - box with a small hole in it

Forsyth&Ponce CSE190, Winter 2011 Biometrics

Parallel lines meet in the image •  vanishing point

Image plane

CSE190

Computer Vision I

Equation of Perspective Projection

Cartesian coordinates: •  We have, by similar triangles, that (x, y, z) -> (f x/z, f y/z, f) •  Establishing an image plane coordinate system at C’ aligned with i

and j, we get

€

(x,y, z)→ ( f xz, f yz)

CSE190

Computer Vision I

A Digression

Homogenous Coordinates and

Camera Matrices

CSE190

Computer Vision I

What is the intersection of two lines in a plane?

A Point

CSE190

Computer Vision I

Do two lines in the plane always intersect at a point?

No, Parallel lines don’t meet at a point.

5

CSE190

Computer Vision I

Can the perspective image of two parallel lines meet at a point?

YES

CSE190

Computer Vision I

Projective geometry provides an elegant means for handling these different situations in a unified way and homogenous coordinates are a way to represent entities (points & lines) in projective spaces.

CSE190

Computer Vision I

Homogenous coordinates •  Our usual coordinate system is called a Euclidean

or affine coordinate system

•  Rotations, translations and projection in Homogenous coordinates can be expressed linearly as matrix multiplies

Euclidean World

3D

Homogenous World

3D

Homogenous Image

2D

Euclidean World

2D

Convert

Convert

Projection

CSE190

Computer Vision I

Changes of coordinates: Euclidean -> Homogenous-> Euclidean

In 2-D •  Euclidean -> Homogenous: (x, y) -> k (x,y,1) •  Homogenous -> Euclidean: (u,v,w) -> (u/w, v/w)

In 3-D •  Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1) •  Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w)

CSE190

Computer Vision I

Homogenous coordinates A way to represent points in a projective space 1.  Add an extra coordinate

e.g., (x,y) -> (x,y,1)=(u,v,w)

2. Impose equivalence relation such that (k not 0)

(u,v,w) ≈ k*(u,v,w) i.e., (x,y,1) ≈ (kx, ky, k)

3. “Point at infinity” – zero for last coordinate e.g., (x,y,0)

•  Why do this? –  Possible to write the

action of a perspective camera as a matrix

–  Possible to represent points “at infinity”

•  Where parallel lines intersect

•  Vanishing points are the projection of points of points at infinity

CSE190

Computer Vision I

The camera matrix

Turn this expression into homogenous coordinates – HC’s for 3D point are

(X,Y,Z,T) – HC’s for point in image

are (U,V,W) €

UVW

"

#

$ $ $

%

&

' ' '

=

1 0 0 00 1 0 00 0 1 f 0

"

#

$ $ $ $

%

&

' ' ' '

XYZT

"

#

$ $ $ $

%

&

' ' ' '

€

(x,y, z)→ ( f xz, f yz)

Perspective Camera Matrix A 3x4 matrix

6

CSE190

Computer Vision I

End of the Digression

CSE190

Computer Vision I

Simplified Camera Models

Perspective Projection

Affine Camera Model

Orthographic Projection


Affine Camera Model

•  Take Perspective projection equation, and perform Taylor Series Expansion about (some point (x0,y0,z0).

•  Drop terms of higher order than linear. •  Resulting expression is affine camera model

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

0

zyx

Appropriate in Neighborhood About (x0,y0,z0)


•  Perspective

•  Perform a Taylor series expansion about (x0, y0, z0)

•  Dropping higher order terms and regrouping.

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

yx

zf

vu

uv

!

"#

$

%&=

fz0

x0y0

!

"##

$

%&&−fz02

x0y0

!

"##

$

%&&z− z0( )+

fz0

10

!

"#

$

%& x − x0( )

+fz0

01

!

"#

$

%& y− y0( )+

12fz03

x0y0

!

"##

$

%&&z− z0( )

2+!

uv

!

"#

$

%& ≈

fz0

x0y0

!

"##

$

%&&+

f / z0 0 − fx0 / z02

0 f / z0 − fy0 / z02

!

"

###

$

%

&&&

xyz

!

"

###

$

%

&&&=Ap+b


uv

!

"#

$

%& ≈

fz0

x0y0

!

"##

$

%&&+

f / z0 0 − fx0 / z02

0 f / z0 − fy0 / z02

!

"

###

$

%

&&&

xyz

!

"

###

$

%

&&&=Ap+b

Rewrite Affine camera model in terms of Homogenous Coordinates

uvw

!

"

###

$

%

&&&≈

f / z0 0 − fx0 / z02 fx0 / z0

0 f / z0 − fy0 / z02 fy0 / z0

0 0 0 1

!

"

####

$

%

&&&&

xyz1

!

"

####

$

%

&&&&

Parallel lines project to parallel lines Ratios of distances are preserved under orthographic projection


uv

!

"#

$

%&=

fz0

xy

!

"##

$

%&&

Orthographic projection Starting with Affine camera mode

Take Taylor series about (0, 0, z0) – a point on optical axis

UVW

!

"

###

$

%

&&&=

f / z0 0 0 00 f / z0 0 00 0 0 1

!

"

####

$

%

&&&&

XYZT

!

"

####

$

%

&&&&

Euclidean Coordinates Homogenous Coordinates

7

CSE190 Computer Vision I

What if camera coordinate system differs from object coordinate system?

{c}

P {W}


Euclidean Coordinate Systems

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⇔++=⇔⎪⎩

⎪⎨

⎧

=

=

=

zyx

zyxOPOPzOPyOPx

Pkjikji

...


A convenient notation

BP = ABR AP+ BOA

–  Points: AP1 –  Leading superscript indicates the coordinate system that the coordinates are with respect to –  Subscript – an identifier

–  Rotation Matrices –  Lower left (Going from this system) –  Upper left (Going to this system)

–  To add vectors, coordinate systems (leading superscript) must agree –  To rotate a vector, points coordinate system must agree with lower left of rotation matrix

RBA


Coordinate Changes: Rigid Transformations

ABAB

AB OPRP +=

Rotation Matrix Translation vector


More about rotations matrices A rotation matrix R has the following properties:

•  Its inverse is equal to its transpose R-1 = RT or RTR = I

•  Its determinant is equal to 1: det(R)=1. Or equivalently:

Rows (or columns) of R form a right-handed orthonormal coordinate system.

•  Even though a rotation matrix is 3x3 with nine numbers, it only has three degrees of freedom, it can be parameterized with three numbers.

•  There are many parameterizations.


Rotation •  About z axis

x' y' z’

=

x y z

cos θ sin θ

0

-sin θ cos θ

0

0 0 1

rot(z,θ) x

y

z

p

p'

θ

Note: z coordinate doesn’t change after rotation

8


Rotation

•  About x axis:

•  About y axis:

x' y' z’

=

x y z

0 cos θ sin θ

0 -sin θ cos θ

1 0 0

x' y' z’

=

x y z

cos θ 0

-sin θ

sin θ 0

cos θ

0 1 0


Roll-Pitch-Yaw ),ˆ(),ˆ(),ˆ( ϕβα krotjrotirotR =

),ˆ(),'ˆ(),''ˆ( ϕβα krotjrotkrotR =

Euler Angles


Rotation •  Rotation by angle θ about (kx, ky, kz),

a unit vector •  (Rodrigues Formula)

x' y’ z’

= x y z

kxkx(1-c)+c kykx(1-c)+kzs kzkx(1-c)-kys

kxky(1-c)-kzs kyky(1-c)+c kzyx(1-c)-kxs

kxkz(1-c)+kys kykz(1-c)-kxs kzkz(1-c)+c

where c = cos θ & s = sin θ

Rotate(k, θ)

x

y

z

θ

k


Block Matrix Multiplication Given

€

A =A11 A12A21 A22

"

# $

%

& ' B =

B11 B12B21 B22

"

# $

%

& '

What is AB ?

⎥⎦

⎤⎢⎣

⎡

++

++=

2222122121221121

2212121121121111

BABABABABABABABA

AB

Homogeneous Representation of Rigid Transformations

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡ +=⎥

⎦

⎤⎢⎣

⎡

11111P

TPOROPRP AB

A

A

TA

BBAA

BABA

B

0

Transformation represented by 4 by 4 Matrix


What if camera coordinate system differs from object coordinate system?

{c}

P {W}

€

wcT = w

cR cOw0T 1

"

# $

%

& '

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

1010000100001

ZYX

fWVU


Intrinsic parameters

u

v

3x3 homogenous matrix Focal length: Principal Point: C’ Units (e.g. pixels) Orientation and position of image coordinate system Pixel Aspect ratio

9


Camera parameters •  Extrinsic Parameters: Since camera may not be at the origin,

there is a rigid transformation between the world coordinates and the camera coordinates

•  Intrinsic parameters: Since scene units (e.g., cm) differ image units (e.g., pixels) and coordinate system may not be centered in image, we capture that with a 3x3 transformation comprised of focal length, principal point, pixel aspect ratio, angle between axes, etc.

UVW

!

"

###

$

%

&&&=

Transformationrepresented byintrinsic parameters

!

"

###

$

%

&&&

1 0 0 00 1 0 00 0 1 0

!

"

###

$

%

&&&

Rigid Transformationrepresented by extrinsic parameters

!

"

###

$

%

&&&

XYZ1

!

"

####

$

%

&&&&

3 x 3 4 x 4 wcT


, estimate intrinsic and extrinsic camera parameters •  See Text book for how to do it. •  Camera Calibration Toolbox for Matlab (Bouguet) http://www.vision.caltech.edu/bouguetj/calib_doc/

Camera Calibration

Documents

Discriminant Functions for the 3 Normal Densitycseweb.ucsd.edu/classes/sp14/cse190-b/lec5.pdfImage Formation Biometrics CSE 190 Lecture 5 CSE190, Spring 2014 Announcements • HW1: