image stitching richard szeliskiimage stitching2 panoramic image mosaics full screen panoramas...

Image Stitching

Richard Szeliski Image Stitching 2

Panoramic Image Mosaics

Full screen panoramas (cubic): http://www.panoramas.dk/ Mars: http://www.panoramas.dk/fullscreen3/f2_mars97.html

2003 New Years Eve: http://www.panoramas.dk/fullscreen3/f1.html

http://www.panoramas.dk/

http://www.panoramas.dk/fullscreen3/f2_mars97.html

http://www.panoramas.dk/fullscreen3/f1.html


Gigapixel panoramas & images

Mapping / Tourism / WWT

Medical Imaging

http://research.microsoft.com/IVM/HDView.htm


Image Mosaics

+ + … +=

Goal: Stitch together several images into a seamless composite


Today’s lecture

Image alignment and stitching• motion models• image warping• point-based alignment• complete mosaics (global alignment)• compositing and blending• ghost and parallax removal


Readings

• Szeliski, CVAA:• Chapter 3.6: Image warping• Chapter 5.1: Feature-based alignment (in preparation)• Chapter 9.1: Motion models• Chapter 9.2: Global alignment• Chapter 9.3: Compositing

• Recognizing Panoramas, Brown & Lowe, ICCV’2003• Szeliski & Shum, SIGGRAPH'97

Motion models


Motion models

What happens when we take two images with a camera and try to align them?

• translation?• rotation?• scale?• affine?• perspective?• VideoMosaic

Image Warping


Image Warping

image filtering: change range of imageg(x) = h(f(x))

image warping: change domain of imageg(x) = f(h(x))

f

x

hf

x

f

x

hf

x


Image Warping

image filtering: change range of imageg(x) = h(f(x))

image warping: change domain of imageg(x) = f(h(x))

h

h

f

f g

g


Parametric (global) warping

Examples of parametric warps:

translation rotation aspect

affineperspective

cylindrical

2D coordinate transformations



Translation

Rotation + translation

Similarity

Affine


, ( , )x yt t x = x + t t

cos sin, ( , )

sin cos x yt t

x = Rx + t

R t

1

x

y

xa b t

s yb a t

x = Rx + t

00 01 02

10 11 12 1

xa a a

ya a a

x =

Homography

A: Projective – mapping between any two PPs with the same center of projection• rectangle should map to arbitrary quadrilateral • parallel lines aren’t• but must preserve straight lines• same as: project, rotate, reproject

called Homography

PP2

PP1

1yx

*********

wwy'wx'

H pp’ To apply a homography H

• Compute p’ = Hp (regular matrix multiply)

• Convert p’ from homogeneous to image coordinates


Projective transformation or Homography

Normalized homogeneous


x = Hx

Homogeneous, only defined up to a scale

sH H

00 01 02

20 21 22

h x h y hx

h x h y h

10 11 12

20 21 22

h x h y hy

h x h y h

1

x

y

x


Image Warping

Given a coordinate transform x’ = h(x) and a source image f(x), how do we compute a transformed image g(x’) = f(h(x))?

f(x) g(x’)x x’

h(x)


Forward Warping

Send each pixel f(x) to its corresponding location x’ = h(x) in g(x’)

f(x) g(x’)x x’

h(x)

• What if pixel lands “between” two pixels?

Forward WarpingNon-integer x’• Nearest integer coordinate

• Aliasing• Jumping

• Splatting • Distribute to 4 neighbors (weighted) and normalize later • Both aliasing and blur

Cracks and holes• Filling hole with nearby neighbors• Further aliasing and blurring

21


Inverse Warping

Get each pixel g(x’) from its corresponding location x’ = h(x) in f(x)

f(x) g(x’)x x’

h(x)

• What if pixel comes from “between” two pixels?


Inverse Warping

Get each pixel g(x’) from its corresponding location x’ = h(x) in f(x)

• What if pixel comes from “between” two pixels?• Answer: resample color value from

interpolated (prefiltered) source image

f(x) g(x’)x x’


Interpolation

Possible interpolation filters:• nearest neighbor• bilinear• bicubic (interpolating)• sinc / FIR

Needed to prevent “jaggies” and “texture crawl”

Interpolation

Bilinear


0

1 0

u uU

u u

0 0( , )u v 1 0( , )u v

1 1( , )u v0 1( , )u v

1f 2f

3f 4f

( , )u v

f

0

1 0

v vV

v v

0 1

3 4

(1 )

(1 )

(1 )

a

b

a b

f f U f U

f f U f U

f f V f V

Bicubic

32

2

1 ( 3) ( 2) 1

( ) ( 1)( 2) 1 2

0

a x a x if x

h x a x x if x

otherwise

ɑ: the derivative at x=1

Interpolation

Window sinc functions


Image Interpolation


All perform high-accuracy low-pass filtering while preserve details and avoid aliasing

Bilinear for speedBicubic and windowed sinc for better visual quality

Decimation

Interpolation: increase the image resolution

Decimation (downsampling): reduce resolution


Samples are convolved with a low-pass filter before downsampled to avoid aliasing

Decimation

Sample rate


r

,

1( , ) ( , ) ,

k l

k kg i j f k l h i j

r r r

( , )h k l : same kernel as interpolation

Commonly use r=2

Decimation


Frequency response

Decimation

• Linear filter gives a relatively poor response• Binomial filter cuts off a lot of frequencies but is

useful for computer vision analysis pyramid• Cubic filter: a=-1 filter has a shaper fall-off than the

a=-0.5 filter• Cosine-windowed sinc function• QMF-9 filter for wavelet denoising and aliases a fair

amount• 9/7 analysis filter form JPEG2000



Prefiltering

Essential for downsampling (decimation) to prevent aliasing

MIP-mapping [Williams’83]:1. build pyramid (but what decimation filter?):

– block averaging– Burt & Adelson (5-tap binomial)– 7-tap wavelet-based filter (better)

2. trilinear interpolation– bilinear within each 2 adjacent levels– linear blend between levels (determined by pixel size)


Prefiltering

Essential for downsampling (decimation) to prevent aliasing

Other possibilities:• summed area tables• elliptically weighted Gaussians (EWA)

[Heckbert’86]

Resmaple on interpolated signal

To avoid blurring or aliasing


Motion models (reprise)


Motion models

Translation

2 unknowns

Affine

6 unknowns

Perspective

8 unknowns

3D rotation

3 unknowns

Planar perspective motion

• Simplest case:• Overlapping photographic prints, scanned pieces• Translation and rotation are usually adequate


1 0 x Rx t

Four unknowns

Planar perspective motion

• Planar Homography


00 01 02

20 21 22

h x h y hx

h x h y h

10 11 12

20 21 22

h x h y hy

h x h y h

Planar Perspective Motion

• Extract features from images• Match features using RANSAC and get inliers• Estimate a homography H given pairs of

corresponding features• Least squares fitting


Solving for homography


1

y

x

ihg

fed

cba

w

wy'

wx'p’ = Hp

Can set scale factor i=1. So, there are 8 unknowns.

Set up a system of linear equations:

Ah = b

where vector of unknowns h = [a,b,c,d,e,f,g,h]T

Need at least 8 eqs, but the more the better…

Solve for h. If overconstrained, solve using least-squares 2

min bAh


Image warping with homographies

image plane in front image plane below


Plane perspective mosaics

• 8-parameter generalization of affine motion– works for pure rotation or planar surfaces

• Limitations:– local minima – slow convergence– difficult to control interactively


Rotational mosaics

• Directly optimize rotation and focal length

• Advantages:– ability to build full-view

panoramas– easier to control interactively– more stable and accurate

estimates

GrandCanyon panorama


3D → 2D Perspective Projection

u

(Xc,Yc,Zc)

ucf

Rotational panoramas

Setting


0 1t = t = 0 ( , ,1)K diag f f

With known f0, f1, solve R10

Rotational panoramas

• More stable for large-scale image stitching algorithms than homography

• How to solve the unknowns?• Update R10 with incremental rotation matrix R(w)

to current estimate R10



Rotations and quaternionsHow do we represent rotation matrices?

1. Axis / angle (n,θ)R = I + sinθ [n] + (1- cosθ) [n]

2 (Rodriguez Formula), with [n] = cross product matrix (see paper)

2. Unit quaternions [Shoemake SIGG’85]q = (n sinθ/2, cosθ/2) = (w,s)quaternion multiplication (division is easy)q0 q1 = (s1w0+ s0w1, s0 s1-w0·w1)


Incremental rotation update

1. Small angle approximationΔR = I + sinθ [n] + (1- cosθ) [n]

2

≈ θ [n] = [ω] linear in ω

2. Update original R matrixR ← R ΔR

Rotational panoramasHow to solve the unknowns of R10?

Compute the incremental rotation vector w,



Rotational mosaic

Projection equations

1. Project from image to 3D ray

(x0,y0,z0) = (u0-uc,v0-vc,f)

2. Rotate the ray by camera motion

(x1,y1,z1) = R01 (x0,y0,z0)

3. Project back into new (source) image

(u1,v1) = (fx1/z1+uc,fy1/z1+vc)

Gap Closing

• Estimate a series of rotation matrices and focal lengths to create large panoramas

• Unfortunately, accumulated errors• Hard to estimate accurate focal length• Rarely produce a closed 360 pararoma


Gap Closing

• Estimate a series of rotation matrices and focal lengths to create large panoramas

• Unfortunately, accumulated errors• Rarely produce a closed 360 pararoma


Wrong focal length: f=510; correct focal length f=468

Gap Closing

• Matching the first image and the last one• Compute the gap angle • Distribute the gap angle evenly across the

whole sequence• Modify rotations by • Update focal length

• Only works for 1D panorama where the camera is continuously turning in the same direction


g

/g imageN

' (1 / 360 )gf f


mosaic PP

Image reprojection

The mosaic has a natural interpretation in 3D• The images are reprojected onto a common plane• The mosaic is formed on this plane

Image Mosaics (Stitching)

[Szeliski & Shum, SIGGRAPH’97]

[Szeliski, FnT CVCG, 2006]


Image Mosaics (stitching)

Blend together several overlapping images into one seamless mosaic (composite)

+ + … +=


Mosaics for Video CodingConvert masked images into a background sprite

for content-based coding

+ + +

=


Establishing correspondences

1. Direct method:• Use generalization of affine motion model

[Szeliski & Shum ’97]

2. Feature-based method• Extract features, match, find consisten inliers

[Lowe ICCV’99; Schmid ICCV’98,Brown&Lowe ICCV’2003]

• Compute R from correspondences(absolute orientation)


Stitching demo


Panoramas

What if you want a 360 field of view?

mosaic Projection Cylinder


Cylindrical panoramas

Steps• Reproject each image onto a cylinder• Blend • Output the resulting mosaic


f = 180 (pixels)

Cylindrical Panoramas

Map image to cylindrical or spherical coordinates• need known focal length

Image 384x300 f = 380f = 280


Determining the focal length

1. Initialize from homography H(see text or [SzSh’97])

2. Use camera’s EXIF tags (approx.)

3. Use a tape measure

4. Ask your instructor

1m4m


3D → 2D Perspective Projection

u

(Xc,Yc,Zc)

ucf


• Map 3D point (X,Y,Z) onto cylinder

Cylindrical projection

XY

Z

unit cylinder

unwrapped cylinder

• Convert to cylindrical coordinates

cylindrical image

• Convert to cylindrical image coordinates

– s defines size of the final image


Cylindrical warping

Given focal length f and image center (xc,yc)

X

YZ

(X,Y,Z)

(sin,h,cos)


Spherical warping

Given focal length f and image center (xc,yc)

X

YZ

(x,y,z)

(sinθcosφ,cosθcosφ,sinφ)

cos φ

φ

cos θ cos φ

sin φ


Radial distortion

Correct for “bending” in wide field of view lenses


Fisheye lens

Extreme “bending” in ultra-wide fields of view


Other types of mosaics

Can mosaic onto any surface if you know the geometry• See NASA’s Visible Earth project for some stunning earth

mosaics– http://earthobservatory.nasa.gov/Newsroom/BlueMarble/

http://earthobservatory.nasa.gov/Newsroom/BlueMarble/

http://earthobservatory.nasa.gov/Newsroom/BlueMarble/


Slit images

y-t slices of the video volume are known as slit images• take a single column of pixels from each input image


Slit images: cyclographs


Slit images: photofinish


Final thought:What is a “panorama”?

Tracking a subject

Repeated (best) shots

Multiple exposures

“Infer” what photographer wants?

image stitching richard szeliskiimage stitching2 panoramic image mosaics full screen panoramas...

Documents

image warping slide

image stitching slide

image warping chapter

hxhx slide

scale slide

hfx image warping

image coordinates

siggraph97 slide