Image stitching

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Image stitching

Digital Visual Effects, Spring 2005 Yung-Yu Chuang

2005/3/30

with slides by Richard Szeliski, Steve Seitz, Matthew Brown and Vaclav Hlavac

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Announcements

• Project #1 was due yesterday.

• Project #2 handout will be available on the web later tomorrow.

• I will set up a webpage for artifact voting soon.

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Outline

• Image stitching

• Motion models

• Direct methods

• Feature-based methods

• Applications

• Project #2

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Image stitching

• Stitching = alignment + blending

geometrical registration

photometric registration

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Applications of image stitching

• Video stabilization

• Video summarization

• Video compression

• Video matting

• Panorama creation

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Video summarization

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Video compression

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Video matting

input video

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Video matting

remove foreground

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Video matting

estimate background

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Video matting

background estimation

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Video matting

alpha matte

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Panorama creation

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Why panorama?

• Are you getting the whole picture?

– Compact Camera FOV = 50 x 35°

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Why panorama?

– Human FOV = 200 x 135°

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Why panorama?

– Human FOV = 200 x 135°

– Panoramic Mosaic = 360 x 180°

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Panorama examples

• Panorama mode in consumer cameras

• Mars:

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

• Earth:

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

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2D motion models

• translation: x’ = x + t x = (x,y)

• rotation: x’ = R x + t

• similarity: x’ = s R x + t

• affine: x’ = A x + t

• perspective: x’ ≅ H x x = (x,y,1) (x is a homogeneous coordinate)

• These all form a nested group (closed under composition w/ inv.)

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2D image transformations

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A pencil of rays contains all views

real

camera synthetic

camera

Can generate any synthetic camera view

as long as it has the same center of projection!

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Mosaic as Image Reprojection

mosaic projection plane

• The images are reprojected onto a common plane

• The mosaic is formed on this plane

• Mosaic is a synthetic wide-angle camera

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Changing camera center

• Does it still work? synthetic PP PP1

PP2

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Planar scene (or far away)

• PP3 is a projection plane of both centers of projection, so we are OK!

• This is how big areal photographs are made

PP1

PP3

PP2

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3D motion models

u

(X,Y,Z) f u

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3D motion models

• Rotational

• Cylindrical

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Direct methods

• Select a motion model and estimate parameters

• Direct methods use pixel-to-pixel matching

• We have covered this last time actually.

• We will show a case study on constructing cylindrical panorama using a direct method.

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A case study: cylindrical panorama

1. Take pictures on a tripod (or handheld) 2. Warp to cylindrical coordinate

3. Compute pairwise alignments using the hierarchical Lucas-Kanade algorithm

4. Fix up the end-to-end alignment 5. Blending

6. Crop the result and import into a viewer

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Taking pictures

Kaidan panoramic tripod head

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Translation model

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A case study: cylindrical panorama

• What if you want a 360° field of view?

mosaic projection cylinder

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– Map 3D point (X,Y,Z) onto cylinder

Cylindrical projection

X Y

Z

unit cylinder

unwrapped cylinder

– Convert to cylindrical coordinates

cylindrical image

– Convert to cylindrical image coordinates

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f = 180 (pixels)

Cylindrical reprojection

f = 380 f = 280

Image 384x300

top-down view Focal length – the dirty secret…

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A simple method for estimating f

We will discuss more accurate methods next time

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Distortion

• Radial distortion of the image

– Caused by imperfect lenses

– Deviations are most noticeable for rays that pass through the edge of the lens

No distortion Pin cushion Barrel

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Radial correction

• Correct for “bending” in wide field of view lenses

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Input images

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Cylindrical warping

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Alignment

• a rotation of the camera is a translation of the cylinder!

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LucasKanadeStep

void LucasKanadeStep(CByteImage& img1, CByteImage& img2, float t[2]) { // Transform the image

Translation(img2, img2t, t);

// Compute the gradients and summed error by comparing img1 and img2t double A[2][2], b[2];

for (int y = 1; y < height-1; y++) { // ignore borders for (int x = 1; x < width-1; x++) {

// If both have full alphas, then compute and accumulate the error double e = img2t.Pixel(x, y, k) - img1.Pixel (x, y, k);

// Accumulate the matrix entries

double gx = 0.5*(img2t.Pixel(x+1, y, k) - img2t.Pixel(x-1, y, k));

double gy = 0.5*(img2t.Pixel(x, y+1, k) - img2t.Pixel(x, y-1, k));

A[0][0] += gx*gx; A[0][1] += gx*gy;

A[1][0] += gx*gy; A[1][1] += gy*gy;

b[0] += e*gx; b[1] += e*gy;

} }

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LucasKanadeStep (cont.)

// Solve for the update At=b and update the vector double det = 1.0 / (A[0][0]*A[1][1] - A[1][0]*A[1][0]);

t[0] += (A[1][1]*b[0] - A[1][0]*b[1]) * det;

t[1] += (A[0][0]*b[1] - A[1][0]*b[0]) * det;

}

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PyramidLucasKanade

void PyramidalLucasKanade(CByteImage& img1, CByteImage& img2, float t[2], int nLevels, int nLucasKanadeSteps)

{

CBytePyramid p1(img1); // Form the two pyramids CBytePyramid p2(img2);

// Process in a coarse-to-fine hierarchy for (int l = nLevels-1; l >= 0; l--)

{

t[0] /= (1 << l); // scale the t vector t[1] /= (1 << l);

CByteImage& i1 = p1[l];

CByteImage& i2 = p2[l];

for (int k = 0; k < nLucasKanadeSteps; k++) LucasKanadeStep(i1, i2, t);

t[0] *= (1 << l); // restore the full scaling t[1] *= (1 << l);

} }

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Gaussian pyramid

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Blending

• Why blending: parallax, lens distortion, scene motion, exposure difference

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Blending

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Blending

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Blending

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Assembling the panorama

• Stitch pairs together, blend, then crop

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Problem: Drift

• Error accumulation

– small errors accumulate over time

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Problem: Drift

• Solution

– add another copy of first image at the end

– there are a bunch of ways to solve this problem

• add displacement of (y₁ – y_n)/(n -1) to each image after the first

• compute a global warp: y’ = y + ax

• run a big optimization problem, incorporating this constraint

– best solution, but more complicated – known as “bundle adjustment”

(x₁,y₁)

• copy of first image

(x_n,y_n)

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End-to-end alignment and crop

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Viewer: panorama

++

++ ++

example: http://www.cs.washington.edu/education/courses/cse590ss/01wi/projects/project1/students/dougz/index.html

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Viewer: texture mapped model

example: http://www.panoramas.dk/

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Feature-based methods

• Only use feature points to estimate parameters

• We will study the “Recognising panorama”

paper published in ICCV 2003

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RANSAC

• RANSAC = Random Sample Consensus

• an algorithm for robust fitting of models in the presence of many data outliers

• Compare to robust statistics

• Given N data points x_i, assume that mjority of them are generated from a model with

parameters Θ, try to recover Θ.

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RANSAC algorithm

Run k times:

(1) draw n samples randomly

(2) fit parameters Θ with these n samples (3) for each of other N-n points, calculate

its distance to the fitted model, count the number of inlier points, c

Output Θ with the largest c

How many times?

How big?

Smaller is better

How to define?

Depends on the problem.

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How to determine k

p: probability of real inliers

P: probability of success after k trials

k

p

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P = 1 − ( 1 − )

n samples are all inliers a failure

failure after k trials

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log(

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log(

p

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k P

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293 0.5

6

97 0.6

6

35 0.5

3

k p

n for P=0.99

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Example: line fitting

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n=2

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Model fitting

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Measure distances

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Count inliers

c=3

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Another trial

c=3

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The best model

c=15

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Recognising Panoramas

• 1D Rotations (θ)

– Ordering ⇒ matching images

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Recognising Panoramas

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Recognising Panoramas

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Recognising Panoramas

• 2D Rotations (q, f)

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Recognising Panoramas

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Recognising Panoramas

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Recognising Panoramas

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Overview

• SIFT Feature Matching

• Image Matching

• Bundle Adjustment

• Multi-band Blending

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Nearest Neighbour Matching

• Find k-NN for each feature

– k ≈ number of overlapping images (we use k = 4)

• Use k-d tree

– k-d tree recursively bi-partitions data at mean in the dimension of maximum variance

– Approximate nearest neighbours found in O(nlogn)

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Overview

• Image Matching

– For each image, use RANSAC to select inlier features from 6 images with most feature matches

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RANSAC for Homography

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RANSAC for Homography

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RANSAC for Homography

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Probabilistic model for verification

• Compare probability that this set of RANSAC inliers/outliers was generated by a

correct/false image match

• Choosing values for p₁, p₀ and p_min

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Finding the panoramas

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Finding the panoramas

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Finding the panoramas

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Finding the panoramas

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Overview

• Image Matching

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• Parameterise each camera by rotation and focal length

• This gives pairwise homographies

Homography for Rotation

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Error function

• Sum of squared projection errors

– n = #images

– I(i) = set of image matches to image i

– F(i, j) = set of feature matches between images i,j – r_ij^k = residual of k^th feature match between images

i,j

• Robust error function

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Overview

• Image Matching

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Multi-band Blending

• Burt & Adelson 1983

– Blend frequency bands over range ∝ λ

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Low frequency (λ > 2 pixels)

High frequency (λ < 2 pixels)

2-band Blending

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Linear Blending

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2-band Blending

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Results

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Direct vs feature-based

• Direct methods use all information and can be very accurate, but they depend on the fragile

“brightness constancy” assumption

• Iterative approaches require initialization

• Not robust to illumination change and noise images

• In early days, direct method is better.

• Feature based methods are now more robust and potentially faster

• Even better, it can recognize panorama without initialization

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Applications of panorama in VFX

• Background plates

• Image-based lighting

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Spiderman 2 (background plate)

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Troy (image-based lighting)

http://www.cgnetworks.com/story_custom.php?story_id=2195&page=4

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Project #2 Image stitching

• Assigned: 3/30

• Due: 11:59pm 4/19

• Work in pairs

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Reference software

• Autostitch

http://www.cs.ubc.ca/~mbrown/autostitch/autostitch.html

• Many others are available online.

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Bells & whistles

• Full SIFT implementation

• Recognizing panorama

• Bundle adjustment

• Handle dynamic objects

• Better blending techniques

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Artifacts

• Take your own pictures and generate a stitched image, be creative.

• http://www.cs.washington.edu/education/courses/cse590ss/01wi/projec ts/project1/students/allen/index.html

• Common focal point

• Rotate your camera to increase vertical FOV

• Tripod

• Fixed exposure?

Tips for taking pictures

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Submission

• You have to turn in your complete source, the executable, a html report and an artifact.

• Report page contains:

description of the project, what do you learn, algorithm, implementation details, results, bells and whistles…

• Artifacts must be made using your own program.

artifacts voting on forum.

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Reference

• Richard Szeliski, Image Alignment and Stitching, unpublished draft, 2005.

• R. Szeliski and H.-Y. Shum. Creating full view panoramic image mosaics and texture-mapped models, SIGGRAPH 1997, pp251-258.

• M. Brown, D. G. Lowe, Recognising Panoramas, ICCV 2003.