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Презентация, доклад на тему Image Stitching

Презентация на тему Image Stitching, из раздела: Разное. Эта презентация содержит 62 слайда(ов). Информативные слайды и изображения помогут Вам заинтересовать аудиторию. Скачать конспект-презентацию на данную тему можно внизу страницы, поделившись ссылкой с помощью социальных кнопок. Также можно добавить наш сайт презентаций в закладки! Презентации взяты из открытого доступа или загружены их авторами, администрация сайта не отвечает за достоверность информации в них. Все права принадлежат авторам презентаций.

Image StitchingAli FarhadiCSE 455	Several slides from Rick Szeliski, Steve Seitz, Derek Hoiem, and Ira Kemelmacher Combine two or more overlapping images to make one larger imageAdd exampleSlide credit: Vaibhav Vaish How to do it?Basic ProcedureTake a sequence of images from the same 1. Take a sequence of images from the same position Rotate the 2. Compute transformation between imagesExtract interest pointsFind MatchesCompute transformation? 3. Shift the images to overlap 4. Blend the two together to create a mosaic 5. Repeat for all images How to do it?Basic ProcedureTake a sequence of images from the same Compute TransformationsExtract interest pointsFind good matches Compute transformation✓Let’s assume we are given Image reprojectionThe mosaic has a natural interpretation in 3DThe images are reprojected ExampleCamera Center Image reprojectionObservationRather than thinking of this as a 3D reprojection, think of Motion modelsWhat happens when we take two images with a camera and try to align them?translation?rotation?scale?affine?Perspective? Recall: Projective transformations(aka homographies) Parametric (global) warpingExamples of parametric warps:translationrotationaspectaffineperspective 2D coordinate transformationstranslation:		x’ = x + t		 x = (x,y)rotation:		x’ = R Image WarpingGiven a coordinate transform x’ = h(x) and a source image Forward WarpingSend each pixel f(x) to its corresponding location x’ = h(x) Forward WarpingSend each pixel f(x) to its corresponding location x’ = h(x) Richard SzeliskiImage StitchingInverse WarpingGet each pixel g(x’) from its corresponding location x’ Richard SzeliskiImage StitchingInverse WarpingGet each pixel g(x’) from its corresponding location x’ InterpolationPossible interpolation filters:nearest neighborbilinearbicubic (interpolating) Motion models Finding the transformationTranslation 	= 	2 degrees of freedomSimilarity 	= 	4 degrees of Plane perspective mosaics8-parameter generalization of affine motionworks for pure rotation or planar Simple case: translations Simple case: translations Simple case: translationsSystem of linear equationsWhat are the knowns? Unknowns?How many unknowns? Simple case: translationsProblem: more equations than unknowns“Overdetermined” system of equationsWe will find the least squares solution Least squares formulationFor each pointwe define the residuals as Least squares formulationGoal: minimize sum of squared residuals“Least squares” solutionFor translations, is equal to mean displacement Least squaresFind t that minimizes To solve, form the normal equations Solving for translationsUsing least squares Affine transformationsHow many unknowns?How many equations per match?How many matches do we need? Affine transformationsResiduals:Cost function: Affine transformationsMatrix form Solving for homographies Solving for homographies Direct Linear TransformsDefines a least squares problem:Since    is only Richard SzeliskiImage StitchingMatching featuresWhat do we do about the “bad” matches? Richard SzeliskiImage StitchingRAndom SAmple ConsensusSelect one match, count inliers Richard SzeliskiImage StitchingRAndom SAmple ConsensusSelect one match, count inliers Richard SzeliskiImage StitchingLeast squares fitFind “average” translation vector RANSAC for estimating homographyRANSAC loop:Select four feature pairs (at random)Compute homography H Simple example: fit a lineRather than homography H (8 numbers)  fit Simple example: fit a linePick 2 pointsFit lineCount inliers3 inliers Simple example: fit a linePick 2 pointsFit lineCount inliers4 inliers Simple example: fit a linePick 2 pointsFit lineCount inliers9 inliers Simple example: fit a linePick 2 pointsFit lineCount inliers8 inliers Simple example: fit a lineUse biggest set of inliersDo least-square fit RANSACRed: 	rejected by 2nd nearest neighbor criterionBlue:	Ransac outliersYellow:	inliers How many rounds? If we have to choose s samples each timewith Richard SzeliskiImage StitchingRotational mosaicsDirectly optimize rotation and focal lengthAdvantages:ability to build full-view Richard SzeliskiImage StitchingRotational mosaicProjection equationsProject from image to 3D ray	(x0,y0,z0) 	= (u0-uc,v0-vc,f)Rotate Computing homographyAssume we have four matched points: How do we compute homography Computing homographyAssume we have matched points with outliers: How do we compute Automatic Image StitchingCompute interest points on each imageFind candidate matchesEstimate homography H RANSAC for HomographyInitial Matched Points RANSAC for HomographyFinal Matched Points RANSAC for Homography

Слайды и текст этой презентации

Слайд 1
Image StitchingAli FarhadiCSE 455	Several slides from Rick Szeliski, Steve Seitz, Derek Hoiem, and Ira Kemelmacher
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Image Stitching

Ali Farhadi
CSE 455





Several slides from Rick Szeliski, Steve Seitz, Derek Hoiem, and Ira Kemelmacher


Слайд 2
Combine two or more overlapping images to make one larger imageAdd exampleSlide credit: Vaibhav Vaish
Текст слайда:

Combine two or more overlapping images to make one larger image

Add example

Slide credit: Vaibhav Vaish


Слайд 3
Текст слайда:

How to do it?

Basic Procedure
Take a sequence of images from the same position
Rotate the camera about its optical center
Compute transformation between second image and first
Shift the second image to overlap with the first
Blend the two together to create a mosaic
If there are more images, repeat


Слайд 4
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1. Take a sequence of images from the same position

Rotate the camera about its optical center


Слайд 5
2. Compute transformation between imagesExtract interest pointsFind MatchesCompute transformation?
Текст слайда:

2. Compute transformation between images

Extract interest points
Find Matches
Compute transformation

































?


Слайд 6
3. Shift the images to overlap
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3. Shift the images to overlap


Слайд 7
4. Blend the two together to create a mosaic
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4. Blend the two together to create a mosaic



Слайд 8
5. Repeat for all images
Текст слайда:

5. Repeat for all images


Слайд 9
Текст слайда:

How to do it?

Basic Procedure
Take a sequence of images from the same position
Rotate the camera about its optical center
Compute transformation between second image and first
Shift the second image to overlap with the first
Blend the two together to create a mosaic
If there are more images, repeat



Слайд 10
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Compute Transformations

Extract interest points
Find good matches
Compute transformation


Let’s assume we are given a set of good matching interest points



Слайд 11
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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


Слайд 12
ExampleCamera Center
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Example

Camera Center



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Image reprojection

Observation
Rather than thinking of this as a 3D reprojection, think of it as a 2D image warp from one image to another


Слайд 14
Motion modelsWhat happens when we take two images with a camera and try to align them?translation?rotation?scale?affine?Perspective?
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Motion models

What happens when we take two images with a camera and try to align them?
translation?
rotation?
scale?
affine?
Perspective?


Слайд 15
Recall: Projective transformations(aka homographies)
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Recall: Projective transformations

(aka homographies)



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Parametric (global) warpingExamples of parametric warps:translationrotationaspectaffineperspective
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Parametric (global) warping

Examples of parametric warps:

translation

rotation

aspect

affine

perspective


Слайд 17
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2D coordinate transformations

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)


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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)


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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?


Слайд 20
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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?

Answer: add “contribution” to several pixels, normalize later (splatting)


Слайд 21
Текст слайда:

Richard Szeliski

Image Stitching

Inverse Warping

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

f(x)

g(x’)

x

x’




h-1(x)

What if pixel comes from “between” two pixels?


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Richard Szeliski

Image Stitching

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 source image

f(x)

g(x’)

x

x’




h-1(x)


Слайд 23
InterpolationPossible interpolation filters:nearest neighborbilinearbicubic (interpolating)
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Interpolation

Possible interpolation filters:
nearest neighbor
bilinear
bicubic (interpolating)


Слайд 24
Motion models
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Motion models


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Finding the transformation

Translation = 2 degrees of freedom
Similarity = 4 degrees of freedom
Affine = 6 degrees of freedom
Homography = 8 degrees of freedom

How many corresponding points do we need to solve?


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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


Слайд 27
Simple case: translations
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Simple case: translations


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Simple case: translations
Текст слайда:


Simple case: translations


Слайд 29
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Simple case: translations

System of linear equations
What are the knowns? Unknowns?
How many unknowns? How many equations (per match)?


Слайд 30
Simple case: translationsProblem: more equations than unknowns“Overdetermined” system of equationsWe will find the least squares solution
Текст слайда:

Simple case: translations

Problem: more equations than unknowns
“Overdetermined” system of equations
We will find the least squares solution


Слайд 31
Least squares formulationFor each pointwe define the residuals as
Текст слайда:

Least squares formulation

For each point



we define the residuals as


Слайд 32
Least squares formulationGoal: minimize sum of squared residuals“Least squares” solutionFor translations, is equal to mean displacement
Текст слайда:

Least squares formulation

Goal: minimize sum of squared residuals




“Least squares” solution

For translations, is equal to mean displacement


Слайд 33
Least squaresFind t that minimizes To solve, form the normal equations
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Least squares

Find t that minimizes


To solve, form the normal equations


Слайд 34
Solving for translationsUsing least squares
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Solving for translations

Using least squares


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Affine transformationsHow many unknowns?How many equations per match?How many matches do we need?
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Affine transformations

How many unknowns?
How many equations per match?
How many matches do we need?



Слайд 36
Affine transformationsResiduals:Cost function:
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Affine transformations

Residuals:



Cost function:


Слайд 37
Affine transformationsMatrix form
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Affine transformations

Matrix form


Слайд 38
Solving for homographies
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Solving for homographies


Слайд 39
Solving for homographies
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Solving for homographies


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Direct Linear Transforms

Defines a least squares problem:

Since is only defined up to scale, solve for unit vector
Solution: = eigenvector of with smallest eigenvalue
Works with 4 or more points


Слайд 41
Richard SzeliskiImage StitchingMatching featuresWhat do we do about the “bad” matches?
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Richard Szeliski

Image Stitching

Matching features

What do we do about the “bad” matches?


Слайд 42
Richard SzeliskiImage StitchingRAndom SAmple ConsensusSelect one match, count inliers
Текст слайда:

Richard Szeliski

Image Stitching

RAndom SAmple Consensus

Select one match, count inliers


Слайд 43
Richard SzeliskiImage StitchingRAndom SAmple ConsensusSelect one match, count inliers
Текст слайда:

Richard Szeliski

Image Stitching

RAndom SAmple Consensus

Select one match, count inliers


Слайд 44
Richard SzeliskiImage StitchingLeast squares fitFind “average” translation vector
Текст слайда:

Richard Szeliski

Image Stitching

Least squares fit

Find “average” translation vector


Слайд 45
Текст слайда:




Слайд 46
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RANSAC for estimating homography

RANSAC loop:
Select four feature pairs (at random)
Compute homography H (exact)
Compute inliers where ||pi’, H pi|| < ε
Keep largest set of inliers
Re-compute least-squares H estimate using all of the inliers

CSE 576, Spring 2008

Structure from Motion




Слайд 47
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Simple example: fit a line

Rather than homography H (8 numbers) fit y=ax+b (2 numbers a, b) to 2D pairs













Слайд 48
Simple example: fit a linePick 2 pointsFit lineCount inliers3 inliers
Текст слайда:

Simple example: fit a line

Pick 2 points
Fit line
Count inliers













3 inliers


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Simple example: fit a linePick 2 pointsFit lineCount inliers4 inliers
Текст слайда:

Simple example: fit a line

Pick 2 points
Fit line
Count inliers












4 inliers


Слайд 50
Simple example: fit a linePick 2 pointsFit lineCount inliers9 inliers
Текст слайда:

Simple example: fit a line

Pick 2 points
Fit line
Count inliers












9 inliers


Слайд 51
Simple example: fit a linePick 2 pointsFit lineCount inliers8 inliers
Текст слайда:

Simple example: fit a line

Pick 2 points
Fit line
Count inliers












8 inliers


Слайд 52
Simple example: fit a lineUse biggest set of inliersDo least-square fit
Текст слайда:

Simple example: fit a line

Use biggest set of inliers
Do least-square fit













Слайд 53
RANSACRed: 	rejected by 2nd nearest neighbor criterionBlue:	Ransac outliersYellow:	inliers
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RANSAC

Red:
rejected by 2nd nearest neighbor criterion
Blue:
Ransac outliers
Yellow:
inliers


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How many rounds?

If we have to choose s samples each time
with an outlier ratio e
and we want the right answer with probability p


Слайд 55
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Richard Szeliski

Image Stitching

Rotational mosaics

Directly optimize rotation and focal length
Advantages:
ability to build full-view panoramas
easier to control interactively
more stable and accurate estimates


Слайд 56
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Richard Szeliski

Image Stitching

Rotational mosaic

Projection equations
Project from image to 3D ray
(x0,y0,z0) = (u0-uc,v0-vc,f)
Rotate the ray by camera motion
(x1,y1,z1) = R01 (x0,y0,z0)
Project back into new (source) image
(u1,v1) = (fx1/z1+uc,fy1/z1+vc)


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Computing homography

Assume we have four matched points: How do we compute homography H?

Normalized DLT
Normalize coordinates for each image
Translate for zero mean
Scale so that average distance to origin is ~sqrt(2)

This makes problem better behaved numerically

Compute using DLT in normalized coordinates
Unnormalize:




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Computing homography

Assume we have matched points with outliers: How do we compute homography H?

Automatic Homography Estimation with RANSAC
Choose number of samples N
Choose 4 random potential matches
Compute H using normalized DLT
Project points from x to x’ for each potentially matching pair:
Count points with projected distance < t
E.g., t = 3 pixels
Repeat steps 2-5 N times
Choose H with most inliers






HZ Tutorial ‘99



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Automatic Image Stitching


Compute interest points on each image

Find candidate matches

Estimate homography H using matched points and RANSAC with normalized DLT

Project each image onto the same surface and blend


Слайд 60
RANSAC for HomographyInitial Matched Points
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RANSAC for Homography

Initial Matched Points


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RANSAC for HomographyFinal Matched Points
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RANSAC for Homography

Final Matched Points


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RANSAC for Homography
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RANSAC for Homography