Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal...
Transcript of Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal...
![Page 1: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/1.jpg)
Geometric Transformations(based on Joan Serrat slides)
Felipe LumbrerasDept. Ciències de la Computació / Centre de Visió per Computador
Universitat Autònoma de Barcelonahttp://www.cvc.uab.es/shared/teach/a102784/
![Page 2: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/2.jpg)
Geometric Transformations
• Introduction
• Mapping computation– Scene Sensor modelling
– Control points
– Semi-automatic control point determination
– Automatic control point determination
• Resampling
• Applications– Registration
– Bird’s view
– Mosaic of images
![Page 3: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/3.jpg)
Introduction
Image transforms according to their formulation :
• point
• local neighbourhood
![Page 4: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/4.jpg)
Introduction
Image transforms according to their formulation :
• global
• arithmetic (+,-,*,/)
u
v
![Page 5: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/5.jpg)
Introduction
Image transforms according to their formulation :
• geometric
transform coordinates, not pixel values
![Page 6: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/6.jpg)
Introduction
Main applications:
• Correct image distortions caused by acquisition devices or geometry
• Compare images
Fields:
• Remote sensing
• Medical imaging
• …
![Page 7: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/7.jpg)
Introduction
Camera distortions
![Page 8: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/8.jpg)
Introduction
Satellite images
![Page 9: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/9.jpg)
Introduction
Satellite images
Orthorectified image:
each pixel “seen
from above”
![Page 10: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/10.jpg)
Introduction
Satellite images
Sensor fusion
IKONOS4m color
1m panchromatic fusion(pansharpening)
+ =
![Page 11: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/11.jpg)
Introduction
Medical Imaging
Frame registration on eye angiographies
original registered
![Page 12: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/12.jpg)
Introduction
Medical Imaging
Frame registration on eye angiographies
original registered
![Page 13: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/13.jpg)
Introduction
Medical Imaging
Frame registration on eye angiographies
original registered
![Page 14: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/14.jpg)
Introduction
A geometric transform has two steps:
1. Computation of the mappingFind the equation relating the coordinates systems of original and result images.
2. Resampling or interpolationAssign values to the pixels of the result image from those of the input image.
![Page 15: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/15.jpg)
Mapping
![Page 16: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/16.jpg)
Introduction (Output to input)
Input to output :problem?
Output to input
![Page 17: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/17.jpg)
Introduction
Zoom
input to output output to input
input
outputs
![Page 18: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/18.jpg)
Introduction
Rotation
input to output output to input
input
outputs
![Page 19: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/19.jpg)
Mapping computation
• Model mathematically the acquisition process and invert it– Example: earth rotation,
effect in satellite images
– Example: rolling shutter
in photogrammetry
𝑥 = 𝑘 𝑅 −𝑅𝑐 𝑋 𝑥 = 𝑘 𝑅 −𝑅(𝑡)𝑐(𝑡) 𝑋rolling shutter
modelglobal shutter
model
![Page 20: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/20.jpg)
Mapping computation
• Adjustment to control points
![Page 21: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/21.jpg)
Mapping computation
• Adjustment to control points
input data: pairs of corresponding control points
𝑘 = 𝑞1 + 𝑞2𝑖 + 𝑞3𝑗 + 𝑞4𝑖2 + 𝑞5𝑖𝑗 + 𝑞6𝑗
2 +⋯
𝑙 = 𝑟1 + 𝑟2𝑖 + 𝑟3𝑗 + 𝑟4𝑖2 + 𝑟5𝑖𝑗 + 𝑟6𝑗
2 +⋯
𝑖𝑚, 𝑗𝑚 , 𝑘𝑚, 𝑙𝑚 , 𝑚 = 1…𝑛
![Page 22: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/22.jpg)
Mapping computation• Adjustment to control points
• Matrix notation
𝑄 𝑖, 𝑗 = 𝑘 = 𝑞1 + 𝑞2𝑖 + 𝑞3𝑗 + 𝑞4𝑖2 + 𝑞5𝑖𝑗 + 𝑞6𝑗
2 +⋯
𝑅 𝑖, 𝑗 = 𝑙 = 𝑟1 + 𝑟2𝑖 + 𝑟3𝑗 + 𝑟4𝑖2 + 𝑟5𝑖𝑗 + 𝑟6𝑗
2 +⋯
min𝑟1…𝑟𝑝
𝑚=1
𝑛
𝑙𝑚 − 𝑅 𝑖𝑚, 𝑗𝑚2
min𝑞1…𝑞𝑝
𝑚=1
𝑛
𝑘𝑚 − 𝑄 𝑖𝑚, 𝑗𝑚2
𝐾 = [𝑘1, 𝑘2, … 𝑘𝑛]𝑇, 𝐿 = [𝑙1, 𝑙2, … 𝑙𝑛]
𝑇 , 𝑄 = [𝑞1, 𝑞2, … 𝑞𝑛]𝑇, 𝑅 = [𝑟1, 𝑟2, … 𝑟𝑛]
𝑇
M=
1 𝑖1 𝑗1 𝑖12 𝑖1𝑗1 𝑗1
2 𝑖13 ⋯
1 𝑖2 𝑗2 𝑖22 𝑖2𝑗2 𝑗2
2 𝑖23 ⋯
⋮⋮
1 𝑖𝑛 𝑗𝑛 𝑖𝑛2 𝑖𝑛𝑗𝑛 𝑗𝑛
2 𝑖𝑛3 ⋯
𝐾 = M𝑄 𝐿 = M𝑅
𝑄 = M𝑇M −1M𝑇𝐾
𝑅 = M𝑇M −1M𝑇𝐿
Solution using pseudoinverse matrix
![Page 23: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/23.jpg)
Mapping computation
• Adjustment to control points
Manually selecting a number of corresponding control points is:
– time consuming
– tedious
– error prone
– imprecise
![Page 24: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/24.jpg)
Mapping computation
• Adjustment to control points
Semi-automatically: given control points on the input image, find the corresponding points on the reference (output)
Similarity measure: correlation, covariance, normalized cross-correlation
![Page 25: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/25.jpg)
Mapping computation
• Adjustment to control points
Automatically: find a set of local features in both images and look for a transformation in a robust way (RANSAC: RANdom SAmple Consensus)
![Page 26: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/26.jpg)
RANSAC
• RANdom SAmple Consensus: iterative method to estimate parameters of a mathematical model from a set of observed data which contains outliers.
1.- Select randomly the minimum number of points required to determine the model parameters. Solve for the parameters of the model.
2.- Determine how many points from the set of all points fit with a predefined tolerance (inliers).
3.- If inliers exceeds a predefined threshold τ, re-estimate the model parameters using all the identified inliers and terminate.
4.- Else repeat steps 1 through 3 a maximum of N times.
source: http://en.wikipedia.org/wiki/RANSAC
![Page 27: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/27.jpg)
Resampling
![Page 28: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/28.jpg)
Resampling
Given a certain mapping, fill in the output image by interpolating the pixel values of the input image at non integer coordinates
![Page 29: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/29.jpg)
Resampling
• Zero order interpolation: nearest neighbour
![Page 30: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/30.jpg)
Resampling
• First order interpolation: in 2D images bilinear
![Page 31: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/31.jpg)
Resampling
• Third order interpolation: in 2D images bicubic
source: https://en.wikipedia.org/wiki/Bicubic_interpolation
![Page 32: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/32.jpg)
Resampling
• Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc 𝑥, 𝑦
sinc 𝑥 =sin(𝑥)
𝑥
Nyquist rate: is the minimum rate at which a finite bandwidth (bandlimited) signal needs to be sampled to retain all of the information.
For a bandwidth of span B, Nyquist frequency = 2 B. If a signal is sampled at regular intervals dt, Nyquist rate = 1/(2dt)
sinc 𝑥, 𝑦
![Page 33: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/33.jpg)
Resampling
• Comparison example180o rotation by reflection (no interpolation needed) versus 36 successive rotations of 5o each
. . .
5° 10° 15°
170° 175° 180°
original
![Page 34: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/34.jpg)
Resampling
• Comparison example3 input images
![Page 35: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/35.jpg)
Resampling
• Biliniar interpolation (2 x 2 pixels)
• Bicubic (4 x 4 pixels)
![Page 36: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/36.jpg)
Resampling
• Lanczos kernel (6 x 6 pixels)
• sinc (x,y) (16 x 16 pixels)
![Page 37: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/37.jpg)
Applications
![Page 38: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/38.jpg)
Applications
• Registration
• Geometric alignment of images in order to compare or fusion pixel
by pixel.
• More used in medical imaging and remote sensing.
• Detects temporal differences or combine different sensors.
• Projections:
similarities, affinities, projective, spherical, cylindrical, warpings
![Page 39: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/39.jpg)
Applications
• Registration : Geometric alignment for 3D fusion CT-MRI
CT MRI
![Page 40: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/40.jpg)
Applications
• Registration : Geometric alignment
Sagital slicesAxial slices
MRI volume
![Page 41: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/41.jpg)
Applications
• Registration :
Geometric alignment
![Page 42: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/42.jpg)
Applications
• Registration : Alignment to stabilize a video sequence
Original sequence Stabilized sequence
![Page 43: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/43.jpg)
Applications
• Registration :
Alignment to stabilize a
video sequence
A arteryV vein
![Page 44: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/44.jpg)
Applications
• Construction of stitching mosaics from individual image
– Mathematical model of a camera: pin-hole, central projection.
– What a homography is?
– Mosaics of stitching images.
![Page 45: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/45.jpg)
Applications
• Mathematical model of a camera: pin-hole, central
projection.
![Page 46: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/46.jpg)
Applications
• Mathematical model of a camera: pin-hole, central
projection.
focal distance
holeimage
plane
![Page 47: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/47.jpg)
Applications
• Mathematical model of a camera: pin-hole, central
projection.
focal distance
variation
![Page 48: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/48.jpg)
Applications
• Mathematical model of a camera: pin-hole, central
projection.
photographic paper (negative)
1 2 3 4
![Page 49: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/49.jpg)
Applications
• Mathematical model of a camera: pin-hole, central projection.
![Page 50: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/50.jpg)
Applications
• What a homography is?
A particular case: central projection of a plane into a another plane
(2D to 2D).
There are other projections: orthographic, weak-perspective
![Page 51: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/51.jpg)
Applications
• What a homography is?
Examples of interest: changing the point of view in the image of a plane.
![Page 52: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/52.jpg)
Applications
• What a homography is?
Examples of interest: changing the point of view
in the image of a plane.
![Page 53: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/53.jpg)
Applications
• Panoramic mosaics
![Page 54: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/54.jpg)
Applications
projecció rectilínia (homografia) 100°
Panoramics of Georges Lagarde (GURL)
![Page 55: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/55.jpg)
Applications
projecció rectilínia (homografia) 130°
Panoramics of Georges Lagarde (GURL)
![Page 56: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/56.jpg)
Applications
projecció rectilínia (homografia) 170°
Panoramics of Georges Lagarde (GURL)
![Page 57: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/57.jpg)
Applications
Panoramics of Georges Lagarde (GURL)
![Page 58: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/58.jpg)
Applications
Panoramics of Georges Lagarde (GURL)
Impossible ?
![Page 59: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/59.jpg)
Applications
Panoramics of Georges Lagarde (GURL)
![Page 60: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/60.jpg)
Applications
• Lane road detection
![Page 61: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/61.jpg)
Applications
• Lane road detection
![Page 62: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/62.jpg)
Applications
• Lane road detection
![Page 63: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/63.jpg)
Applications
• Lane road detection
![Page 64: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/64.jpg)
Applications
• Lane road detection
s
−60°
+60°
Radon transform
θ
s
θi
g(s,θi)
![Page 65: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/65.jpg)
Applications
• Lane road detection
![Page 66: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/66.jpg)
Applications
• Distances in a plane
![Page 67: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/67.jpg)
Applications
• Distances in a plane
![Page 68: Visio per Computador presentació - UAB Barcelona · • Ideal interpolation for bandlimited signal sampled at Nyquist rate: convolution with sinc , sinc = sin( ) Nyquist rate: is](https://reader031.fdocuments.net/reader031/viewer/2022013021/5edb127609ac2c67fa68c38c/html5/thumbnails/68.jpg)
Mapas obtenido mediante imágenes aéreasTFG: Rubén Pérez Conte (2014)