computer vision 4

8/13/2019 Computer Vision 4

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Copyright 2001. Howie Choset,Renata Melamud, Al Costa,

Vincent Lee-Shue, Sean Piper,

Ryan de Jonckheere. All RightsReserved

Computer Vision


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2008 Howie Choset

Projection

P : R 3 -> R 2 or P : R 3 -> Z 2

Recover third dimension or justinfer stuff


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ImagesDiscrete representation of a continuous function

Pixel: Picture Element cell of constant color in a digital image An image is a two dimensional array of pixelsPixel: numeric value representing a uniform portion of an image

Grayscale All pixels represent the intensity of light in an image, be it red,green, blue, or another color

Like holding a piece of transparent colored plastic over your eyesIntensity of light in a pixel is stored as a number, generally 0..255

inclusiveColorBinary

Three grayscale images layered on top of eachother with each layerindicating the intensity of a specific color light, generally red, green,and blue (RGB)Third dimension in a digital image


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ImagesBinary images: Two color image

Pixel is only one byte of informationIndicates if the intensity of color is above or below some

nominal valueThresholding

ResolutionNumber of pixels across in horizontalNumber of pixels in the verticalNumber of layers used for color

Often measured in bits per pixel (bpp) where each color uses 8bits of data

Ex: 640x480x24bpp


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OpticsFocal length

Length f of projection through lens onimage plane

InversionProjection on image plane is inverted

f

Image planeObject


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Projection on the image planeSize of an image on the image plane is inversely proportional tothe distance from the focal point

h h

f

d

f h

d h

Focal point

h

h fd

f h

d h

Focal point

By conceptually moving the image plane, we can eliminate thenegative sign

Image plane

Image plane


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Move to three dimensionsSimilarity holds when three dimensions areconsidered

x

y

z

Image plane

f

(x,y,z)

(x,y,z)

z z

y y

f x

x x



Perspective1 Point Perspective

Using similar triangles, it is possible to determine

the relative sizes of objects in an imageGiven a calibrated camera (predetermine amathematical relationship between size on theimage plane and the actual object)

f

Image planeObject



Preliminary Math:

TrigonometryPythagorean Theorem

a2 + b 2 = c 2

Law of sines

Law of cosinescba

sinsinsin

cos2222 abbac


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Grayscale vs. Binary imageGrayscale Binary threshold



ThresholdingPurposeTrying to find areas of high color intensityHighlights locations of different features of the image (noticeMonas eyes) Image compression, use fewer bits to encode a pixel

How doneDecide on a value Scan every pixel in the image

If it is greater than , make it 255If it is less than , make it 0

Picking a good Often 128 is a good value to start withUse a histogram to determine values based on color frequencyfeatures



HistogramMeasure the number of pixels of differentvalues in an image.

Yields information such as the brightness ofan image, important color features,possibilities of color elimination forcompression

ThresholdingMake pixels above a value one color and valuesbelow that value a different colorBinary threshold often used to transform agrayscale image into black and white

Also usable for compression and feature extraction



Monas Histogram

0 255



ConnectivityTwo conventions on considering two pixelsnext to each other

To eliminate the ambiguity, we could define

the shape of a pixel to be a hexagon

8 point connectivity All pixels sharing a sideor corner are consideredadjacent

4 point connectivityOnly pixels sharing a sideare considered adjacent



Object location -

SegmentationOne method of locating an object is throughthe use of a wave front

Wavefront Assume a binary image with values of 0 or 11. Choose 1 st pixel with value 1, make it a 22. For each neighbor, if it is also a 1, make it a 2as well3. Repeat step two for each neighbor until thereare no neighbors with value 14. All pixels with a value 2 are are a continuousobject



CentroidsUse the region filled image from aboveCompute the area of the region

Number of pixels with the same number value (n)

Sum all of the x coords with the same pixel value. Do thesame for y coordsDivide each sum by n and the resulting x, y coord is thecentroid



Edge detectionScanline: one row of pixels in an imageTake the first derivative of a scanline

The derivative becomes nonzero when an

edge (pixels change values) is encountered



Implementing 1 st derivative

edge detection digitallyDerivative is defined asWith a scan line, the run (x c) is 1, and the

rise (f(x) f(c)) is B[m+1] B[m]This becomeswhere I is the resulting image of edges

This is really just a dot product of the vector[-1 1] repeated each pixel in the resultingimage

c xc f x f

c x )()(

lim

][1]1[1][ m Bm Bm I

11]1[][][ m Bm Bm I



More Math: ConvolutionThis operation of moving a mask acrossan image has a name, calledconvolutionIn order to mathematically apply a filterto a signal, we must use convolution

If you know Laplace transforms, this is amultiplication in the Laplace domain



Convolution: Analogd t h xt y )()()(

Given a symmetric h (common in image processing), simplifies to

d h xt y )()()(

h(t) = [-1 1]

Move across the signal x(possibly a scanline in animage)



Convolution: Digitalk

k hk n xn y ][][][

More useful in image processing on a digital computer

x[n] is a pixel in an image, y[n] is the resulting pixel0 2 2 0 1 1 3 0 1 1

40 2 2 0 1 1 3

0 1 1

0 2 2 0 1 1 3

0 1 14 2

1.

2.


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Convolution: Two-dimensionalRotate your mask 180 degrees about the origin (if you were

doing correct convolution, but since we are doing the otherconvolution, you can skip this step.Do the same dot product operation, this time using matricesinstead of vectorsRepeat the dot product for every pixel in the resulting imageIn the boundary case around the edges of the image there aretwo options

extend the original image out using the pixel values at the edgeMake the resulting image y smaller than the original and dontcompute pixels where the mask would extend beyond the edge ofthe original

0 0

),(),(),( 0000m n

nmhnnmm xnm y


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Convolution: Old and New Analog

Digital

Two dimensional, digital

0 0

),(),(),( 0000m n

nnmmhnm xnm y

d t h xt y )()()(

k

k nhk xn y ][][][

d ht xt y )()()(

k

k hk n xn y ][][][

0 0

),(),(),( 0000m n

nmhnnmm xnm y


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Filters, Masks, TransformsEdge detection

Wide masks

SmoothingObject detection


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Wide MasksWider masks lead to uncertainty about location of edgeCan detect more gradual edges

Convolution with Gaussian distribution


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Wide MasksWider masks lead to uncertainty about location of edgeCan detect more gradual edges

Convolution with Gaussian distributionEdge detection on a signal thatwas convolved with a Gaussian


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Cat Video


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Stereo VisionWay of calculating depth from two dimensionalimages using two cameras

d and y are unknowns, and can be determinedprocessing and c is known

Camera 1

Camera 2

d

y

c

d

ycd

tan

tan

tan

tantantan

d

c y

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