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ME5286 – Lecture 4

#1

Lecture 4: Spatial Domain Transformations

Saad J [email protected]


2nd Robotics Quiz#2

• Friday April 14• Place: Keller 3-125


#3

Last Vision Lecture

• Digital Image Representation– Sampling – Quantization

• Color Fundamentals– Color Spaces


#4

Any Questions for HW#1


#5

Outline of this Lecture

• Spatial Domain Transformation– Point Processing Transformations

• Pixel Mapping• Histogram Processing

– Area/Mask Processing Transformations• Image Filtering

– Frame Processing Transformations– Geometric Transformations


Digital Image Processing

• Pixel Processing: change range of image• g(x) = h(f(x))

f

x

hf

x

f

x

hf

x

• Geometric Transform: change domain of imageg(x) = f(h(x))


Digital Image Processing

h

h

f

f g

g

• Pixel Processing : change range of image• g(x) = h(f(x))

• Geometric Transform : change domain of imageg(x) = f(h(x))


Spatial Domain Methods

Point Processing

Area/Mask Processing


Point Processing• The simplest kind of range transformations are

independent of the position x,y:• g = t(f)

• This is called point processing.

• What can they do?• What’s the form of t?

• Important: every pixel is mapped independently – spatial information completely lost!


#10

Point Processing Methods• The most primitive, yet essential, image processing operations.• Intensity transformations that convert an old pixel into a new

pixel based on some predefined function.• Operate on a pixel based solely on that pixel’s value.• Used primarily for image enhancement.

Transformation function


Identity Transformation


Negative Image

• O(r,c) = 255-I(r,c)


Negative Image


Contrast Stretching or Compression

• Stretch gray-level ranges where we desire more information (slope > 1).

• Compress gray-level ranges thatare of little interest(0 < slope < 1).


Point Pocessing Methods• Thresholding: Special case of contrast compression


#16

Point Processing Methods• Intensity-level

slicing– Highlight a specific

range of gray-levels only

– Acts as double thresholding


Bit-level Slicing• Highlighting the contribution made by a

specific bit.• For pgm images, each pixel is represented

by 8 bits.• Each bit-plane is a binary image



Logarithmic transformation• Non-linear transformations

– We may use any function, provided that is gives a one-to-one or many-to-one (i.e., single-valued) mapping.

• Enhance details in the darker regions of an image at the expense of detail in brighter regions.

rcrTs 1logcompress

stretch


Log Stretching


Exponential transformation• Reverse effect of that obtained using logarithmic

mapping.

compress

stretch


Basic Point Processing


Power-law transformations


Image Enhancement


Contrast Stretching


Image Histogram

#26


#27

Image Intensity Histogram

• The histogram of a digital image with gray levels from 0 to L-1 is a discrete function h(rk)=nk, where:

– rk is the kth gray level– nk is the # pixels in the image with that gray level– n is the total number of pixels in the image– k = 0, 1, 2, …, L-1

• Normalized histogram: p(rk)=nk/n– sum of all components = 1


Histogram provides a global description of the appearance ofthe image.

If we consider the gray values in the image as realizations of arandom variable R, with some probability density, histogramprovides an approximation to this probability density. In otherwords,

)()Pr( kk rprR


#29

Building Distributions

50 70 100 110 150

60

80

70

40

Number of Pixels

Intensity of Green Channel

0

80 80( 100) 0.2540 60 2 70 80 320GP t


Example: Image Histograms

• An image histogram is a plot of the gray-level frequencies (i.e., the number of pixels in the image that have that gray level).


Example: Image Histograms

• Divide frequencies by total number of pixels to represent as probabilities.

Nnp kk /


Some Typical HistogramsThe shape of a histogram provides useful information forcontrast enhancement.

Dark image

im = imread(‘pic1.tif’) ;figure ; imshow(im) ; [h,n] = hist(double(im(:))) ; [h,n] = hist(double(im(:),100) ; bar(n,h) ;Imhist(im) ; % image histogram function with image proc toolbox


Bright image

Low contrast image


High contrast image


Image Histograms


Properties of Image Histograms

• Histograms clustered at the low end correspond to dark images.• Histograms clustered at the high end correspond to bright

images.


Properties of Image Histograms• Histograms with small spread correspond to low contrast

images (i.e., mostly dark, mostly bright, or mostly gray).• Histograms with wide spread correspond to high contrast

images.


Properties of Image HistogramsLow contrast High contrast


#39

Histogram Processing

• The shape of the histogram of an image does provide useful info about the possibility for contrast enhancement.

• Types of processing:

Histogram equalizationHistogram matching (specification)Local enhancement


#40

Point Processing Methods• Histogram equalization

– Low contrast images are usually mostly dark, mostly light, or mostly gray.

– High contrast images have large regions of dark and large regions of white.

– Good contrast images exhibit a wide range of pixel values (i.e., no single gray level dominates the image).


#41


– is a transformation that stretches the contrast by redistributing the gray-level values uniformly.

– is fully automatic compared to other contrast stretching techniques.


#42

Histogram Equalization• As mentioned above, for gray levels that take on

discrete values, we deal with probabilities: pr(rk)=nk/n, k=0,1,.., L-1

nk is the number of pixels with rk gray leveln is the total number of pixelsk= 0,1,…,255

– The plot of pr(rk) versus rk is called a histogram and the technique used for obtaining a uniform histogram is known as histogram equalization (or histogram linearization).


How do we determine this grey scale transformation function?

Assume our grey levels are continuous and have been normalized to lie between 0 (black) and 1 (white).

We must find a transformation T that maps grey values r in the input image F to grey values s = T(r) in the transformed image .

It is assumed that

T is single valued and monotonically increasing, and

for

The inverse transformation from s to r is given by :

r = T-1(s).


An example of such a transfer function is illustrated in the Figure


• T(r) maps [0,1] into [0,1] (preserves the range of allowedGray values). • In terms of histograms, the output image will have allgray values in “equal proportion” .


#46

Histogram Equalization

• Histogram equalization(HE) results are similar to contrast stretching but offer the advantage of full automation, since HE automatically determines a transformation function to produce a new image with a uniform histogram.

)()(0 0

j

k

j

k

jr

jkk rp

nn

rTs

http://homepages.inf.ed.ac.uk/rbf/HIPR2/histeq.htm


Step 1:For images with discrete gray values, compute:

nnrp k

kin )( 10 kr 10 Lk

L: Total number of gray levelsnk: Number of pixels with gray value rk

n: Total number of pixels in the image

Step 2: compute the discrete version of the previous transformation :

k

jjinkk rprTs

0)()( 10 Lk

How to implement histogram equalization?


Histogram Equalization

• A fully automatic gray-level stretching technique.


#49


– In practice, the histogram might not become totally flat !


#50

Histogram Specification

• Histogram equalization does not allow interactive image enhancement and generates only one result: an approximation to a uniform histogram.

• Sometimes though, we need to be able to specify particular histogram shapes capable of highlighting certain gray-level ranges.


#51


• The procedure for histogram-specification based enhancement is:

– Equalize the levels of the original image using:

k

j

jk n

nrTs

0

)(

n: total number of pixels,

nj: number of pixels with gray level rj,

L: number of discrete gray levels


#52


– Specify the desired density function and obtain the transformation function G(z):

z

i

iz

z nnwpzGv

00

)()(

– Apply the inverse transformation function z=G-1(s) to the levels obtained in step 1.

pz: specified desirable PDF for output


#53


• The new, processed version of the original image consists of gray levels characterized by the specified density pz(z).

)]([ )( 11 rTGzsGz In essence:


Input image

Uniform image

Output images=T(r) v=G(z)

z=H(r)

Approach of derivation

= G-1(v=s=T(r))


#55


• The principal difficulty in applying the histogram specification method to image enhancement lies in being able to construct a meaningful histogram.


#56


– Either a particular probability density function (such as a Gaussian density) is specified and then a histogram is formed by digitizing the given function,

– Or a histogram shape is specified on a graphic device and then is fed into the processor executing the histogram specification algorithm.


#57

Local Enhancement

– Desire to enhance a local region– Local Enhancement is achieved by applying

histogram processing ( Equalization ) on a local region instead of the whole image


Histogram Color Processing

• can apply histogram equalization to color images

• don't want to apply it using the RGB color model - equalizing R, G, and B bands independently causes color shifts

• must convert to a color model that separates intensity information from color information (e.g. HSI)

• can then apply histogram equalization on the intensity band


Area Mask ProcessingOR

Image Filtering

#59


Spatial Domain Methods

Point Processing

Area/Mask Processing


#61

Area/Mask Processing Methods• A pixel value is computed from its old value and the

values of pixels in its vicinity.• More costly operations than simple point processes, but

more powerful.• What is a Mask?

– A mask is a small matrix whose values are called weights.– Each mask has an origin, which is usually one of its positions.– The origins of symmetric masks are usually their center pixel

position.


#62

Area/Mask Processing Methods• Examples:

• Applying masks to images (filtering)– The application of a mask to an input image produces an

output image of the same size as the input.– For linear filters, the operation is a linear operation on the

image pixels


Linear Mask Processing Methods

• Two main linear Mask Operation or Spatial Filtering methods:– Correlation– Convolution


Correlation

OutputImage

w(i,j)

f(i,j)

/2 /2

/2 /2( , ) ( , ) ( , ) ( , ) ( , )

K K

s K t Kg x y w x y f x y w s t f x s y t

g(i,j)


Convolution

• Similar to correlation except that the mask is first flipped both horizontally and vertically.

Note: if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then convolution is equivalent to correlation!

/2 /2

/2 /2( , ) ( , ) ( , ) ( , ) ( , )

K K

s K t Kg x y w x y f x y w s t f x s y t


#66

Area/Mask Processing Methods• Convolution

1) For each pixel in the input image, the mask is conceptually placed on top of the image with its origin lying on that pixel.

2) The values of each input image pixel under the mask are multiplied by the values of the corresponding mask weights.

3) The results are summed together to yield a single output value that is placed in the output image at the location of the pixel being processed on the input.


Example

Correlation:

Convolution:


#68

Area/Mask Processing Methods• Cross Correlation

– Correlation applies the mask directly to the image without flipping it.– It is often used in applications where it is necessary to measure the

similarity between images or parts of images.– If the mask is symmetric (i.e., the flipped mask is the same as the original

one) then the results of convolution and correlation are the same.


#69

Area/Mask Processing Methods• Normalization of mask weights

• Practical problems– How to treat the image borders?– Time increases exponentially with

mask size.

– The sum of weights in the convolution mask affect the overall intensity of the resulting image.

– Many convolution masks have coefficients that sum to 1 (convolved image will have the same average intensity as the original one).

– Some masks have negative weights and sum to 0.– Pixels with negative values may be generated using masks with negative weights.– Negative values are mapped to the positive range through appropriate

normalization.


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

[.,.]h[.,.]f

Image filtering111

111

111

],[g


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk


0 10 20 30 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk


0 10 20 30 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

?

],[],[],[,

lnkmflkgnmhlk


0 10 20 30 30

50

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111

111

111

],[g

Credit: S. Seitz

?

],[],[],[,

lnkmflkgnmhlk


0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20 30 30 30 20 10

0 20 40 60 60 60 40 20

0 30 60 90 90 90 60 30

0 30 50 80 80 90 60 30

0 30 50 80 80 90 60 30

0 20 30 50 50 60 40 20

10 20 30 30 30 30 20 10

10 10 10 0 0 0 0 0

[.,.]h[.,.]f

Image filtering111111111],[g

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk


What does it do?• Replaces each pixel with an

average of its neighborhood

• Achieve smoothing effect (remove sharp features)

111

111

111

Slide credit: David Lowe (UBC)

],[g

Box Filter


Smoothing with box filter


Practice with linear filters

000010000

Original

?

Source: D. Lowe



000010000

Original Filtered (no change)

Source: D. Lowe



000100000

Original

?

Source: D. Lowe



000100000

Original Shifted leftBy 1 pixel

Source: D. Lowe


#84

Area/Mask Processing Methods• Smoothing (or Low-Pass) Filters

• Averaging or mean filter– Elements of mask must be positive.– Size of mask degree of smoothing.

– Useful for noise reduction and image blurring.– Remove the finer details of an image.



Original

111111111

000020000 - ?

(Note that filter sums to 1)

Source: D. Lowe



Original

111111111

000020000 -

Sharpening filter- Accentuates differences with local average

Source: D. Lowe


Sharpening

Source: D. Lowe


Other filters

-101

-202

-101

Vertical Edge(absolute value)

Other Filter


Other filters

-1-2-1

000

121

Horizontal Edge(absolute value)

OtherFilter


Basic gradient filters

000

10-1

000

0-10

000

010

10-1

or

Horizontal Gradient Vertical Gradient

1

0

-1

or


#91

Area/Mask Processing Methods• Averaging or mean filter – cont.

– Not the best choice


#92

Area/Mask Processing Methods• Gaussian filters – properties:

– Most common natural model (there are cells in eye that perform Gaussian filtering)

– Smooth function, it has infinite number of derivatives– Fourier Transform of Gaussian is Gaussian.– Convolution of a Gaussian with itself is a Gaussian.– Gaussian smoothing can be implemented efficiently because

the kernel is separable


Gaussian filter

* =Input image f

Filter hOutput image g

Compute empirically


Gaussian vs. mean filters

What does real blur look like?


• Spatially-weighted average

0.003 0.013 0.022 0.013 0.0030.013 0.059 0.097 0.059 0.0130.022 0.097 0.159 0.097 0.0220.013 0.059 0.097 0.059 0.0130.003 0.013 0.022 0.013 0.003

5 x 5, = 1

Slide credit: Christopher Rasmussen

Important filter: Gaussian


Gaussian filters• What parameters matter here?• Variance of Gaussian: determines extent of

smoothing

Source: K. Grauman


#97

Area/Mask Processing Methods• The value of determines the degree of smoothing.• As increases, the size of the mask must also increase

if we are to sample the Gaussian satisfactorily.• Rule – choose:

height = width = 5 (subtends 98.76% of the area)


Smoothing with box filter


Smoothing with Gaussian filter


Smoothing with a Gaussian

…

Parameter σ is the “scale” / “width” / “spread” of the Gaussian kernel, and controls the amount of smoothing.

Source: K. Grauman


#101

Area/Mask Processing Methods• Sharpening (or High-Pass) Filters

– Used to emphasize the fine details of an image (has the opposite effect of smoothing).

– Points of high contrast can be detected by computing intensity differences in local image regions.

– Weights of the mask are both positive and negative.– When the mask is over an area of constant or slowly varying

gray level, the result of convolution will be close to zero.– When gray level is varying rapidly within the neighborhood,

the result of convolution will be a large number.– Typically, such points form the border between different

objects or scene parts


#102

Area/Mask Processing Methods• Sharpening (or High-Pass) Filters – cont.


#103

Area/Mask Processing Methods• Sharpening using derivatives

• Computing the derivative of an image has as a result the sharpening of the image.

• The most common way to differentiate an image is by using the gradient.


#104

Area/Mask Processing Methods• Sharpening using derivatives – cont.

• The gradient can be approximated by finite differences which can be implemented efficiently as masks.

• Examples of masks based on gradient approximations with finite differences:


How do we choose the elements of a mask?

• Typically, by sampling certain functions.

Gaussian1st derivativeof Gaussian

2nd derivativeof Gaussian


Filters

• Smoothing (i.e., low-pass filters)– Reduce noise and eliminate small details.– The elements of the mask must be positive.– Sum of mask elements is 1 (after

normalization)

Gaussian


Filters• Sharpening (i.e., high-pass filters)

– Highlight fine detail or enhance detail that has been blurred.

– The elements of the mask contain both positiveand negative weights.

– Sum of the mask weights is 0 (after normalization)

1st derivativeof Gaussian

2nd derivativeof Gaussian


Sharpening Filters (High Pass Filtering)

• Useful for emphasizing transitions in image intensity (e.g., edges).


Sharpening Filters

• Note that the response of high-pass filtering might be negative.• Values must be re-mapped to [0, 255]

sharpened imagesoriginal image


Sharpening Filters: Unsharp Masking

• Obtain a sharp image by subtracting a lowpass filtered (i.e., smoothed) image from the original image:

- =


Sharpening Filters: Derivatives

• Taking the derivative of an image results in sharpening the image.

• The derivative of an image can be computed using the gradient.


Sharpening Filters: Derivatives

• The gradient is a vector which has magnitude and direction:

| | | |f fx y

or

(approximation)


Sharpening Filters: Derivatives• Magnitude: provides information about

edge strength.

• Direction: perpendicular to the direction of the edge.


Sharpening Filters: Gradient Computation

• Approximate gradient using finite differences:

sensitive to horizontal edges

sensitive to vertical edges

Δx


Sharpening Filters: Gradient Computation (cont’d)


Example

fx

fy



• We can implement and using masks:

• Example: approximate gradient at z5

(x+1/2,y)

(x,y+1/2) **

good approximationat (x+1/2,y)

good approximationat (x,y+1/2)



• A different approximation of the gradient:

•We can implement and using the following masks:

*

(x+1/2,y+1/2)good approximation



• Example: approximate gradient at z5

• Other approximations

Sobel


Example

fy

fx


Sharpening Filters: Laplacian

The Laplacian (2nd derivative) is defined as:

(dot product)

Approximatederivatives:


Sharpening Filters: LaplacianLaplacian Mask


Handling Pixels Close to Boundaries

pad with zeroes

or

0 0 0 ……………………….0

0 0 0 ……

……

……

……

….0


Handling Pixels Close to Boundaries

What to do about image borders:

black fixed periodic reflected


Linear vs Non-Linear Spatial Filtering

• A filtering method is linear when the output is a weighted sum of the input pixels.

• Methods that do not satisfy the above property are called non-linear.– e.g.,


#126

Area/Mask Processing Methods• Median filter (non-linear filter) .

– Replace each pixel value by the median of the gray-levels in the neighborhood of the pixels


Smoothing Filters: Median Filtering(non-linear)

• Very effective for removing “salt and pepper” noise (i.e., random occurrences of black and white pixels).

averagingmedian filtering


Remember• Intensity operations can yield pixel values

outside of the range [0 – 255].• You should convert values back to the range

[0 – 255] to ensure that the image is displayed properly.

• Consider the following mapping?

[fmin – fmax] [ 0 – 255]


Geometric Transformations

#129


#130

Geometric Transformations• Modify the arrangement of pixels based on some

geometric transformation.


What are geometric transformations?


Translation


Translation and rotation


Scale


Similarity transformations

Similarity transform (4 DoF) = translation + rotation + scale


Aspect ratio


Shear


Affine transformations

Affine transform (6 DoF) = translation + rotation + scale + aspect ratio + shear


Geometric Transformations

100

]1[]1[

3231

2221

1211

tttttt

wvyx

Affine Transformationy=v sinθ + w cosθ


#140

Geometric Transformations• Some Practical Problems

1) Transformed pixel coordinates might not lie within the bounds of the image.

2) Transformed pixel coordinates can be non-integer.3) There might be no pixels in the input image that map to

certain pixel locations in the transformed image (one-to-one correspondence can be lost).


#141

Geometric Transformations• Problem (3): Forward vs. Inverse Mapping

– To guarantee that a value is generated for every pixel in the output image, we must consider each output pixel in turn and use the inverse mapping to determine the position in the input image.


#142

Geometric Transformations• Problem (2): Image Interpolation

– Interpolation is the process of generating integer coordinates for a transformed pixel by examining its surrounding pixels.

• Zero-order interpolation (or nearest-neighbor)


#143

Geometric Transformations• First-order interpolation

• Higher-order interpolation schemes are more sophisticated but also more time consuming


Frame Processing

#144


#145

Frame Processing Methods• Generate a pixel value based on an operation involving two or

more images or Frames.• Each output pixel is usually located at the same position in the

input image.• Attractive in Video Processing

– Useful for combining information in two images.


#146

Frame Processing Methods• Subtraction

O(r, c) = |I1(r, c) - I2(r, c)|

– Useful for "change detection“


#147

Frame Processing Methods• Averaging

– Image quality can be improved by averaging a number of images together.


#148

Frame Processing Methods


#149

Review

• Spatial Domain Transformation– Point Processing for Enhancement– Area/Mask Processing Transformations

• Image Filter for Blurring, Enhancement

– Frame Processing Transformations• Operations on multiple images

– Geometric Transformations• Image Warping or Operations on the image axes

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