Computer Vision : CISC 4/689 Note to self David Jacob’s notes: Convolution Correlation Derivatives Separability

Computer Vision : CISC 4/689

Note to self

David Jacob’s notes:ConvolutionCorrelationDerivativesSeparability


Linear Filters

• General process:– Form new image whose pixels

are a weighted sum of original pixel values, using the same set of weights at each point.

• Properties– Output is a linear function of

the input

– Output is a shift-invariant function of the input (i.e. shift the input image two pixels to the left, the output is shifted two pixels to the left)

• Example: smoothing by averaging– form the average of pixels in a

neighbourhood

• Example: smoothing with a Gaussian– form a weighted average of

pixels in a neighbourhood

• Example: finding a derivative– form a weighted average of

pixels in a neighbourhood


Convolution

• Represent these weights as an image, F

• F is usually called the kernel

• Operation is called convolution– it’s associative

• Result is:

Rij H i u, j vFuvu,v


Convolution

• Formalization of idea of overlap (product) of two functions as one is shifted across the other

• The area under convolution is product of areas under the functions

• The convolution of two Gaussians is another Gaussian

courtesy of mathworld.wolfram.com


Convolution Notes

• Note the assumption that both f

and g are continuous and defined everywhere

• Properties– Commutative

– Associative


Discrete 2-D Convolution

• Sum instead of integral: I’ = K ¤ I (book uses ¤¤ for 2-D convolution) is defined by:

I’(u, v) = §x, y I(u ¡ x, v ¡ y) K(x, y)

where the kernel K approximates the impulse function f by sampling from it where it is non-zero

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

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

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


Example: Smoothing by Averaging


Smoothing with a Gaussian

• Smoothing with an average actually doesn’t compare at all well with a defocussed lens– Most obvious difference is that

a single point of light viewed in a defocussed lens looks like a fuzzy blob; but the averaging process would give a little square.

• A Gaussian gives a good model of a fuzzy blob


exp x2 y2

22

An Isotropic Gaussian

• The picture shows a smoothing kernel proportional to

(which is a reasonable model of a circularly symmetric fuzzy blob)


Smoothing with a Gaussian


Problem: Image Noise

from Forsyth & Ponce


Solution: Smoothing (Low-Pass) Filters

• If object reflectance changes slowly and noise at each pixel is independent, then we want to replace each pixel with something like the average of neighbors

– Disadvantage: Sharp (high-frequency) features lost

7 x 7 averaging neighborhood

Original image


Smoothing Filters: Details

• Filter types– Mean filter (box)– Median (nonlinear)– Gaussian

• Can specify linear operation by shifting kernel over image and taking product

111

111

111

3 x 3 box filter kernel


Gaussian Kernel

• Idea: Weight contributions of neighboring pixels by nearness

• Smooth roll-off reduces “ringing” seen in box filter

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


Gaussian Smoothing

• In theory, the Gaussian distribution is non-zero everywhere, which would require an infinitely large convolution mask, but in practice it is effectively zero more than about three standard deviations from the mean, and so we can truncate the mask at this point

• Discrete approximation to Gaussian function with sd = 1.4• (gets normalized so that sum is 1.0)

Source: http://www.cee.hw.ac.uk/hipr/html/gsmooth.html


Gaussian Smoothing

• Once a suitable mask has been calculated, then the Gaussian smoothing can be performed using standard convolution methods. The convolution can in fact be performed fairly quickly since the equation for the 2-D isotropic Gaussian shown above is separable into x and y components. Thus the 2-D convolution can be performed by first convolving with a 1-D Gaussian in the x direction, and then convolving with another 1-D Gaussian in the y direction. (The Gaussian is in fact the only completely circularly symmetric operator which can be decomposed in such a way.) Below shows the 1-D x component mask that would be used to produce the full mask shown in previous slide. The y component is exactly the same but is oriented vertically.

A further way to compute a Gaussian smoothing with a large standard deviation is to convolve an image several times with a smaller Gaussian. While this is computationally complex, it can have applicability if the processing is carried out using a hardware pipeline.

http://www.cee.hw.ac.uk/hipr/html/convolve.html


Gaussian Smoothing

• One of the principle justifications for using the Gaussian as a smoothing filter is due to its frequency response. Most convolution based smoothing filters act as lowpass frequency filters. This means that their effect is to remove high spatial frequency components from an image. The frequency response of a convolution filter, i.e. its effect on different spatial frequencies, can be seen by taking the Fourier transform of the filter.

•

• Frequency responses of Box (i.e. mean) filter (width 7 pixels) and Gaussian filter ( = 3 pixels). Shown in cycles per pixel, i.e. for one pixel.

• Mean filter exhibits oscillations in its response (within a pixel), whereas, Gaussian filter’s response is Gaussian. So you can be sure of what range of frequencies are present when Gaussian filter is applied.

http://www.cee.hw.ac.uk/hipr/html/freqfilt.html

http://www.cee.hw.ac.uk/hipr/html/freqfilt.html

http://www.cee.hw.ac.uk/hipr/html/fourier.html


Example: Gaussian Smoothing

= 1 = 3

7 x 7kernel

Original image Box filter


Correlation

• Same as convolution, a dot product, thus it is largest when the pattern matches (vectors are parallel), this can be used to find texture patterns

• This yields a value that is +ve when the image region looks like the filter kernel, and small and –ve when it is opposite. Can be squared if pattern reversal doesn’t matter.

• Since value maybe large if image is bright, so divide by root sum of squares of image region and filter. (dot of unit vectors)

• Some ways to interpret what the kernel is doing– As a template being matched by correlation – As simply a set of weights on the corresponding image pixels


Normalized correlation

• Think of filters of a dot product– now measure the angle

– i.e normalised correlation output is filter output, divided by root sum of squares of values over which filter lies

• Tricks:– ensure that filter has a zero

response to a constant region (helps reduce response to irrelevant background)

– subtract image average when computing the normalizing constant (i.e. subtract the image mean in the neighborhood)

– absolute value deals with contrast reversal


Positive responses


Positive responses


Figure from “Computer Vision for Interactive Computer Graphics,” W.Freeman et al, IEEE Computer Graphics and Applications, 1998 copyright 1998, IEEE


Dealing with Image Boundaries

• Only convolve with interior– Shrinks image

• Zero-padding– Results in spurious gradients

• Border replication• Symmetric: Reflect image at border b so that

I(b + i) = I(b ¡ i)– Results in spurious 2nd-derivatives


Step 1

3

2

1

2

2

1

3

2

32

21

22

32 5

3

2

1

2

2

1

3

2

32

21

22

32

1-2-1

24-1

111

1-1-1

12-1

111

zero

-pad

ded


Step 2

3

2

1

2

2

1

3

2

32

21

22

32 45

3

2

1

2

2

1

3

2

32

21

22

32

3-1-2

24-2

111

1-1-1

12-1

111


Step 3

3

2

1

2

2

1

3

2

32

21

22

32 4 45

3

2

1

2

2

1

3

2

32

21

22

32

3-3-1

34-2

111

1-1-1

12-1

111


Step 4

3

2

1

2

2

1

3

2

32

21

22

32 4 4 -25

3

2

1

2

2

1

3

2

32

21

22

32

1-3-3

16-2

111

1-1-1

12-1

111


Step 5

3

2

1

2

2

1

3

2

32

21

22

32 4 4

9

-25

3

2

1

2

2

1

3

2

32

21

22

32

2-2-1

14-1

221

1-1-1

12-1

111


Step 6

3

2

1

2

2

1

3

2

32

21

22

32

6

4 4

9

-25

3

2

1

2

2

1

3

2

32

21

22

32

1-2-2

32-2

222

1-1-1

12-1

111


and so on…


Final Result

2 2 2 3

2 1 3 3

2 2 1 2

1 3 2 2 12

7

6

4

8

6

14

4

59

59

511

-25

I’I

1-1-1

12-1

111

Why is I’ large in some places and small in others?


Linear Shift Invariance

• Possible properties of f– Superposition: f(I1 + I2) = f(I1) + f(I2)– Scaling: f(®I) = ®f(I)– Shift invariance:

f(Shift(I, k)) = Shift (f(I), k)

A system with these properties is performing convolution


Imaging Systems

• An imaging system describes a functional transformation f of an image due to…– Physics: A real-world phenomenon such as blurring from defocus or fish-eye

lens distortion– Filtering: A transformation we apply in order to

• Undo or mitigate the bad effects of a physical system (e.g., deblur, undistort, etc.)

• Emphasize or highlight particular image properties (e.g., color similarity, edges, etc.)

• Noise reduction!!

I I’f


What’s Not a Convolution?

• Nonlinear systems– E.g., radial distortion of fish-eye lens is not LSI because geometric

transformation depends on pixel location

f

courtesy of M. Fiala


Filtering in Matlab

• imfilter(I, K)filters image I with kernel K– Default filtering is correlation (no kernel rotation)– Can set options on border handling

• corr2, conv2 are the generic versions• Kernel creation

– Custom (create a matrix)– fspecial function


Differentiation and convolution

• Recall

• Now this is linear and shift invariant, so must be the result of a convolution.

• We could approximate this as

(which is obviously a convolution; it’s not a very good way to do things, as we shall see)

fx

lim 0

f x , y

f x,y

fx

f xn1,y f xn , y

x


Finite differences

Partial derivative in x-direction: gives vertical stripes.mid-gray = 0, dark gray = -ve, light gray = +veKernel = (1,0,-1)


Noise

• Simplest noise model– independent stationary additive

Gaussian noise

– the noise value at each pixel is given by an independent draw from the same normal probability distribution

• Issues– this model allows noise values

that could be greater than maximum camera output or less than zero

– for small standard deviations, this isn’t too much of a problem - it’s a fairly good model

– independence may not be justified (e.g. damage to lens)

– may not be stationary (e.g. thermal gradients in the ccd)


sigma=1


sigma=16


Finite differences and noise

• Finite difference filters respond strongly to noise– obvious reason: image noise

results in pixels that look very different from their neighbours

• Generally, the larger the noise the stronger the response

• What is to be done?– intuitively, most pixels in

images look quite a lot like their neighbors

– this is true even at an edge; along the edge they’re similar, across the edge they’re not

– suggests that smoothing the image should help, by forcing pixels different to their neighbors (=noise pixels?) to look more like neighbors


Finite differences responding to noise

Increasing noise ->(this is zero mean additive Gaussian noise)


Filter responses (of noise) are correlated

• Filtered noise is sometimes useful– looks like some natural textures, can be used to simulate fire, etc.





Smoothing reduces noise

• Generally expect pixels to “be like” their neighbours– surfaces turn slowly

– relatively few reflectance changes

• Generally expect noise processes to be independent from pixel to pixel

• Implies that smoothing suppresses noise, for appropriate noise models

• Scale– the parameter in the symmetric

Gaussian

– as this parameter goes up, more pixels are involved in the average

– and the image gets more blurred

– and noise is more effectively suppressed


The effects of smoothing Each row shows smoothingwith Gaussians of differentwidth; each column showsdifferent realisations of an image of Gaussian noise.


Gradients and edges

• Points of sharp change in an image are interesting:– change in reflectance

– change in object

– change in illumination

– noise

• Sometimes called edge points

• General strategy– determine image gradient

– now mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).


The Gradient and Edges

• Consider image intensities as a 2-D height function I(x, y). Then the image gradient is the vector field defined by:

• Definition of an edge– Line segment separating regions of contrasting intensity– Location: Where gradient magnitude is high – Direction: Orthogonal to the gradient


Edge Causes

• Depth discontinuity

• Surface orientation discontinuity

• Reflectance discontinuity (i.e., change in surface material properties)

• Illumination discontinuity (e.g., shadow)


Edges

• Edge normal: unit vector in the direction of maximum intensity change.

• Edge direction: unit vector perpendicular to the edge normal.

• Edge position: the image position at which the edge is located.

• Edge strength: related to the local image contrast along the normal.


Edges

• Searching for Edges:– Filter: Smooth image

– Enhance: Apply numerical derivative approximation

– Detect: Threshold to find strong edges

– Localize/analyze: Reject spurious edges, include weak but justified edges


Edge Detection• An edge point can be regarded as a

point in an image where a discontinuity (in gradient) occurs across some line. A discontinuity may be classified as one of five types

Gradient Discontinuity -- where the gradient of the pixel values changes across a line. This type of discontinuity can be classed as roof edges, ramp edges convex edges concave edges, by noting the sign of the component of the gradient perpendicular to the edge on either side of the edge. Ramp edges have the same signs in the gradient components on either side of the discontinuity, while roof edges have opposite signs in the gradient components.

A Jump or Step Discontinuity -- where pixel values themselves change suddenly across some line. A Bar Discontinuity -- where pixel values rapidly increase then decrease again (or vice versa) across some line.

Source: http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/MARSHALL/node28.html

http://homepages.inf.ed.ac.uk/rbf/CVonline/


Edges Models

Step edge Ramp edge Roof edge (line)

Edge models are useful for designing optima edge detectors

Courtesy: R. C. Gonzalez & R. E. Woods


Step and Ramp edge detection

• Two typical ways to find step and ramp edges. Look for:– Pixels with large (in absolute value) first order derivatives– Zero-crossings of second derivatives



Step edge detection: First Derivative Operators

• Method: Differentiate and find extrema

• Examples– Sobel operator (Matlab: edge(I, ‘sobel’))

– Prewitt, Roberts cross

– Derivative of Gaussian

-1-2-1

000

121

-101

-202

-101

Sobel x Sobel y

Book uses thisformat


Sobel Edge Filtering Example

1 0 -1

2 0 -2

1 0 -1

0 0 2 2

0 0 2 2

0 0 2 2

0 0 2 2

Rotate

10-1

20-2

10-1


Step 1

0

0

1

2

2

1

2

2

22

20

20

22 0

0

0

0

0

2

2

2

2

20

20

20

20

00-1

00-2

10-1

10-1

20-2

10-1


Step 2

0

0

1

2

2

2

3

2

22

20

20

22 60

0

0

0

0

2

2

2

2

20

20

20

20

200

400

10-1

10-1

20-2

10-1


Step 3

0

0

1

2

2

2

3

2

30

20

20

30 6 60

0

0

0

0

2

2

2

2

20

20

20

20

200

400

10-1

10-1

20-2

10-1


Step 4

0

0

0

0

2

2

3

2

30

20

20

30 6 6 -60

0

0

0

0

2

2

2

2

20

20

20

20

10-2

20-4

10-1

10-1

20-2

10-1

edgeeffectfrom zero-padding


Sobel Edge Filtering Example: Result

6

8

8

6

6

8

8

6

-80

-60

-80

-60

(pad with zeroes again, replace the boundarywith zeros) and then we threshold…


Sobel Edge Detection: Gradient Approximation

Horizontal Vertical

-1-2-1

000

121

-101

-202

-101

Note anisotropy of edge finding


Sobel

• These can then be combined together to find the absolute magnitude of the gradient at each point and the orientation of that gradient. The gradient magnitude is given by:

• an approximate magnitude is computed using:

which is much faster to compute.

• The angle of orientation of the edge (relative to the pixel grid) giving rise to the spatial gradient is given by:

In this case, orientation 0 is taken to mean that the direction of maximum contrast from black to white runs from left to right on the image, and other angles are measured anti-clockwise from this.



Mag. Of 1st

Derivative andzero-crossing of2nd derivative


Ramp edges

• 1st derivative is non-zero along the entire ramp

• 2nd derivative nonzero only at onset and end

• Most image edges are ramps rather than step edges

• 1st order derivatives produce thicker edges


Step edge

• Second derivative produces a double edge response

• The sign of the second order derivative can be used to locate edges


More..

• Noise: The 2nd derivative response is much stronger than the 1st derivative response

• Line (roof edge): 2nd derivative response is again stronger than the 1st derivative response

• Point detection: second order derivatives have a stronger response to isolated points

• What second order derivative operator can we use?


Laplacian (derive..)

The implementationsare rotationallyinvariant only inincrements such as45 or 90 degrees.

Turn by 45 and add.

Weights from analytical eqns.


Scale is important

So far, we donot have scaleFor Laplacian, i.e, extent forLaplacian.


Derivative of Gaussian

Documents

Computer Vision : CISC 4/689 Note to self David Jacob’s notes: Convolution Correlation Derivatives Separability