95.4501

Wilf LaLonde ©2012Comp 4501

95.4501

Filters


• A filter is a matrix of weights centered on a specific pixel in an image and used to produce a weighted average as follows.

• The center weight is multiplied with the pixel, the other weights are multiplied with corresponding neighbor pixels.

• The results are added and divided by the sum of the weights (or, avoid the divide by using normalized weights; i.e., pre-divided).

What’s a Filter?

[ 0 0 0 ][ 0 1 0 ][ 0 0 0 ]

3x3 identity filter


• This filtering operation applied to each pixel of an image is called a convolution (if the filter is symmetrical) or correlation otherwise.

• More complex filters, that can use fancier functions, exist as well.

What’s a Filter?


If the sampler can be indexed via image sized texture coordinates (otherwise, *pixelSize ).

float3 fillterResult = float4 (0.0, 0.0, 0.0);for (int i = -1; i <= 1; i++) {

for (int j = -1; j <= 1; j++) { fillterResult +=

sampler (uv.xy + float2 (i,j) ).xyz *filterWeight [i,j];

}}

How filters Get Used: Let Compiler Loop Unroll

Filter result is the answer: assuming normalized weights


If the sampler can be indexed via image sized texture coordinates (otherwise, *pixelSize ).

float3 fillterResult = sampler (uv.xy + float2 (-1,-1)).xyz * filterWeight [-1,-1] +sampler (uv.xy + float2 (-1, 0)).xyz * filterWeight [-1, 0] +sampler (uv.xy + float2 (-1,+1)).xyz * filterWeight [-1,+1] +sampler (uv.xy + float2 ( 0,-1)).xyz * filterWeight [ 0,-1] +sampler (uv.xy + float2 ( 0, 0)).xyz * filterWeight [ 0, 0] +sampler (uv.xy + float2 ( 0,+1)).xyz * filterWeight [ 0,+1] +sampler (uv.xy + float2 (+1,-1)).xyz * filterWeight [+1,-1] +sampler (uv.xy + float2 (+1, 0)).xyz * filterWeight [+1, 0] +sampler (uv.xy + float2 (+1,+1)).xyz * filterWeight [+1,+1];

How filters Get Used: Unroll Loop Yourself

Filter result is the answer: assuming normalized weights


• An odd size filter looks cleaner but even size works too (consistently applying right and down, for example)...

• sum of normalized weights 1 brighter image• sum of normalized weights 1 darker image

A Few Observations

Weight 0.25 Weight 0.25

Weight 0.25 Weight 0.25

[x, y] [x+1, y]

[x, y+1] [x+1, y+1] right and up for OpenGL

right and down for DirectX


• Indexing off the end is can be handled with • 0 weight• automatically via a clamping sampler

• Filtered results are sometimes clamped to the bounds of the application; e.g., 0 and 1 for color.

A Few Observations


A Blur Filter (Minimal Blur)

[ 0 1 0 ][ 1 1 1 ][ 0 1 0 ]

3x3 blur filter

from LODEV.org

Use normalizing factor1/5 = 0.2

1 5


A Blur Filter (More Noticeable Blur)

[ 0 0 1 0 0 ][ 0 1 1 1 0 ] [ 1 1 1 1 1 ] [ 0 1 1 1 0 ][ 0 0 1 0 0 ] 5x5 blur filter

from LODEV.org

113



A 45 Degree Motion Blur Filter

[ 1 0 0 0 0 0 0 0 0 ][ 0 1 0 0 0 0 0 0 0 ] [ 0 0 1 0 0 0 0 0 0 ] [ 0 0 0 1 0 0 0 0 0 ] [ 0 0 0 0 1 0 0 0 0 ] [ 0 0 0 0 0 1 0 0 0 ][ 0 0 0 0 0 0 1 0 0][ 0 0 0 0 0 0 0 1 0 ][ 0 0 0 0 0 0 0 0 1 ] 9x9 motion blur filter

from LODEV.org


1 9


A Horizontal Edge Finding Filter

[ 0 0 0 0 0 ][ 0 0 0 0 0 ] [-1 -1 2 0 0 ] [ 0 0 0 0 0 ][ 0 0 0 0 0 ] 5x5 horizontal edge finding filter

from LODEV.org

dark since weights sum to 0

deliberately non-symmetric just to see


A Vertical Edge Finding Filter

[ 0 0 -1 0 0 ][ 0 0 -1 0 0 ] [ 0 0 4 0 0 ] [ 0 0 -1 0 0 ][ 0 0 -1 0 0 ] 5x5 vertical edge finding filter

from LODEV.org



A 45 Degree Edge Finding Filter

[-1 0 0 0 0 ][ 0 -2 0 0 0 ] [ 0 0 6 0 0 ] [ 0 0 0 -2 0 ][ 0 0 0 0 -1 ] 5x5 45 degree edge finding filter

from LODEV.org



An Edge Detection Filter

[-1 -1 -1][-1 8 -1][-1 -1 -1]

3x3 edge detection filter

from LODEV.org



A Sharpening Filter

[-1 -1 -1][-1 9 -1][-1 -1 -1]

3x3 sharpening filter

from LODEV.org

note that sum is 1


A More Subtle Sharpening Filter

[-1 -1 -1 -1 -1][-1 2 2 2 -1] [-1 2 8 2 -1] [-1 2 2 2 -1][-1 -1 -1 -1 -1] 5x5 subtle shapening filter

from LODEV.org

1 8

Use normalizing factor 1/8 = 0.125


An Excessive Sharpening Filter

[1 1 1][1 -7 1][1 1 1] 3x3 excessive sharpening filter

from LODEV.org

note that sum is 1


A 45 Degree Embossing Filter

[-1 -1 0][-1 0 1][ 0 1 1] 3x3 45 degree embossing filter

from LODEV.org

0.5 +


A 45 Degree Embossing GRAY SCALED Filter

[-1 -1 0][-1 0 1][0 1 1] 3x3 45 degree embossing filter

from LODEV.org

NO CHANGE IN FILTER BUT MAKE GREEN AND BLUE = RED 0.5 +


A More Exaggerated Emboss Filter

[-1 -1 -1 -1 0][-1 -1 -1 0 1] [-1 -1 0 1 1] [-1 0 1 1 1][ 0 1 1 1 1] 5x5 exaggerated emboss filter

from LODEV.org

0.5 +


A Mean Filter (Average or blur removes PEPPER)

[ 1 1 1 ][ 1 1 1 ][ 1 1 1 ] 3x3 mean filter

removes PEPPER by bluring

from LODEV.org

1 9

Use normalizing factor 1/9 = 0.111

Also called a BOX FILTER


A Median Filter (Uses Middle in Sorted Result)

[ 1 1 1 ][ 1 1 1 ][ 1 1 1 ]

Slightly better lookingde-PEPPERING and

blurring (I can’t see it)

from LODEV.org

the middle value after x-sorting and y-sorting

1 9


A Median Filter

3x3 5x5

9x9 15x15


Gaussian Filters

• Based on the gaussian distribution


180

A Crude Approximation of A Gaussian Filter


Another One

1464141624164624362464162416414641

2561

Source:Stephen Chenney University of Wisconsin


A More Exact Gaussian Filter For = 0.84

0.00000067 0.00002292 0.00019117 0.00038771 0.00019117 0.00002292 0.00000067

0.00002292 0.00078633 0.00655965 0.01330373 0.00655965 0.00078633 0.00002292

0.00019117 0.00655965 0.05472157 0.11098164 0.05472157 0.00655965 0.00019117

0.00038771 0.01330373 0.11098164 0.22508352 0.11098164 0.01330373 0.00038771

0.00019117 0.00655965 0.05472157 0.11098164 0.05472157 0.00655965 0.00019117

0.00002292 0.00078633 0.00655965 0.01330373 0.00655965 0.00078633 0.00002292

0.00000067 0.00002292 0.00019117 0.00038771 0.00019117 0.00002292 0.00000067


Gaussian Filter Uses

• Noise reduction blur...


• Provides random sample points where each point is at least distance r apart...

Poisson Filter (Randomized Points)


• Provide image size nxn, the minimum distance r between samples (e.g., r = 1.8 pixels), and the maximum number of attempts k per sample (e.g., k = 30).

• Initialize a 2D nxn grid with -1, a list of samples initially empty, and a stack of unprocessed indices.

• Randomly choose a sample x0, add x0 to samples, and 0 to indices.

Algorithm To Build Random 2D Samples

continued on next slide


While indices is not empty

Remove i from indices.for (j = 0; j < k; j++) {

p = generate random point between radius r and 2r around xi.if (p is further than r from each point in

samples) { Add p to samples and its index to indices}

}

Algorithm To Build Random 2D Samples

Fast Poisson Disk Sampling in Arbitrary Dimensions, Bridson, R., ACM SIGGRAPH 2007 Sketches Program

Wilf: There

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float3 poissonSample (sampler texture, float2 uv, float2 pixelSize, float discRadius) {

float2 offsets = {float2 (...), float2 (...), ...};float average = tex2D (texture, uv);for (int tap = 0; tap < 12; tap++) {

average += tex2D (texture, uv + offsets [tap] * (discRadius *

pixelSize);}return average / 13.0;

}

Can Find Prebuilt Poisson Filters on Internet

Heat and Haze Post-Processing Effects, Oat and Tatarchuk, Game Programming Gems 4, 2004

next slide


float2 offsets = {float2 (-0.326212, -0.40581), float2 (-0.840144, -0.07358), float2 (-0.695914, 0.457137), float2 (-0.203345, 0.620716), float2 (0.96234, -0.194983), float2 (0.473434, -0.480026), float2 (0.519456, 0.767022), float2 (0.185461, -0.893124), float2 (0.507431, 0.064425), float2 (0.89642, 0.412458), float2 (-0.32194, -0.932615), float2 (-0.791559, -0.59771),

};

Rest of poissonSample Shader Function


• Relates the filter capability to what happens in the frequency domain (fourier transforms)

• Low-pass filter lets low frequencies through which eliminates speckles and sharp discontinuities.

• High-pass filter lets high frequencies through, an edge detector.

Engineering Terminology



Box Filter

• Box filters by averaging neighbors (so it smooths)

• In frequency domain, keeps low frequencies and attenuates high frequencies (so it’s a low-pass filter)

111111111

91

Spatial domain: box frequency domain: sinc


Bartlett Filter

• Triangle shaped filter in spatial domain (attenuates high frequencies less than a gaussian filter).

• In frequency domain, product of two box filters (so attenuates high frequencies more than a box).

1232124642369632464212321

811

spatial domain: triangle frequency domain: sinc2



• A filter is a matrix of weights centered on a specific pixel in an image and used to produce some sort of weighted average.

• A host of different effects result from weighting the filters differently...

Conclusion

95.4501

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normalized weights

matrix of weights

filterswilf lalonde

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object orinted programming

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corresponding neighbor