Jonathan Dinger

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Traffic footage example

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Important step in video analysis Background subtraction is often used

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Uses pixelwise computations Performance could be better Better segmentation = better traffic

detection

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Use interaction between neighboring pixels Keep objects segmented together Better segmentations

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G = (V,E) Two vertices in V

called source and sink ◦ s and t, respectively

Remaining vertices called M

Vertices in M connected to both s and t (T-links)

Vertices in M connected to neighboring vertices (N-links)

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T-links are uni-directional N-links are bi-directional Each edge has a weight

◦ also known as a capacity Each pixel has its own

vertex

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Path◦ List of vertices connected by edges

s-t cut◦ Removal of edges such that all vertices have a

path to either the source or the sink, but not both Flow

◦ Each edge has a capacity◦ That much flow can be pushed through each edge◦ Flow through a graph is the cumulative amount of

flow going from the source to the sink

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Minimum cut/maximum flow◦ A cut with the smallest cost (weight)◦ Maximum flow that can be pushed from source to

sink (capacity)◦ By max-flow min-cut theorem,

these are equal Graph cuts

◦ Minimum cut on a graph

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Method to find maximum flow Augmenting path

◦ Path where flow can be increased in all edges between vertices in path

Run search1.Find augmenting path from source to sink2.Add more flow to that path3.Loop back to 1.

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Step 1. Step 2.

Step 3. Step 4.

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Step 5. Step 6.

Step 7. Step 8.

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Two separate implementations◦ Our implementation◦ Kolmogorov’s implementation

http://www.cs.ucl.ac.uk/staff/V.Kolmogorov/software/maxflow-v3.01.src.tar.gz

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Downsampled images Cut performed over smaller area Then upsample, and perform cut over band Faster than graph cuts Less detailed than graph cuts

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Downsampling images loses information Use Laplacian pyramid to store lost data

Computes differencebetween image and imagegained by downsamplingand upsampling again

Add back some of the lost detail◦ Resegment in areas where detail was lost

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BGC faster, less accurate GC slightly slower, more accurate

Graph cut Banded graph cut

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3 or more labels α-β swap

◦ Loop through label pairs◦ Run graph cut on current pair of labels

If current label of a pixel is not one of the pair, do not use pixel in graph cut

◦ Graph cuts will swap some pixels with label α to label β and vice versa

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Graph cut on each image Background image computed per pixel

◦ N is the number of images, xi is the grayscale value of the current pixel, and μ is the average grayscale value over all image frames

N

iixN 1

1

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Find variance of each pixel

N, xi, and μ are as above σ2 is the variance Threshold the variance If variance is below the threshold, do not

include pixel in graph cut◦ Assume non-varying pixels are background pixels◦ Avoid divide-by-zero errors in weights

N

iixN 1

22 )(1

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Exponentials

x is the grayscale value of the current pixel◦ μ and σ2 are as above

β is a constant that forces the two functions to be equal at α standard deviations

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Absolute differences

x and μ are as above L is the maximum possible distance

between x and μ, so for grayscale images

K is a shift constant that forces f3 and f4 to be equal at α standard deviations

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Simple Grayscale Differences

x is the grayscale value of the pixel. Cm and Cn are two grayscale values used as a basis for segmentation

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Distance

◦ Euclidean distance between two neighboring pixels with coordinates (x1, y1) and (x2, y2)

Smoothing (similarity)

◦ x and y are the grayscale values of two neighboring pixels.

◦ is the maximum possible difference between the pixel values

◦ γ is a modifier that defines the amount of smoothing that takes place

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Image Graph cut Banded graph cut

Augmented BGC Our graph cut Multi-way cut

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Computing times for cut results in milliseconds (ms)

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Image Background Variance

Exponential cut Absolute differencecut

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ExponentialImage Absolute difference

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ExponentialImage Absolute difference

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ExponentialImage Background subtraction

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ExponentialImage Absolute difference

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ExponentialImage Absolute difference

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Absolute differenceImage Background subtraction

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Image Graph cut BackgroundSubtraction

Segmentation performance without smoothing comparable to background subtraction◦ Background subtraction is faster, easier

Smoothing model ◦ Segments larger pieces of vehicles into one

section◦ Vehicle segmentations more “solid”

Absolute difference T-link weights combined with smoothing N-link weights give best results

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Use multi-way cuts to add shadow segmentation

Extend to RGB

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