Sparse Representation based Anomaly Detection with Enhanced Local Dictionaries Sovan Biswas and R. Venkatesh Babu Video Analytics Lab, Indian Institute of Science, Bangalore, India Quantitative Results: ROC

Generally, anomaly (e) is defined as e ∝ dist(X , y ) ◮ In sparse framework, the anomaly is defined as e ∝ ky − Dαk2 where, dictionary is defined as D = f (X ) ◮ So, if data Xi is related to Xj by Ψji X˜j = Ψji Xi T T ˜ ˜ subject to Xj Xj = Xj Xj ◮ Then, Xi ≈ Di α ⇒ Ψji Xi ≈ Ψji Di α ˜jα ⇒ ˜ j = Ψji Di X˜j ≈ D D Dictionary Enhancement Ψji has the following properties det(Ψji ) = 1 or −1 −1 ◮ Ψij = Ψ ji ◮

As,

T Xj Xj T Xj Xj T Qj Λj Qj

T ˜ ˜ = Xj Xj T = (Ψji Xi )(Ψji Xi ) T T = Ψji Qi Λi Qi Ψji

Estimating Ψji ,

Qj = Ψji Qi −1 Ψji = Qi Qj subject to kΛi − Λj k2 ≤ ǫ ˜ j ] where D ˜ j = Ψji Di ◮ Enhanced dictionary Dj = [Dj , D ◮

The Proposed Approach Features for each dense space-time cubes ◮ Foreground pixel occupancy ◮ Histogram of optical flow (HOF) ◮ Flow magnitude ◮ Learn local dictionary for each region during ◮ Dictionary enhancement from neighborhood dictionaries ◮ l1-minimization solving to obtain sparse α 1 λ2 2 2 min ky − Dαk2 + λ1kαk1 + kαk2 α 2 2 ◮ Final anomaly is obtained as: e = w1 ∗ ky − Dαk2 + w2 ∗ ky − Dαk2 + w3 ∗ kWαk1 where, w3 ≤ w1 ≤ w2.

True Positve Rate (recall)

0.6 0.4 0.2

The Proposed Approach Sparse LSA MDT MPPCA Social Force

0.5 False Positve Rate

1

True Positve Rate (recall)

ROC (AUC: 85.95%, EER: 20.38%) 1 0.8 0.6 0.4 0.2 0 0

With Dictionary Enhancement Without Dictionary Enhancement

0.5 False Positve Rate

ROC (AUC: 85.95%, EER: 20.38%) 1 0.8 0.6 0.4 0.2 0 0

Ped1: Frame level anomaly

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0.8

0 0

Proposed Framework

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ROC (AUC: 85.85%, EER: 19.22%) 1

Ped1: Effect of enhancement

0.5 False Positve Rate

1

ROC (AUC: 50.63%, EER: 48.92%) 1 The Proposed Approach Sparse MDT MPPCA

0.8 0.6 0.4 0.2 0 0

1

The Proposed Approach MDT MPPCA Social Force

Ped2: Frame level anomaly True Positve Rate (recall)

Motion based anomaly detection using sparse representation over normal dictionary. ◮ Enhancing the local dictionaries based on the similarity of usual behavior with its spatial neighbors. ◮

True Positve Rate (recall)

Objective

0.5 False Positve Rate

1

Ped1: Pixel level anomaly

Quantitative Results: Detection Accuracy Approaches Ped1 (EER) Ped2 (EER) SF 31% 42% MPPCA 40% 30% SF-MPPCA 32% 36% MDT 25% 25% Sparse 19% LSA 16% Ours (No Enhancement) 19.53% 21.26% Ours (With Enhancement) 19.22% 20.38%

Frame Level Comparison Approaches

RD

Detection Speed AUC (frame per sec.) 17.9% 20.5% 44% 0.04 fps 46.1% 0.25 fps

SF 21% MPPCA 18% MDT 45% Sparse 46% Our (With 51.02% 50.63% Enhancement)

∼ 3 fps

Pixel Level Comparison on Ped1

Qualitative Results

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E-mail: [email protected], [email protected]

(a) Video Sequence Ped1 Test 001

(b) Video Sequence Ped1 Test 019

Conclusion Proposed enhancing the local dictionaries with the help of spatial neighbors to improve anomaly detection. Website: val.serc.iisc.ernet.in