**3.1 Wallflower dataset** <sup>4</sup>

We have chosen this particular dataset provided by Toyama et al. Toyama et al. (1999) because of how frequent its use is in this field. This frequency is due to its faithful representation of real-life situations typical of scenes susceptible to video surveillance. Moreover, it consists of seven video sequences, with each sequence presenting one of the difficulties a practical task is

<sup>4</sup> http://research.microsoft.com/en-us/um/people/jckrumm/wallflower/ testimages.htm


 

%"&5! " ! "&1+ .& !4./.,4

&%&0.," -%.4"1

\*"%"!"&+,!-./0 "0 "-&. ." "&,"%%"12,&0.3

total pixels classified as background by the method:

cumulative score by method (resp. sequence).

%.4"18'\*!4 .. )

for Background Subtraction: Systematic Evaluation and Comparative Analysis

"1"-6,. %"#1 7!.

Fig. 1. Performance on the Wallflower dataset. The left (resp. right) figure concern the

*DR* <sup>=</sup> *TP*

*Precision* <sup>=</sup> *TP*

A good performance is obtained when the detection rate is high without altering the precision.

*<sup>F</sup>* <sup>=</sup> <sup>2</sup> <sup>×</sup> *DR* <sup>×</sup> *Precision*

Table 2 shows the results obtained by the different algorithms on each sequence. For each sequence, the first column shows the original image and the corresponding ground truth.

The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate (DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask. Fig. 1 shows two cumulative histograms of F-measure: The left (resp. right) figure concern the cumulative score by method (resp. sequence). PCP gives the best result followed by RSL, IALM, TFOCS, and Bayesian RPCA. This ranking has to be taken with prrecaution because a poor performance on one sequence influences the overall F-measure and then modifies the rank for just one sequence. For example, the Bayesian obtained a bad score because of the following assumption: the background has necessarily a bigger area than the foreground. It happen in the sequences Camouflage and Light Switch. In the first case, the person hides more than half of the screen space. In the second case, when the person switch on the light all the pixels are affected and the algorithm exchanges

We also computed the F-measure used in (Maddalena & Petrosino (2010)) as follows:

the foreground and background. PCP seems to be robust for all critical situations.

Precision gives the percentage of corrected pixels classified as background as compared at the


 

\$/0 1(

*TP* <sup>+</sup> *FN* (15)

*TP* <sup>+</sup> *FP* (16)

*DR* <sup>+</sup> *Precision* (17)

!"&.

\*'
,

22 30

32(

!"#

\$%
&

045 5
&

'
!






(!
%)
\*+,

<sup>229</sup> Robust Principal Component Analysis


 -


 -







!"#
\$%& '()


 -


 -


likely to encounter (i.e illumination changes, dynamic backgrounds). The size of the images is 160 × 120 pixels. A brief description of the Wallflower image sequences can be made as follows:


For each sequence, the ground truth is provided for one image when the algorithm has to show its robustness to a specific change in the scene. Thus, the performance is evaluated against hand-segmented ground truth. Four terms are used in the evaluation:



Table 1. Measure for performance evalutation

Table 1 illustrates how to compute these different terms. Then, we computed the following metrics: the detection rate, the precision and the F-measure. Detection rate gives the percentage of corrected pixels classified as background when compared with the total number of background pixels in the ground truth:

6 Will-be-set-by-IN-TECH

likely to encounter (i.e illumination changes, dynamic backgrounds). The size of the images is 160 × 120 pixels. A brief description of the Wallflower image sequences can be made as

• **Moved Object (MO)**: A person enters into a room, makes a phone call, and leaves. The phone and the chair are left in a different position. This video contains 1747 images. • **Time of Day (TOD)**: The light in a room gradually changes from dark to bright. Then, a

• **Light Switch (LS)**: A room scene begins with the lights on. Then a person enters the room and turns off the lights for a long period. Later, a person walks in the room and switches

• **Waving Trees (WT)**: A tree is swaying and a person walks in front of the tree. This video

• **Camouflage (C)**: A person walks in front of a monitor, which has rolling interference bars on the screen. The bars include similar color to the person's clothing. This video contains

• **Bootstrapping (B)**: The image sequence shows a busy cafeteria and each frame contains

• **Foreground Aperture (FA)**: A person with uniformly colored shirt wakes up and begins to

For each sequence, the ground truth is provided for one image when the algorithm has to show its robustness to a specific change in the scene. Thus, the performance is evaluated

• True Positive (TP) is the number of foreground pixels that are correctly marked as

• False Positive (FP) is the number of background pixels that are wrongly marked as

• True Negative (TN) is the number of background pixels that are correctly marked as

• False Negative (FN) is the number of foreground pixels that are wrongly marked as

Ground Truth Foreground TP FN

Table 1 illustrates how to compute these different terms. Then, we computed the following metrics: the detection rate, the precision and the F-measure. Detection rate gives the percentage of corrected pixels classified as background when compared with the total number

Background FP TN

Algorithm Foreground Background

against hand-segmented ground truth. Four terms are used in the evaluation:

person enters the room and sits down. This video contains 5890 images.

on the light. This video contains 2715 images.

people. This video contains 3055 images.

Table 1. Measure for performance evalutation

of background pixels in the ground truth:

move slowly. This video contains 2113 images.

contains 287 images.

353 images.

foreground.

foreground.

background.

background.

follows:

Fig. 1. Performance on the Wallflower dataset. The left (resp. right) figure concern the cumulative score by method (resp. sequence).

$$DR = \frac{TP}{TP + FN} \tag{15}$$

Precision gives the percentage of corrected pixels classified as background as compared at the total pixels classified as background by the method:

$$Precision = \frac{TP}{TP + FP} \tag{16}$$

A good performance is obtained when the detection rate is high without altering the precision. We also computed the F-measure used in (Maddalena & Petrosino (2010)) as follows:

$$F = \frac{2 \times DR \times Precision}{DR + Precision} \tag{17}$$

Table 2 shows the results obtained by the different algorithms on each sequence. For each sequence, the first column shows the original image and the corresponding ground truth.

The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate (DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask. Fig. 1 shows two cumulative histograms of F-measure: The left (resp. right) figure concern the cumulative score by method (resp. sequence). PCP gives the best result followed by RSL, IALM, TFOCS, and Bayesian RPCA. This ranking has to be taken with prrecaution because a poor performance on one sequence influences the overall F-measure and then modifies the rank for just one sequence. For example, the Bayesian obtained a bad score because of the following assumption: the background has necessarily a bigger area than the foreground. It happen in the sequences Camouflage and Light Switch. In the first case, the person hides more than half of the screen space. In the second case, when the person switch on the light all the pixels are affected and the algorithm exchanges the foreground and background. PCP seems to be robust for all critical situations.

**Original / GT** TFOCS IALM PCP RSL SUB Bayesian

<sup>231</sup> Robust Principal Component Analysis

92.5 / 89.3 **90**.**92**

95.3 / 94.6 **94**.**97**

96.0 / 96.1 **96**.**10**

Table 3. **Shah dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian*

**Original / GT** TFOCS IALM PCP RSL SUB Bayesian

58.8 / 81.0 **68**.**20**

87.3 / 83.5 **87**.**42**

Table 4. **Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian*

*RPCA. From top to bottom: level crossing (309), level crossing (395), level crossing (462)*

90.5 / 90.7 **90**.**64**

95.6 / 94.9 **95**.**28**

56.4 / 86.6 **68**.**37**

60.3 / 81.1 **69**.**19**

92.3 / 83.7 **87**.**82**

84.9 / 90.6 **87**.**67**

86.1 / 94.1 **89**.**99**

75.0 / 66.7 **70**.**67**

57.5 / 61.6 **59**.**52**

82.6 / 81.4 **82**.**03**

86.9 / 95.2 **90**.**91**

89.1 / 95.2 **92**.**10**

86.2 / 93.7 **89**.**82**

51.9 / 67.5 **58**.**73**

79.8 / 80.3 **80**.**06**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

92.3 / 87.2 **89**.**71**

65.2 / 96.0 **77**.**68**

51.7 / 97.5 **67**.**62**

67.6 / 66.0 **66**.**82**

86.7 / 79.9 **83**.**22**

*RPCA. From top to bottom: campus (1650), campus (1812)*

94.4 / 92.3 **93**.**40**

for Background Subtraction: Systematic Evaluation and Comparative Analysis

63.4 / 96.7 **76**.**60**

45.4 / 97.1 **61**.**93**

70.2 / 65.8 **68**.**00**

87.4 / 85.3 **86**.**37**

Table 2. **Wallflower dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: camouflage (251), foreground aperture (489), light switch (1865), moved object (985), time of day (1850), waving trees (247).*

230 Principal Component Analysis Robust Principal Component Analysis for Background Subtraction: Systematic Evaluation and Comparative Analysis <sup>9</sup> <sup>231</sup> Robust Principal Component Analysis for Background Subtraction: Systematic Evaluation and Comparative Analysis

8 Will-be-set-by-IN-TECH

**Original / GT** TFOCS IALM PCP RSL SUB Bayesian

89.8 / 65.0 **75**.**43**

73.1 / 71.0 **72**.**07**

74.4 / 67.6 **70**.**86**

85.1 / 77.5 **81**.**18**

81.5 / 91.9 **86**.**40**

Table 2. **Wallflower dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: camouflage (251), foreground aperture (489), light switch*

93.7 / 89.9 **91**.**78**

69.2 / 80.3 **74**.**37**

100 / 16.5 **28**.**36**

75.7 / 75.7 **75**.**73**

89.4 / 89.9 **89**.**69**

... / ... ... ... / ... ... ... / ... ... ... / ... ... ... / ... ... ... / ... ...

95.6 / 96.2 **95**.**92**

51.8 / 81.8 **63**.**46**

100 / 16.5 **28**.**36**

60.7 / 88.6 **72**.**06**

75.5 / 92.5 **83**.**21**

06.0 / 06.8 **06**.**41**

53.6 / 81.2 **64**.**58**

37.3 / 08.0 **13**.**19**

37.8 / 94.1 **53**.**94**

81.3 / 89.1 **85**.**09**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

14.1 / 47.7 **21**.**77**

38.8 / 74.3 **51**.**03**

70.9 / 79.0 **74**.**73**

87.1 / 63.8 **73**.**75**

34.8 / 80.1 **48**.**58**

13.9 / 53.4 **22**.**02**

51.9 / 76.7 **61**.**92**

66.0 / 81.9 **73**.**16**

79.9 / 81.1 **80**.**56**

27.6 / 78.6 **40**.**88**

*(1865), moved object (985), time of day (1850), waving trees (247).*

Table 3. **Shah dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: level crossing (309), level crossing (395), level crossing (462)*

Table 4. **Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: campus (1650), campus (1812)*

• **Curtain**: A person presents a course in a meeting room with a moving curtain. This

<sup>233</sup> Robust Principal Component Analysis

• **Escalator**: This image sequence shows a busy hall where an escalator is used by people.

• **Airport**: This image sequence shows a busy hall of an airport and each frame contains

• **Shopping Mall**: This image sequence shows a busy shopping center and each frame

• **Restaurant**: This sequence comes from the wallflower dataset and shows a busy cafeteria.

The sequences Campus, Water Surface and Curtain present dynamic backgrounds whereas the sequences Restaurant, Airport, Shopping Mall show bootstrapping issues. For each sequence, the ground truth is provided for twenty images when algorithms have to show their robustness. Table 4 shows the results obtained by the different algorithms on the sequence campus. Table 5 presents the results on the dynamic background on the sequences Water Surface, Curtain, Escalator, whereas table 6 presents the result on bootstrapping issues on the sequences Airport, Shopping mall, Restaurant. For each table, the first column shows the original image and the corresponding ground truth. The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate

(DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask.

Fig. 3 and 4 shows the two cumulative histograms of F-measure respectively for dynamic background and bootstrapping issues. In each case, Bayesian RPCA gives best results




,!- ./0

 


)\*+

#
&'(

%

11 2/

215(

 

34 4
\*







!"
#\$%

Regarding the code, we have used the following implementation in *MATLAB*: RSL provided by F. De La Torre7, PCP provided by C. Qiu8, IALM provided by M. Chen and A. Ganesh9,


 -


 -


sequence contains 23893 images.

This video contains 3055 images.

followed by PCP, TFOCS, IALM and RSL.


 

%+&-

**3.4 Implementation and time issues**

%+& 5. & #, 5. & #,

&%&0.," -%.4"1

\*"%"!"&+,!-./0 "0 "-&. ." "&,"%%"12,&0.3

%.4"17'\*!4 .. )

& . & . 6,0." 

6,0." -

Fig. 3. Performance on the Li dataset (Dynamic backgrounds).






!"#
\$%& '()


 -


 -


This sequence contains 4787 images.

people. This sequence contains 3584 images.

contains people. This sequence contains 1286 images.

for Background Subtraction: Systematic Evaluation and Comparative Analysis

Fig. 2. Performance on the Shah dataset. The left (resp. right) figure concern the cumulative score by method (resp. sequence).

#### **3.2 Shah's dataset** <sup>5</sup>

This sequence involved a camera mounted on a tall tripod and comes from Sheikh & Shah (2005). It contains 500 images and the corresponding GT. The wind caused the tripod to sway back and forth causing nominal motion in the scene. Table 3 shows the results obtained by the different algorithms on three images of the sequence: Frame 309 that contains a walking person, frame 395 when a car arrived the scene and frame 462 when the same car left the scene. For each frame, the first column shows the original image and the corresponding ground truth. The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate (DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask. Fig. 2 shows two cumulative histograms of F-measure: as in previous performance evaluation. PCP gives the best result followed by Bayesian RPCA, RSL, TFOCS and IALM. We can notice that the Bayesian give better performance on this dataset because none of moving object are bigger than the background area.

#### **3.3 Li's dataset** <sup>6</sup>

This dataset provided by Li et al. (2004) consists of nine video sequences, which each sequence presenting dynamic backgrounds or illumination changes. The size of the images is 176\*144 pixels. Among this dataset, we have chosen seven sequences that are the following ones:


<sup>5</sup> http://www.cs.cmu.edu/~yaser/new\_backgroundsubtraction.htm

<sup>6</sup> http://perception.i2r.a-star.edu.sg/bk\_model/bk\_index.html

10 Will-be-set-by-IN-TECH


 

%-

score by method (resp. sequence).

**3.2 Shah's dataset** <sup>5</sup>

area.

**3.3 Li's dataset** <sup>6</sup>

contains 1439 images.

contains 633 images.

\*"%"!"&+,!-./0 "0 "-&. ." "&,"%%"12,&0.3

%.4"15'\*!4 .. )

&%&0.," -%.4"1

% %

Fig. 2. Performance on the Shah dataset. The left (resp. right) figure concern the cumulative

This sequence involved a camera mounted on a tall tripod and comes from Sheikh & Shah (2005). It contains 500 images and the corresponding GT. The wind caused the tripod to sway back and forth causing nominal motion in the scene. Table 3 shows the results obtained by the different algorithms on three images of the sequence: Frame 309 that contains a walking person, frame 395 when a car arrived the scene and frame 462 when the same car left the scene. For each frame, the first column shows the original image and the corresponding ground truth. The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate (DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask. Fig. 2 shows two cumulative histograms of F-measure: as in previous performance evaluation. PCP gives the best result followed by Bayesian RPCA, RSL, TFOCS and IALM. We can notice that the Bayesian give better performance on this dataset because none of moving object are bigger than the background

This dataset provided by Li et al. (2004) consists of nine video sequences, which each sequence presenting dynamic backgrounds or illumination changes. The size of the images is 176\*144 pixels. Among this dataset, we have chosen seven sequences that are the following ones:

• **Campus**: Persons walk and vehicles pass on a road in front of waving trees. This sequence

• **Water Surface**: A person arrives in front of the sea. There are many waves. This sequence

<sup>5</sup> http://www.cs.cmu.edu/~yaser/new\_backgroundsubtraction.htm <sup>6</sup> http://perception.i2r.a-star.edu.sg/bk\_model/bk\_index.html


() \*+,



.-


!"#
\$%&'#

 

/0 0%"'






 


 -


 -







!"#
\$%& '()


 -

 -




The sequences Campus, Water Surface and Curtain present dynamic backgrounds whereas the sequences Restaurant, Airport, Shopping Mall show bootstrapping issues. For each sequence, the ground truth is provided for twenty images when algorithms have to show their robustness. Table 4 shows the results obtained by the different algorithms on the sequence campus. Table 5 presents the results on the dynamic background on the sequences Water Surface, Curtain, Escalator, whereas table 6 presents the result on bootstrapping issues on the sequences Airport, Shopping mall, Restaurant. For each table, the first column shows the original image and the corresponding ground truth. The second part presents the sparse matrix in the first row and the optimal foreground mask in the second row. The detection rate (DR), Precision (Prec) and F-measure (F) are indicated below each foreground mask.

Fig. 3 and 4 shows the two cumulative histograms of F-measure respectively for dynamic background and bootstrapping issues. In each case, Bayesian RPCA gives best results followed by PCP, TFOCS, IALM and RSL.

Fig. 3. Performance on the Li dataset (Dynamic backgrounds).

#### **3.4 Implementation and time issues**

Regarding the code, we have used the following implementation in *MATLAB*: RSL provided by F. De La Torre7, PCP provided by C. Qiu8, IALM provided by M. Chen and A. Ganesh9,

**Original / GT** TFOCS IALM PCP RSL SUB Bayesian

<sup>235</sup> Robust Principal Component Analysis

70.7 / 86.2 **77**.**74**

64.7 / 70.8 **67**.**63**

80.6 / 84.6 **82**.**60**

69.3 / 84.1 **76**.**07**

80.2 / 90.9 **85**.**25**

67.2 / 83.3 **74**.**4**

Table 6. **(Perception) Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: Airport (2926), Airport (3409), shopping mall (1862),*

66.8 / 63.7 **65**.**26**

67.7 / 52.3 **59**.**07**

55.3 / 88.7 **68**.**17**

65.6 / 53.0 **58**.**64**

71.1 / 61.0 **65**.**70**

69.9 / 68.8 **69**.**38**

76.1 / 76.4 **76**.**25**

63.9 / 59.0 **61**.**38**

78.7 / 80.3 **79**.**55**

66.7 / 86.4 **75**.**31**

68.3 / 98.2 **80**.**58**

67.6 / 76.0 **71**.**58**

71.9 / 86.3 **78**.**48**

64.9 / 68.7 **66**.**83**

80.1 / 81.7 **80**.**94**

68.5 / 83.0 **75**.**13**

82.1 / 88.5 **85**.**23**

83.4 / 65.6 **73**.**52**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

68.1 / 80.3 **73**.**75**

66.5 / 75.1 **70**.**60**

57.9 / 91.3 **70**.**94**

49.1 / 73.6 **58**.**96**

83.4 / 80.8 **82**.**21**

74.2 / 70.1 **72**.**17**

*shopping mall (1980) Restaurant (1842), Restaurant (2832).*

69.3 / 79.9 **74**.**26**

for Background Subtraction: Systematic Evaluation and Comparative Analysis

66.3 / 76.2 **70**.**93**

46.9 / 94.6 **62**.**74**

36.2 / 85.5 **50**.**89**

77.4 / 91.8 **83**.**88**

72.0 / 75.5 **73**.**73**

Table 5. **Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: water surface (1523), water surface (1594), curtain (22772), curtain (23257), escalator (2978), escalator (3260)*

12 Will-be-set-by-IN-TECH

**Original / GT** TFOCS IALM PCP RSL SUB Bayesian

86.8 / 93.0 **89**.**84**

42.4 / 47.9 **45**.**00**

86.2 / 96.2 **90**.**96**

81.1 / 84.8 **82**.**95**

80.3 / 87.5 **83**.**76**

69.6 / 69.0 **69**.**35**

Table 5. **Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: water surface (1523), water surface (1594), curtain (22772), curtain*

51.1 / 61.4 **55**.**84**

26.6 / 48.5 **34**.**42**

85.6 / 97.2 **91**.**04**

78.2 / 64.5 **70**.**73**

86.5 / 53.7 **66**.**31**

73.8 / 58.0 **64**.**96**

85.8 / 94.6 **90**.**02**

78.6 / 99.1 **87**.**68**

85.4 / 97.5 **91**.**08**

86.3 / 74.9 **80**.**22**

74.6 / 90.1 **81**.**64**

36.2 / 35.5 **35**.**91**

85.4 / 93.2 **89**.**15**

88.6 / 94.9 **91**.**66**

89.5 / 91.1 **90**.**34**

83.5 / 90.9 **87**.**09**

72.3 / 86.4 **78**.**77**

52.1 / 64.1 **57**.**54**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

*DR* / *Prec* **F**

72.9 / 89.5 **80**.**38**

70.2 / 24.5 **36**.**39**

90.0 / 89.7 **89**.**89**

76.2 / 79.7 **77**.**98**

75.1 / 84.2 **79**.**45**

65.8 / 70.4 **68**.**09**

*(23257), escalator (2978), escalator (3260)*

72.5 / 96.3 **82**.**77**

34.9 / 28.4 **31**.**34**

90.3 / 95.8 **93**.**01**

69.8 / 84.8 **76**.**59**

80.7 / 80.3 **80**.**54**

72.1 / 68.7 **70**.**40**

Table 6. **(Perception) Li dataset:** *From left to right: Ground Truth, TFOCS, IALM, PCP, RSL, SUB, Bayesian RPCA. From top to bottom: Airport (2926), Airport (3409), shopping mall (1862), shopping mall (1980) Restaurant (1842), Restaurant (2832).*

**5. References**

*Symposium, IPDPS 2011* .

*ComputerVision* pp. 57–92.

1(3): 219–237.

*Information Systems* .

*on Computer Science* 2(3): 223–234.

*Scientific Publishing* 4(2): 181–189.

*Transaction on Image Processing* .

*and Applications, NCA* pp. 1–8.

*Psychometrika* 1: 211–218.

Anderson, M., Ballard, G., Demme, J. & Keutzer, K. (2011). Communication-avoiding

<sup>237</sup> Robust Principal Component Analysis

for Background Subtraction: Systematic Evaluation and Comparative Analysis

Becker, S., Candes, E. & Grant, M. (2011). Tfocs: Flexible first-order methods for rank

Black, M. & Rangarajan, A. (1996). On the unification of line processes, outlier rejection,

Bouwmans, T. (2009). Subspace learning for background modeling: A survey, *Recent Patents*

Bouwmans, T., Baf, F. E. & Vachon, B. (2008). Background modeling using mixture of

Bouwmans, T., Baf, F. E. & Vachon, B. (2010). Statistical background modeling for foreground

Bucak, S., Gunsel, B. & Gursoy, O. (2007). Incremental non-negative matrix factorization

Candes, E., Li, X., Ma, Y. & Wright, J. (2009). Robust principal component analysis?, *Preprint* . Ding, X., He, L. & Carin, L. (2011). Bayesian robust principal component analysis, *IEEE*

Eckart, C. & Young, G. (1936). The approximation of one matrix by another of lower rank,

Li, L., Huang, W., Gu, I. & Tian, Q. (2004). Statistical modeling of complex backgrounds for foreground object detection, *IEEE Transaction on Image Processing* pp. 1459–1472. Li, X., Hu, W., Zhang, Z. & Zhang, X. (2008). Robust foreground segmentation based on two effective background models, *Multimedia Information Retrieval (MIR)* pp. 223–228. Lin, Z., Chen, M., Wu, L. & Ma, Y. (2009). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices, *UIUC Technical Report* . Maddalena, L. & Petrosino, A. (2010). A fuzzy spatial coherence-based approach to

Mu, Y., Dong, J., Yuan, X. & Yan, S. (2011). Accelerated low-rank visual recovery by random

Oliver, N., Rosario, B. & Pentland, A. (1999). A bayesian computer vision system for modeling human interactions, *International Conference on Vision Systems, ICVS 1998* . Sheikh, Y. & Shah, M. (2005). Bayesian modeling of dynamic scenes for object detection, *IEEE Transactions on Pattern Analysis and Machine Intelligence* 27: 1778–1792. Torre, F. D. L. & Black, M. (2001). Robust principal component analysis for computer vision,

Torre, F. D. L. & Black, M. (2003). A framework for robust subspace learning, *International*

Toyama, K., Krumm, J., Brumitt, B. & Meyers, B. (1999). Wallflower: Principles and practice

of background maintenance, *International Conference on Computer Vision* pp. 255–261.

projection, *International Conference on Computer Vision, CVPR 2011* .

*International Conference on Computer Vision, ICCV 2001* .

*Journal on Computer Vision* pp. 117–142.

qr decomposition for gpu, *IEEE International Parallel and Distributed Processing*

minimization, *Low-rank Matrix Optimization Symposium, SIAM Conf. on Optimization* .

and robust statistics with applications in early vision., *International Journal of*

gaussians for foreground detection - a survey, *Recent Patents on Computer Science*

detection: A survey, *Handbook of Pattern Recognition and Computer Vision, World*

for dynamic background modeling, *International Workshop on Pattern Recognition in*

background foreground separation for moving object detection, *Neural Computing*

Fig. 4. Performance on the Li dataset (Bootstrap issues).

TFOCS provided by S. Becker10 and Bayesian provided by X. Ding11. Additionally, a 5 <sup>×</sup> <sup>5</sup> median filter is postprocessed in order to suppress peak noise. The thresholding value is automatically choose for maximize the F-measure.

For time issues, the current implementations are faraway to achieve real-time. Indeed, the computing of the backgrounds take few hours for a training sequence with 200 frames for each algorithm. This time can be reduced by *C/Cuda* implementation as suggested in (Mu et al. (2011);Anderson et al. (2011)).
