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{{redirect|TF Algorithm}}
[[File:Street in Sendai.png|thumb|400px|The TF algorithm produces the background image from a video of a street with many pedestrians crossing.]]
The '''
By assuming that the background image is shown in the majority of the video, the algorithm is able to generate a good background image of a video in <math>O(R)</math>-time using only a small number of [[binary operations]] and Boolean Bit operations, which require a small amount of memory and has built-in operators found in many programming languages such as [[C]], [[C++]], and [[Java]].<ref name="TF">{{cite journal | last1 = Teknomo | first1 = Kardi | last2 = Fernandez | first2 = Proceso| title = Background Image Generation Using Boolean Operations | url =https://arxiv.org/abs/1510.00889}}</ref><ref name="PCTF">{{cite journal | last1 = Abu | first1 = Patricia Angela | last2 = Fernandez | first2 = Proceso| title = Performance Comparison of the Teknomo-Fernandez Algorithm on the RGB and HSV Colour Spaces | url =https://www.semanticscholar.org/paper/Performance-comparison-of-the-Teknomo-Fernandez-al-Abu-Fernandez/c45c7e300e2bbc800f269ddfe22596a8fe7b301f}}</ref><ref name="ITF" />
==History==
[[File:Sample Colored Result.png|thumb|500px|The TF
People tracking from videos usually involves some form of [[background subtraction]] to segment foreground from background. Once foreground images are extracted, then desired algorithms (such as those for [[motion tracking]], [[object tracking]], and [[facial recognition]]) may be executed using these images.<ref name="TF" /><ref name="ITF">{{cite thesis |last=Abu|first=Patricia Angela|date=March 2015|title=Improving the
However, [[background subtraction]] requires that the background image is already available<ref name="EHSV" /> and unfortunately, this is not always the case. Traditionally, the background image is searched for manually or automatically from the video images when there are no objects. More recently, automatic background generation through [[object detection]], [[medial filtering]], [[medoid filtering]], [[approximated median filtering]], [[linear predictive filter]], [[non-parametric model]], [[Kalman filter]], and [[adaptive smoothening]] have been suggested; however, most of these methods have high computational complexity and are resource-intensive. <ref name="TF" /><ref name="RTTF">{{cite conference |url=https://www.researchgate.net/publication/298791390_Modifying_the_Teknomo-Fernandez_Algorithm_for_Accurate_Real-Time_Background_Subtraction |title=Modifying the
The
==Assumptions==
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Generally, however, the algorithm will certainly work whenever the following single important assumption holds: <blockquote>For each pixel position, the majority of the pixel values in the entire video contain the pixel value of the actual background image (at that position).<ref name="TF" /></blockquote>As long as each part of the background is shown in the majority of the video, the entire background image needs not to appear in any of its frames. The algorithm is expected to work accurately.<ref name="TF" />
==Background
===Equations===
# For three frames of image sequence <math>x_1</math>, <math>x_2</math>, and <math>x_3</math>, the background image <math>B</math> is obtained using <math>B = x_3(x_1\oplus x_2)+x_1x_2 </math><ref name="TF" />
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# For three images, the background image <math>B</math> can be taken as the value <math>\bar{x_1}x_2x_3+x_1\bar{x_2}x_3+x_1x_2\bar{x_3}+x_1x_2x_3</math> <ref name="TF" />
===Background
At the first level, three frames are selected at random from the image sequence to produce a background image by combining them using the first equation. This yields a better background image at the second level. The procedure is repeated until desired level <math>L</math>.<ref name="TF" />
==Theoretical
At level <math>l</math>, the probability <math>
The table below gives the computed probability values across several levels using some specific initial probabilities. It can be observed that even if the modal bit at the considered position is at a low 60% of the frames, the probability of accurate modal bit determination is already more than 99% at 6 levels.<ref name="TF" />
[[File:Probability Table.png|inline|center|400px|This table gives the computed probability values across several levels using some specific initial probabilities. It can be observed that even if the modal bit at the considered position is at a low 60% of the frames, the probability of accurate modal bit determination is already more than 99% at
==Space
The space requirement of the
==Time
The entire algorithm runs in <math>O(R</math>)-time, only depending on the resolution of the image. Computing the modal bit for each bit can be done in <math>O(1)</math>-time while the computation of the resulting image from the three given images can be done in <math>O(R)</math>-time. The number of the images to be processed in <math>L</math> levels is <math>O(3^L)</math>. However, since <math>L \le 6</math>, then this is actually <math>O(1)</math>, thus the algorithm runs in <math>O(R)</math>.<ref name="TF" />
==Variants==
A variant of the
==Applications==
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* [[Motion capture]]
* [[Human-computer interaction]]
* Content
* Traffic monitoring
* Real-time [[
==References==
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* {{cite thesis |last=Chu |first=Varian Sherwin B. |title=Background image reconstruction using random frame sampling and logical bit operations |date=2013 |publisher=Ateneo de Manila University }}
*{{cite thesis |last=Abu |first=Patricia Angela R. |title=Improving the Teknomo-Fernandez Background Image Modeling Algorithm for Foreground Segmentation |date=2015 |publisher=Ateneo de Manila University }}
==External links==
*[https://arxiv.org/abs/1510.00889 Background Image Generation Using Boolean Operations]
* [https://www.researchgate.net/publication/273391116_A_Monte-Carlo-based_Algorithm_for_Background_Generation A Monte-Carlo-based Algorithm for Background Generation]
[[Category:Mathematical examples]]
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