Michelson - Revision history

imported>Projects221: /* Detector */

2013-03-21T06:13:23Z

Detector

← Older revision		Revision as of 06:13, 21 March 2013
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	As will be explained below, it's possible that certain channels present little information about the correlations present in the image. Therefore, to limit false positives only one of the channels has to exhibit correlations to be cleared of suspicion. It's unlikely that an forged image will be forged in such a way that it exhibits correlations in one channel but not the other, so this should theoretically not limit detection accuracy significantly.		As will be explained below, it's possible that certain channels present little information about the correlations present in the image. Therefore, to limit false positives only one of the channels has to exhibit correlations to be cleared of suspicion. It's unlikely that an forged image will be forged in such a way that it exhibits correlations in one channel but not the other, so this should theoretically not limit detection accuracy significantly.

	There's a choice than can be made regarding what to do with the scores within a given channel. I explored two variants. The first is the 'one-peak method'. For this mode, only one of the peaks needs to be above a certain threshold in order for the block to pass. The rationale for this mode is that the existence of only one peak above the threshold is indicative or correlations in the image. However, this metric is susceptible to noise, as it can simply be by chance that a sample from the peak location is a factor above the average.		There's a choice than can be made regarding what to do with the scores within a given channel. I explored two variants. The first is the 'one-peak' method. For this mode, only one of the peaks needs to be above a certain threshold in order for the block to pass. The rationale for this mode is that the existence of only one peak above the threshold is indicative or correlations in the image. However, this metric is susceptible to noise, as it can simply be by chance that a sample from the peak location is a factor above the average.

	The other method is the 'all-peak' method. This method requires that all of the peaks be above the threshold. This method is much less susceptible to noise, and allows for lower thresholds to be set while still maintaining reliability. However, it also has a tendency to produce false positives.		The other method is the 'all-peak' method. This method requires that all of the peaks be above the threshold. This method is much less susceptible to noise, and allows for lower thresholds to be set while still maintaining reliability. However, it also has a tendency to produce false positives.

imported>Projects221: /* Detector */

2013-03-21T06:12:43Z

Detector

← Older revision		Revision as of 06:12, 21 March 2013
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	The detector design follows directly from the method and metric described above. After the EM algorithm is performed on the picture, a sliding window (that moves in units of half the blocksize) is used to examine blocksize chunks. The scores according to the previously defined metric are then computed for each channel.		The detector design follows directly from the method and metric described above. After the EM algorithm is performed on the picture, a sliding window (that moves in units of half the blocksize) is used to examine blocksize chunks. The scores according to the previously defined metric are then computed for each channel.

	As will be explained below, it's possible that certain channels present little information about the correlations present in the image. Therefore, to limit false positives only one of the channels has to exhibit correlations. It's unlikely that an forged image will be forged in such a way that it exhibits correlations in one channel but not the other, so this should theoretically not limit detection accuracy significantly.		As will be explained below, it's possible that certain channels present little information about the correlations present in the image. Therefore, to limit false positives only one of the channels has to exhibit correlations to be cleared of suspicion. It's unlikely that an forged image will be forged in such a way that it exhibits correlations in one channel but not the other, so this should theoretically not limit detection accuracy significantly.

	There's a choice than can be made regarding what to do with the scores within a given channel. I explored two variants. The first is the 'one-peak method'. For this mode, only one of the peaks needs to be above a certain threshold in order for the block to pass. The rationale for this mode is that the existence of only one peak above the threshold is indicative or correlations in the image. However, this metric is susceptible to noise, as it can simply be by chance that a sample from the peak location is a factor above the average.		There's a choice than can be made regarding what to do with the scores within a given channel. I explored two variants. The first is the 'one-peak method'. For this mode, only one of the peaks needs to be above a certain threshold in order for the block to pass. The rationale for this mode is that the existence of only one peak above the threshold is indicative or correlations in the image. However, this metric is susceptible to noise, as it can simply be by chance that a sample from the peak location is a factor above the average.

imported>Projects221: /* Blocked Detection Mechanism */

2013-03-21T06:11:00Z

Blocked Detection Mechanism

← Older revision		Revision as of 06:11, 21 March 2013
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	== Blocked Detection Mechanism ==		== Blocked Detection Mechanism ==

	It's unlikely for a forger to alter nearly the entirety of an image. On the contrary, forgeries may often alter only very small portions of an image. As such, examining a probability map for an entire image is unlikely to reveal much information about a forgery, as the effect of the uncorrelated region will be averaged out over the entire image. As such, it becomes necessary to examine the image in smaller blocks. This leads to two interesting potential avenues of exploration.		It's unlikely for a forger to alter nearly the entirety of an image. On the contrary, forgeries may often alter only very small portions of an image. As such, examining the probability map for an entire image at once is unlikely to reveal much information about a forgery, as the effect of the uncorrelated region will be averaged out over the entire image. As such, it becomes necessary to examine the image in smaller blocks. This leads to two interesting potential avenues of exploration.

	=== Approach A ===		=== Approach A ===

imported>Projects221: /* Application to Detection */

2013-03-21T06:09:49Z

Application to Detection

← Older revision		Revision as of 06:09, 21 March 2013
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	=== Application to Detection ===		=== Application to Detection ===
	[[File:realTransform.png \| thumb\| left \| Transform of a correlated region of the P-map]]		[[File:realTransform.png \| thumb\| left \| Transform of a correlated region of the P-map]]
	[[File:probmap.png \| frame \| right\| Probability Map with an uninterpolated region in the upper right corner. Map is represented as a regular picture~~. Color~~ indicates probability for a given channel (brighter is more probable)]]		[[File:probmap.png \| frame \| right\| Probability Map with an uninterpolated region in the upper right corner. Map is represented as a regular picture: therefore color indicates probability for a given channel (brighter is more probable)]]

	Of primary interest when it comes to the results of the E-M algorithm is the probability map (P(x,y) above) produced. This gives a per-channel indication of the likelihood of each pixel being correlated with its neighbors. Naturally, large regions of low probability indicate high potential for forgery. However, it's not always trivial to visually detect issues with the probability map, and is generally not conducive to automatic detection.		Of primary interest when it comes to the results of the E-M algorithm is the probability map (P(x,y) above) produced. This gives a per-channel indication of the likelihood of each pixel being correlated with its neighbors. Naturally, large regions of low probability indicate high potential for forgery. However, it's not always trivial to visually detect issues with the probability map, and is generally not conducive to automatic detection.

imported>Projects221: /* Application to Detection */

2013-03-21T06:09:10Z

Application to Detection

← Older revision		Revision as of 06:09, 21 March 2013
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	[[File:probmap.png \| frame \| right\| Probability Map with an uninterpolated region in the upper right corner. Map is represented as a regular picture. Color indicates probability for a given channel (brighter is more probable)]]		[[File:probmap.png \| frame \| right\| Probability Map with an uninterpolated region in the upper right corner. Map is represented as a regular picture. Color indicates probability for a given channel (brighter is more probable)]]

	Of primary interest when it comes to the results of the E-M algorithm is the probability map produced. This gives a per-channel indication of the likelihood of each pixel being correlated with its neighbors. Naturally, large regions of low probability indicate high potential for forgery. However, it's not always trivial to visually detect issues with the probability map, and is generally not conducive to automatic detection.		Of primary interest when it comes to the results of the E-M algorithm is the probability map (P(x,y) above) produced. This gives a per-channel indication of the likelihood of each pixel being correlated with its neighbors. Naturally, large regions of low probability indicate high potential for forgery. However, it's not always trivial to visually detect issues with the probability map, and is generally not conducive to automatic detection.
	[[File:Faketransform.png\|thumb\| left \| Transform of an uncorrelated region of the P-map]]		[[File:Faketransform.png\|thumb\| left \| Transform of an uncorrelated region of the P-map]]
	Luckily, the presence of correlations and patterns of probability result in certain peaks in the frequency domain. Therefore, we can perform a Fourier transform on the data in order to better visualize the presence of correlations. Below is an example of the transformation from a non-forged and forged image. This provides a much cleaner method of identifying regions of suspicion in a probability map.		Luckily, the presence of correlations and patterns of probability result in certain peaks in the frequency domain. Therefore, we can perform a Fourier transform on the data in order to better visualize the presence of correlations. Below is an example of the transformation from a non-forged and forged image. This provides a much cleaner method of identifying regions of suspicion in a probability map.

imported>Projects221: /* M-step */

2013-03-21T06:08:48Z

M-step

← Older revision		Revision as of 06:08, 21 March 2013
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	and p<sub>0</sub> is our prior on our distribution, here equal to 1 over the range of possible values for f.		and p<sub>0</sub> is our prior on our distribution, here equal to 1 over the range of possible values for f.

	We repeat the E and M steps until our solution stops changing more than a small (where small is at the discretion of the implementer) amount. The algorithm is guaranteed to converge, however what it converges too may not be a local minima. Starting with random values helps mitigate pathological cases.[4]		We repeat the E and M steps until our solution stops changing more than a small (where small is at the discretion of the implementer) amount. The algorithm is guaranteed to converge, however what it converges too may not be a local minima. Starting with random values helps mitigate pathological cases[4].

	=== Application to Detection ===		=== Application to Detection ===

imported>Projects221: /* E-Step */

2013-03-21T06:08:12Z

E-Step

← Older revision		Revision as of 06:08, 21 March 2013
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	In order to estimate both a probability and a model, the EM algorithm has two stages. The first stage, the E-stage, attempts to estimate the probability that our samples fall into a proposed correlation model, as opposed to being totally uncorrelated.		In order to estimate both a probability and a model, the EM algorithm has two stages. The first stage, the E-stage, attempts to estimate the probability that our samples fall into a proposed correlation model, as opposed to being totally uncorrelated.

	We represent our model by a matrix α; the central element of alpha represents the sample we're investigating, and the surrounding elements represent the weighting applied to the surrounding pixels (for example, α<sub>11</sub> would represent the weighting of the pixel one down and one to the right of the inspected pixel). Our assumption is that our pixels are perfectly correlated with their neighbours according to alpha, with variations caused by random noise:		We represent our model by a matrix α; the central element of alpha represents the sample we're investigating, and the surrounding elements represent the weighting applied to the surrounding pixels (for example, α<sub>11</sub> would represent the weighting of the pixel one down and one to the right of the inspected pixel). Our assumption is that our pixels are perfectly correlated with their neighbours according to alpha, with variations caused by random noise (here represented by n):

	:<math>f(x,y) = \sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v)+n(x,y)</math>		:<math>f(x,y) = \sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v)+n(x,y)</math>

imported>Projects221: /* E-Step */

2013-03-21T06:07:42Z

E-Step

← Older revision		Revision as of 06:07, 21 March 2013
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	:<math>r(x,y) = f(x,y) - \sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v)</math>		:<math>r(x,y) = f(x,y) - \sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v)</math>

	We then calculate the probability that a given sample is explained by the model assuming that ~~the variation~~ are caused by Gaussian noise:		We then calculate the probability that the deviation of a given sample is explained by the model assuming that any variations are caused by Gaussian noise:

	:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \left[-\frac{ \left(r^2(x,y) \right)}{2\sigma^{2}} \right] </math>		:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \left[-\frac{ \left(r^2(x,y) \right)}{2\sigma^{2}} \right] </math>

imported>Projects221: /* E-Step */

2013-03-21T06:07:10Z

E-Step

← Older revision		Revision as of 06:07, 21 March 2013
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	We then calculate the probability that a given sample is explained by the model assuming that the variation are caused by Gaussian noise:		We then calculate the probability that a given sample is explained by the model assuming that the variation are caused by Gaussian noise:

	:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \left[-\frac{ \left(~~f(x,y)-\sum_{u,v=-N}~~^~~{N} \alpha_{u,v}f~~(x+u,y+v) \right)^2}{2\sigma^{2}} \right] </math>		:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \left[-\frac{ \left(r^2(x,y) \right)}{2\sigma^{2}} \right] </math>

	=== M-step ===		=== M-step ===

imported>Projects221: /* E-Step */

2013-03-21T06:05:35Z

E-Step

← Older revision		Revision as of 06:05, 21 March 2013
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	We then calculate the probability that a given sample is explained by the model assuming that the variation are caused by Gaussian noise:		We then calculate the probability that a given sample is explained by the model assuming that the variation are caused by Gaussian noise:

	:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \~~right~~[-\frac{ \~~right~~(f(x,y)-\sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v) \~~left~~)^2}{2\sigma^{2}} \~~left~~] </math>		:<math>P(x,y) = \frac{1}{\sigma \sqrt{2\pi}}exp \left[-\frac{ \left(f(x,y)-\sum_{u,v=-N}^{N} \alpha_{u,v}f(x+u,y+v) \right)^2}{2\sigma^{2}} \right] </math>

	=== M-step ===		=== M-step ===