Campiotti - Revision history

imported>Projects221: /* Separation Attempts */

2013-03-21T18:25:49Z

Separation Attempts

← Older revision		Revision as of 18:25, 21 March 2013
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	* Gamma correction coefficient		* Gamma correction coefficient

	Several metrics were examined for each of the above combinations, including means, minimums, ~~as well as~~ the difference between these two quantities. None of these metrics produced separable distributions with any measurable improvement over the method presented above (and many of the methods were much worse).		Several metrics were examined for each of the above combinations, including means, minimums, and the difference between these two quantities. None of these metrics produced separable distributions with any measurable improvement over the method presented above (and many of the methods were much worse).

	=== Classification Results ===		=== Classification Results ===

imported>Projects221: /* Similarity Measure */

2013-03-21T18:19:50Z

Similarity Measure

← Older revision		Revision as of 18:19, 21 March 2013
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	[[File:Equation6.png \|left\|]]		[[File:Equation6.png \|left\|]]
	<br>(6)[1]<br><br><br>		<br>(6) [1]<br><br><br>

	As shown in equation (7), [1] proposes a measure of similarity between the two Fourier transforms as the inner product of their magnitudes. Magnitudes are used in order for the similarity measure to be phase-invariant (i.e. so that the synthesized probability map is not required to align perfectly with the EM-generated probability map).		As shown in equation (7), [1] proposes a measure of similarity between the two Fourier transforms as the inner product of their magnitudes. Magnitudes are used in order for the similarity measure to be phase-invariant (i.e. so that the synthesized probability map is not required to align perfectly with the EM-generated probability map).

	[[File:Equation7.png \|left\|]]		[[File:Equation7.png \|left\|]]
	<br>(7)[1]<br><br>		<br>(7) [1]<br><br>

	The authors of [1] propose setting an empirically determined similarity measure threshold, above which a color channel is determined to be CFA interpolated and thus not forged. Unfortunately, this similarity measure has too large a range for the image set in question in order to reliably generate a threshold that could be used to determine forgeries. The cosine similarity (equation 8) is proposed as an alternative measure of similarity, which takes into account the magnitudes of the vectors in question and generates a similarity measure in the range [0,1]. A phase-invariant version of this measure was used in this project.		The authors of [1] propose setting an empirically determined similarity measure threshold, above which a color channel is determined to be CFA interpolated and thus not forged. Unfortunately, this similarity measure has too large a range for the image set in question in order to reliably generate a threshold that could be used to determine forgeries. The cosine similarity (equation 8) is proposed as an alternative measure of similarity, which takes into account the magnitudes of the vectors in question and generates a similarity measure in the range [0,1]. A phase-invariant version of this measure was used in this project.

	[[File:Equation8.png \|left\|]]		[[File:Equation8.png \|left\|]]
	<br><br>(8)[4]<br><br><br>		<br><br>(8) [4]<br><br><br>

	= Results =		= Results =

imported>Projects221: /* Expectation/Maximization Algorithm */

2013-03-21T18:17:46Z

Expectation/Maximization Algorithm

← Older revision		Revision as of 18:17, 21 March 2013
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	<br><br>(4) [1]<br><br><br>		<br><br>(4) [1]<br><br><br>

	Each term in the sum is weighted by the probability that the pixel is interpolated, given by equation (2). If these probabilities were exactly correct (i.e. 1 for interpolated pixels and 0 for measured pixels), only pixels that were truly interpolated would contribute to the total error. As shown in [2], equation (4) is the characteristic error equation for iteratively reweighted least squares, which have solutions for the α<sub>u,v</sub> terms of the form		Each term in the sum is weighted by the probability that the pixel is interpolated, given by equation (2). If these probabilities were exactly correct (i.e. 1 for interpolated pixels and 0 for measured pixels), only pixels that were truly interpolated would contribute to the total error. As shown in [3], equation (4) is the characteristic error equation for iteratively reweighted least squares, which have solutions for the α<sub>u,v</sub> terms of the form

	[[File:Equation5.png \|left\|]]		[[File:Equation5.png \|left\|]]
	<br>(5) [3]<br><br>		<br>(5) [3]<br><br>

	where t here represents the current iteration.		where t here represents the current iteration. The EM algorithm is iterated over until a stable estimate for the α<sub>u,v</sub> terms is achieved.

	== Similarity Measure ==		== Similarity Measure ==

imported>Projects221: /* Maximization Step */

2013-03-21T18:16:21Z

Maximization Step

← Older revision		Revision as of 18:16, 21 March 2013
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	=== Maximization Step ===		=== Maximization Step ===

	In the maximization step, weighted least squares is used to determine the α<sub>u,v</sub> terms that ~~minimizes~~ the total error over the entire image between the actual pixel values and the estimates for the pixel values determined by the linear model.		In the maximization step, weighted least squares is used to determine the α<sub>u,v</sub> terms that minimize the total error over the entire image between the actual pixel values and the estimates for the pixel values determined by the linear model.

	[[File:Equation4.png \|left\|]]		[[File:Equation4.png \|left\|]]
	<br><br>(4)[1]<br><br><br>		<br><br>(4) [1]<br><br><br>

	Each term in the sum is weighted by the probability that the pixel is interpolated, given by equation (2). If these probabilities were exactly correct (i.e. 1 for interpolated pixels and 0 for measured pixels), only pixels that were truly interpolated would contribute to the total error. As shown in [2], equation (4) is the characteristic error equation for iteratively reweighted least squares, which have solutions for the α<sub>u,v</sub> terms of the form		Each term in the sum is weighted by the probability that the pixel is interpolated, given by equation (2). If these probabilities were exactly correct (i.e. 1 for interpolated pixels and 0 for measured pixels), only pixels that were truly interpolated would contribute to the total error. As shown in [2], equation (4) is the characteristic error equation for iteratively reweighted least squares, which have solutions for the α<sub>u,v</sub> terms of the form

	[[File:Equation5.png \|left\|]]		[[File:Equation5.png \|left\|]]
	<br>(5)[3]<br><br>		<br>(5) [3]<br><br>

	where t here represents the current iteration.		where t here represents the current iteration.

imported>Projects221: /* Expectation Step */

2013-03-21T18:14:31Z

Expectation Step

← Older revision		Revision as of 18:14, 21 March 2013
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	[[File:Equation1.png \|left\|]]		[[File:Equation1.png \|left\|]]
	<br>(1)[1]<br><br><br>		<br>(1) [1]<br><br><br>

	where f(x,y) denotes the pixel value for a given color channel, α<sub>u,v</sub> denotes the weighting of each neighbor, N is the chosen neighborhood size, and n(x,y) is an error (i.e. noise) term. As described in [1], the set of interpolated pixels is deemed M<sub>1</sub> and Bayes's rule can be used to determine whether a given pixel is part of this set		where f(x,y) denotes the pixel value for a given color channel, α<sub>u,v</sub> denotes the weighting of each neighbor, N is the chosen neighborhood size, and n(x,y) is an error (i.e. noise) term. As described in [1], the set of interpolated pixels is deemed M<sub>1</sub> and Bayes' rule can be used to determine whether a given pixel is part of this set.

	[[File:Equation2.png \|left\|]]		[[File:Equation2.png \|left\|]]
	<br>(2)[1]<br><br><br>		<br>(2) [1]<br><br><br>

	It is assumed that there is equal probability (i.e. 0.5) that a given pixel was interpolated (i.e. part of M<sub>1</sub>) or measured (i.e. part of M<sub>2</sub>). This assumption may not be valid given the scarcity of the red and blue pixels in a Bayer array, but was never tested. For pixels in M<sub>2</sub>, it is assumed that there is a uniform distribution of values (i.e. each value has a probability of 1/256 for an 8-bit color channel). The conditional probability for a pixel in M<sub>1</sub> is assumed to be Gaussian and centered around the linear estimate of the pixel value. This is shown in equation (3).		It is assumed that there is equal probability (i.e. 0.5) that a given pixel was interpolated (i.e. part of M<sub>1</sub>) or measured (i.e. part of M<sub>2</sub>). This assumption may not be valid given the scarcity of the red and blue pixels in a Bayer array, but was never tested. For pixels in M<sub>2</sub>, it is assumed that there is a uniform distribution of values (i.e. each value has a probability of 1/256 for an 8-bit color channel). The conditional probability for a pixel in M<sub>1</sub> is assumed to be Gaussian and centered around the linear estimate of the pixel value. This is shown in equation (3).

	[[File:Equation3.png \|left\|]]		[[File:Equation3.png \|left\|]]
	<br>(3)[1]<br><br><br>		<br>(3) [1]<br><br><br>

	=== Maximization Step ===		=== Maximization Step ===

imported>Projects221: /* Appendix I - Code and Data */

2013-03-21T12:04:06Z

Appendix I - Code and Data

← Older revision		Revision as of 12:04, 21 March 2013
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	MATLAB Student Version (R2011a)		MATLAB Student Version (R2011a)

	= Appendix I - Code ~~and Data~~ =		= Appendix I - Code =

	~~== Code ==~~		[[File:CampiottiCodeFile.zip]]
	[[File:~~CodeFile.zip]]~~

	~~== Data ==~~
	~~[[File:DataFile~~.zip]]

imported>Projects221: /* Results */

2013-03-21T11:49:17Z

Results

@@ Line 75: / Line 75: @@
 = Results =
-== Training Set ==
+== Training Image Set ==
 === Expectation/Maximization Algorithm ===
@@ Line 134: / Line 134: @@
 === Classification Results ===
-== Test Set ==
+The thresholds chosen for the training set of images were applied to a test set of images to determine if the same success rate could be achieved. The test set consisted of 10 real images and 10 of their tampered counterparts. The results are shown in Table 3. As shown, the algorithm incorrectly identified 90% of the tampered images and only 10% of the untampered images. This indicates that the thresholds established previously were likely too low for this set of images.
 = Conclusions =

imported>Projects221: /* Results */

2013-03-21T11:11:20Z

Results

← Older revision		Revision as of 11:11, 21 March 2013
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	As shown in these histograms, no bimodal distribution was present for this metric (however, the tampered distribution is shifted slightly lower than the untampered distribution in all cases). This indicates the problem shown in Figures 8 - 10 is more prevalent than not and reliable separation thresholds cannot be determined.		As shown in these histograms, no bimodal distribution was present for this metric (however, the tampered distribution is shifted slightly lower than the untampered distribution in all cases). This indicates the problem shown in Figures 8 - 10 is more prevalent than not and reliable separation thresholds cannot be determined.

			==== Separation Attempts ====

			Due to the success of the authors' proposed similarity measure in [1], many attempts were made to find a metric that would allow reliable separation of untampered and tampered images using similarity measures related to the authors' proposal. In addition to the phase-invariant cosine similarity already discussed, among the methods attempted were:

			* Phase-variant cosine similarity
			* Similarity measure proposed in [1] (both phase-invariant and phase-variant)
			* Vector projection (both phase-invariant and phase-variant)

			For each method, several parameters of the EM and similarity measure algorithms were swept, including:

			* Window size
			* Convergence parameter, epsilon
			* High pass filter cutoff frequency (including no high pass filter)
			* Gamma correction coefficient

			Several metrics were examined for each of the above combinations, including means, minimums, as well as the difference between these two quantities. None of these metrics produced separable distributions with any measurable improvement over the method presented above (and many of the methods were much worse).

			=== Classification Results ===



	== Test Set ==		== Test Set ==

imported>Projects221: /* Similarity Measure */

2013-03-21T10:54:27Z

Similarity Measure

← Older revision		Revision as of 10:54, 21 March 2013
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	<br>(6)[1]<br><br><br>		<br>(6)[1]<br><br><br>

	As shown in equation (7), [1] proposes a measure of similarity between the two Fourier transforms as the inner product of their magnitudes. Magnitudes are used in order for the similarity measure to be phase invariant (i.e. so that the synthesized probability map is not required to align perfectly with the EM-generated probability map).		As shown in equation (7), [1] proposes a measure of similarity between the two Fourier transforms as the inner product of their magnitudes. Magnitudes are used in order for the similarity measure to be phase-invariant (i.e. so that the synthesized probability map is not required to align perfectly with the EM-generated probability map).

	[[File:Equation7.png \|left\|]]		[[File:Equation7.png \|left\|]]
	<br>(7)[1]<br><br>		<br>(7)[1]<br><br>

	The authors of [1] propose setting an empirically determined similarity measure threshold, above which a color channel is determined to be CFA interpolated and thus not forged. ~~As will be discussed in the Results section~~, this similarity measure has too large a range for the image set in question in order to reliably generate a threshold that could be used to determine forgeries. The cosine similarity (equation 8) is proposed as an alternative measure of similarity, which takes into account the magnitudes of the vectors in question and generates a similarity measure in the range [0,1].		The authors of [1] propose setting an empirically determined similarity measure threshold, above which a color channel is determined to be CFA interpolated and thus not forged. Unfortunately, this similarity measure has too large a range for the image set in question in order to reliably generate a threshold that could be used to determine forgeries. The cosine similarity (equation 8) is proposed as an alternative measure of similarity, which takes into account the magnitudes of the vectors in question and generates a similarity measure in the range [0,1]. A phase-invariant version of this measure was used in this project.

	[[File:Equation8.png \|left\|]]		[[File:Equation8.png \|left\|]]
	<br><br>(8)[4]<br><br><br>		<br><br>(8)[4]<br><br><br>


	= Results =		= Results =

imported>Projects221: /* Results */

2013-03-21T10:51:02Z

Results

← Older revision		Revision as of 10:51, 21 March 2013
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	\|}		\|}

			In many cases, however, generating a separability threshold was not possible for one or more of the color channels. Figures 8 - 10 illustrate this point for a difficult image.

			{\| class="wikitable" style="margin: 1em auto 1em auto;"
			\|[[File:CampiottiFigure8.png \|thumb\|392px\| Figure 8. Red channel cosine similarity (untampered = red, tampered = blue)]]
			\|[[File:CampiottiFigure9.png \|thumb\|392px\| Figure 9. Green channel cosine similarity (untampered = red, tampered = blue)]]
			\|[[File:CampiottiFigure10.png \|thumb\|392px\| Figure 10. Blue channel cosine similarity (untampered = red, tampered = blue)]]
			\|}

			Histograms of the minimum cosine similarities across all images for each image channel were generated in the hopes of finding a bimodal distribution that could provide guidance on how to choose thresholds that would maximize classification success. These histograms are shown in Figures 11, 12, and 13 for the red, green, and blue channels, respectively.

			{\| class="wikitable" style="margin: 1em auto 1em auto;"
			\|[[File:CampiottiFigure11.png \|thumb\|392px\| Figure 11. Red channel minimum cosine similarity distribution (untampered = red, tampered = blue)]]
			\|[[File:CampiottiFigure12.png \|thumb\|392px\| Figure 12. Green channel minimum cosine similarity distribution (untampered = red, tampered = blue)]]
			\|[[File:CampiottiFigure13.png \|thumb\|392px\| Figure 13. Blue channel minimum cosine similarity distribution (untampered = red, tampered = blue)]]
			\|}

			As shown in these histograms, no bimodal distribution was present for this metric (however, the tampered distribution is shifted slightly lower than the untampered distribution in all cases). This indicates the problem shown in Figures 8 - 10 is more prevalent than not and reliable separation thresholds cannot be determined.

	== Test Set ==		== Test Set ==