Privacy Preserving Morphological Operations for Digital Images - Revision history

Gturcas: /* Experiments and conclusions */

2021-03-07T11:07:35Z

Experiments and conclusions

2021-03-07T11:07:22Z

Other morphological operations

2021-03-07T10:56:03Z

Other morphological operations

2021-03-07T10:55:05Z

Other morphological operations

2021-03-07T10:54:24Z

Other morphological operations

2021-03-07T10:53:53Z

Other morphological operations

2021-03-07T10:11:08Z

The dilation operation

2021-03-07T10:08:02Z

The dilation operation

2021-03-07T09:58:27Z

The erosion operation

2021-03-07T09:52:59Z

The erosion operation

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 The authors used an implementation of the [[certSGN]] encryption scheme to compute morphological primitives over an encrypted image, and also compared their results with other LWE based schemes such as BGV (HElib) and BFV (Microsoft SEAL).
-The experiments show that for certian structural elements the [[certSGN]] implementation presents optimizations of 300x up to 10000x in execution times of morphological primitives and in 2.87x to 55x in memory optimization over the LWE schemes.
+The experiments show that for certian structural elements the [[certSGN]] implementation presents optimizations of 300x up to 10000x in execution times of morphological primitives and in 2.87x to 55x in memory optimization over the LWE schemes. The details of the experiments can be found in the paper.
 == References ==

← Older revision		Revision as of 11:07, 7 March 2021
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	<center> [[File:Openingclosing.png]] </center>		<center> [[File:Openingclosing.png]] </center>
		+
		+	== Experiments and conclusions ==
		+
		+	The authors used an implementation of the [[certSGN]] encryption scheme to compute morphological primitives over an encrypted image, and also compared their results with other LWE based schemes such as BGV (HElib) and BFV (Microsoft SEAL).
		+
		+	The experiments show that for certian structural elements the [[certSGN]] implementation presents optimizations of 300x up to 10000x in execution times of morphological primitives and in 2.87x to 55x in memory optimization over the LWE schemes.
		+
		+	The details of the experiments can be found in the paper.

	== References ==		== References ==

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 . <math> I_0 </math> is the dilation of <math> I </math> by <math> S</math>.
 . <math> S' </math> is the symmetric of <math> S </math> defined above.
 . The closing <math> I </math> is the erosion of <math> I_0 </math> with the structural element <math> S' </math>.
 == References ==

← Older revision		Revision as of 10:55, 7 March 2021
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	1. <math> I_0 </math> is the dilation of <math> I </math> by <math> S</math>.		1. <math> I_0 </math> is the dilation of <math> I </math> by <math> S</math>.
		+
		+	<center> [[File:Openingclosing.png]] </center>

	2. <math> S' </math> is the symmetric of <math> S </math> defined above.		2. <math> S' </math> is the symmetric of <math> S </math> defined above.

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 The opening of <math>I</math> with <math> S </math> is performed in the following way:
 . <math>I_0 </math> is the erosion of <math> I </math> by <math>S</math>.
 . Let <math> S' = \{-s \mid s \in S \} </math> the symmetric of <math> S</math> with respect to the origin (its center).
 . The opening <math>I' </math> is the dilation of <math> I_0 </math> with the structural element <math> S' </math>.
@@ Line 87: / Line 90: @@
 . <math> I_0 </math> is the dilation of <math> I </math> by <math> S</math>.
 . <math> S' </math> is the symmetric of <math> S </math> defined above.
 . The closing <math> I </math> is the erosion of <math> I_0 </math> with the structural element <math> S' </math>.
 == References ==

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 === Other morphological operations  ===
-Using the primitives described above one can perform edge dection, as the <math> \mathbb F_2 </math> difference between the initial image and its eroded version. This is described in the image below
+Using the primitives described above one can perform edge dection, as the <math> \mathbb F_2 </math> difference between the initial image and its eroded version. This is described in the image below.
 <center> [[File:Edgedetection.png]] </center>
 == References ==

← Older revision		Revision as of 10:11, 7 March 2021
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	This expression can be simplified if one represents binary images as matrices whose <math> 0 </math> entries represent objects and <math> 1 </math> represents background.		This expression can be simplified if one represents binary images as matrices whose <math> 0 </math> entries represent objects and <math> 1 </math> represents background.
		+
		+	=== Other morphological operations ===
		+
		+	Using the primitives described above one can perform edge dection, as the <math> \mathbb F_2 </math> difference between the initial image and its eroded version. This is described in the image below
		+	<center> [[File:Edgedetection.png]] </center>

	== References ==		== References ==

← Older revision		Revision as of 10:08, 7 March 2021
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	<center>[[File:Dilation.png]] </center>		<center>[[File:Dilation.png]] </center>
		+
		+	Using the notations in the previous subsection, the entries of a binary matrix associated to a dilated image can be computed from the following equality.
		+
		+	<center><math> 1-A'[i][j] = \prod_{s \in S_p} (1- A[s_x][s_y]).</math></center>
		+
		+	This expression can be simplified if one represents binary images as matrices whose <math> 0 </math> entries represent objects and <math> 1 </math> represents background.

	== References ==		== References ==

← Older revision		Revision as of 09:58, 7 March 2021
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	This computation can be done in an encrypted manner if the encryption scheme is homomorphic with respect to multiplication, like the [[certSGN]] scheme.		This computation can be done in an encrypted manner if the encryption scheme is homomorphic with respect to multiplication, like the [[certSGN]] scheme.
		+
		+	=== The dilation operation ===
		+
		+	The dilation of <math> I</math> with the structural element <math> S </math> is defined as
		+	<math>D(I,S) = I \oplus S = \{p=(x,y) \in \mathbb Z^2 \mid S_p \cap I \neq \emptyset \}.</math>
		+
		+	That is, the dilation retains all the pixels <math> p </math> for which the translated structured element <math> S_p </math> intersects non-trivially with the image. Below we see the effects of a dilation with a structured element <math>S</math>.
		+
		+	<center>[[File:Dilation.png]] </center>

	== References ==		== References ==

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 <center><math>A'[i][j] = \prod\limits_{s \in S_p} A[s_x][s_y].</math></center>
-This computation can be done in an encrypted manner if the encryption scheme is homomorphic with respect to multiplication, like the [[CSGN]] scheme.
+This computation can be done in an encrypted manner if the encryption scheme is homomorphic with respect to multiplication, like the [[certSGN]] scheme.
 == References ==