retouche complexité

[canny.git] / ourapproach.tex

diff --git a/ourapproach.tex b/ourapproach.tex
index f86c8aead96b49254e4e77ee2f28cb6730fd5857..ec73f7d712d90f71cb1924872ca278adfa6a9686 100644 (file)
--- a/ourapproach.tex
+++ b/ourapproach.tex
@@ -3,39 +3,41 @@ four main steps: the data encryption (Sect.~\ref{sub:bbs}),
 the cover pixel selection (Sect.~\ref{sub:edge}),
 the adaptive payload considerations (Sect.~\ref{sub:adaptive}),
 and how the distortion has been minimized (Sect.~\ref{sub:stc}).
 the cover pixel selection (Sect.~\ref{sub:edge}),
 the adaptive payload considerations (Sect.~\ref{sub:adaptive}),
 and how the distortion has been minimized (Sect.~\ref{sub:stc}).

-The message extraction is finally presented  (Sect.\ref{sub:extract}) and a running example ends this section (Sect.~\ref{sub:xpl}). 
+The message extraction is then presented  (Sect.~\ref{sub:extract}) while a running example ends this section (Sect.~\ref{sub:xpl}). 

 
 
 The flowcharts given in Fig.~\ref{fig:sch}
 summarize our steganography scheme denoted by
 
 
 The flowcharts given in Fig.~\ref{fig:sch}
 summarize our steganography scheme denoted by

-STABYLO, which stands for STeganography with cAnny, Bbs, binarY embedding at LOw cost.
-What follows are successively details of the inner steps and flows inside 
-both the embedding stage (Fig.~\ref{fig:sch:emb}) 
-and the extraction one (Fig.~\ref{fig:sch:ext}).
+STABYLO, which stands for STe\-ga\-no\-gra\-phy with 
+Adaptive, Bbs, binarY embedding at LOw cost.
+What follows are successively some details of the inner steps and the flows both inside 
+ the embedding stage (Fig.~\ref{fig:sch:emb}) 
+and inside the extraction one (Fig.~\ref{fig:sch:ext}).

 Let us first focus on the data embedding. 
 
 Let us first focus on the data embedding. 
 

-\begin{figure*}[t]
+\begin{figure*}%[t]

   \begin{center}
   \begin{center}

-    \subfloat[Data Embedding.]{
+    \subfloat[Data Embedding]{

       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.5]{emb.ps}
+          \includegraphics[scale=0.45]{emb}

         \end{center}
       \end{minipage}
       \label{fig:sch:emb}
         \end{center}
       \end{minipage}
       \label{fig:sch:emb}

-    }%\hfill
-    \subfloat[Data Extraction.]{
+    } 
+
+    \subfloat[Data Extraction]{

       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\textwidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.5]{rec.ps}
+          \includegraphics[scale=0.45]{rec}

         \end{center}
       \end{minipage}
       \label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       \label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{The STABYLO Scheme.}
+  \caption{The STABYLO scheme}

   \label{fig:sch}
 \end{figure*}
 
   \label{fig:sch}
 \end{figure*}
 


@@ -46,18 +48,18 @@ Let us first focus on the data embedding.
 
 
 
 
 
 

-\subsection{Security Considerations}\label{sub:bbs}
-Among methods of message encryption/decryption 
+\subsection{Security considerations}\label{sub:bbs}
+Among the methods of  message encryption/decryption 

 (see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
 we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
 that is based on the Blum Blum Shub~\cite{DBLP:conf/crypto/ShubBB82} 
 pseudorandom number generator (PRNG) and the 
 XOR binary function.
 (see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
 we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
 that is based on the Blum Blum Shub~\cite{DBLP:conf/crypto/ShubBB82} 
 pseudorandom number generator (PRNG) and the 
 XOR binary function.

-It has been indeed proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG 
+It has been proven~\cite{DBLP:conf/crypto/ShubBB82} that this PRNG 

 has the property of cryptographical security, \textit{i.e.}, 
 for any sequence of $L$ output bits $x_i$, $x_{i+1}$, \ldots, $x_{i+L-1}$,
 there is no algorithm, whose time complexity is polynomial  in $L$, and 
 has the property of cryptographical security, \textit{i.e.}, 
 for any sequence of $L$ output bits $x_i$, $x_{i+1}$, \ldots, $x_{i+L-1}$,
 there is no algorithm, whose time complexity is polynomial  in $L$, and 

-which allows to find $x_{i-1}$ and $x_{i+L}$ with a probability greater
+which allows to find $x_{i-1}$ or $x_{i+L}$ with a probability greater

 than $1/2$.
 Equivalent formulations of such a property can
 be found. They all lead to the fact that,
 than $1/2$.
 Equivalent formulations of such a property can
 be found. They all lead to the fact that,


@@ -69,7 +71,7 @@ Starting thus with a key $k$ and the message \textit{mess} to hide,
 this step computes a message $m$, which is the encrypted version  of \textit{mess}.
 
 
 this step computes a message $m$, which is the encrypted version  of \textit{mess}.
 
 

-\subsection{Edge-Based Image Steganography}\label{sub:edge}
+\subsection{Edge-based image steganography}\label{sub:edge}

 
 
 The edge-based image
 
 
 The edge-based image


@@ -88,18 +90,18 @@ Many techniques have been proposed in the literature to  detect
 edges in  images (whose noise has been initially reduced). 
 They can be separated in two categories: first and second order detection
 methods on the one hand, and fuzzy detectors on the other  hand~\cite{KF11}.
 edges in  images (whose noise has been initially reduced). 
 They can be separated in two categories: first and second order detection
 methods on the one hand, and fuzzy detectors on the other  hand~\cite{KF11}.

-In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, \ldots, 
+In first order methods like Sobel, Canny~\cite{Canny:1986:CAE:11274.11275}, and so on, 

 a first-order derivative (gradient magnitude, etc.) is computed 
 to search for local maxima, whereas in second order ones, zero crossings in a second-order derivative, like the Laplacian computed from the image,
 are searched in order to find edges.
 a first-order derivative (gradient magnitude, etc.) is computed 
 to search for local maxima, whereas in second order ones, zero crossings in a second-order derivative, like the Laplacian computed from the image,
 are searched in order to find edges.

-As for as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges.
+As far as fuzzy edge methods are concerned, they are obviously based on fuzzy logic to highlight edges.

 
 

-Canny filters, on their parts, are an old family of algorithms still remaining a state-of-the-art edge detector. They can be well approximated by first-order derivatives of Gaussians.
-As the Canny algorithm is well known and studied, fast, and implementable
-on many  kinds of architectures like FPGAs, smartphones,  desktop machines, and
+Canny filters, on their parts, are an old family of algorithms still remaining a state of the art edge detector. They can be well-approximated by first-order derivatives of Gaussians.
+As the Canny algorithm is fast, well known, has been studied in depth, and is implementable
+on many  kinds of architectures like FPGAs, smart phones,  desktop machines, and

 GPUs, we have chosen this edge detector for illustrative purpose.
 
 GPUs, we have chosen this edge detector for illustrative purpose.
 

-\JFC{il faudrait comparer les complexites des algo fuzy and canny}
+%\JFC{il faudrait comparer les complexites des algo fuzy and canny}

 
 
 This edge detection is applied on a filtered version of the image given 
 
 
 This edge detection is applied on a filtered version of the image given 


@@ -111,15 +113,15 @@ If set with the same value $b$, the edge detection returns thus the same
 set of pixels for both the cover and the stego image.   
 In our flowcharts, this is represented by ``edgeDetection(b bits)''.
 Then only the 2 LSBs of pixels in the set of edges are returned if $b$ is 6, 
 set of pixels for both the cover and the stego image.   
 In our flowcharts, this is represented by ``edgeDetection(b bits)''.
 Then only the 2 LSBs of pixels in the set of edges are returned if $b$ is 6, 

-and the LSB of pixels if $b$ is 7.
+and the LSBs of pixels if $b$ is 7.

 
 
 
 
 
 Let $x$ be the sequence of these bits. 
 
 
 
 
 
 Let $x$ be the sequence of these bits. 

-The next  section section presentsd how our scheme 
-adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
+The next  section presents how to adapt our scheme 
+  when the size of $x$  is not sufficient for the message $m$ to embed.

 
 
  
 
 
  


@@ -127,57 +129,49 @@ adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
 
 
 
 
 
 

-\subsection{Adaptive Embedding Rate}\label{sub:adaptive}
-Two strategies have been developed in our scheme, 
+\subsection{Adaptive embedding rate}\label{sub:adaptive}
+Two strategies have been developed in our approach, 

 depending on the embedding rate that is either \emph{adaptive} or \emph{fixed}.
 In the former the embedding rate depends on the number of edge pixels.
 The higher it is, the larger the message length that can be inserted is.
 Practically, a set of edge pixels is computed according to the 
 depending on the embedding rate that is either \emph{adaptive} or \emph{fixed}.
 In the former the embedding rate depends on the number of edge pixels.
 The higher it is, the larger the message length that can be inserted is.
 Practically, a set of edge pixels is computed according to the 

-Canny algorithm with an high threshold.
+Canny algorithm with a high threshold.

 The message length is thus defined to be less than 
 half of this set cardinality.
 The message length is thus defined to be less than 
 half of this set cardinality.

-If $x$ is then to short for $m$, the message is splitted into sufficient parts. 
+If $x$ is too short for $m$, the message is split into sufficient parts
+and a new cover image should be used for the remaining part of the message. 
+
+ 

 In the latter, the embedding rate is defined as a percentage between the 
 number of modified pixels and the length of the bit message.
 This is the classical approach adopted in steganography.
 Practically, the Canny algorithm generates  
 a set of edge pixels related to a threshold that is decreasing 
 until its cardinality
 In the latter, the embedding rate is defined as a percentage between the 
 number of modified pixels and the length of the bit message.
 This is the classical approach adopted in steganography.
 Practically, the Canny algorithm generates  
 a set of edge pixels related to a threshold that is decreasing 
 until its cardinality

-is sufficient. 
+is sufficient. Even in this situation, our scheme is adapting 
+its algorithm to meet all the user's requirements. 

 
 
 
 

-
-Two methods may further be applied to select bits that 
-will be modified. 
+Once the map of possibly modified pixels is computed, 
+two methods may further be applied to extract bits that 
+are really modified. 

 The first one randomly chooses the subset of pixels to modify by 
 The first one randomly chooses the subset of pixels to modify by 

-applying the BBS PRNG again. This method is further denoted  as to \emph{sample}.
-The second one is a direct application of the 
-STC algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
-It  is further referred to as \emph{adaptive+STC} and is detailled in the nex section.
+applying the BBS PRNG again. This method is further denoted  as a \emph{sample}.
+Once this set is selected, a classical LSB replacement is applied to embed the 
+stego content.
+The second method considers the last significant bits of all the pixels 
+inside the previous map. It next directly applies the STC 
+algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
+It  is further referred to as \emph{STC} and is detailed in the next section.

 
 
 
 
 
 
 
 
 
 

-% First of all, let us discuss about compexity of edge detetction methods.
-% Let then $M$ and $N$ be the dimension of the original image. 
-% According to~\cite{Hu:2007:HPE:1282866.1282944},
-% even if the fuzzy logic based edge detection methods~\cite{Tyan1993} 
-% have promising results, its complexity is in $C_3 \times O(M \times N)$
-% whereas the complexity on the Canny method~\cite{Canny:1986:CAE:11274.11275} 
-% is in $C_1 \times O(M \times N)$ where  $C_1 < C_3$.
-% \JFC{Verifier ceci...}
-% In experiments detailled in this article, the Canny method has been retained 
-% but the whole approach can be updated to consider 
-% the fuzzy logic edge detector.   
-
-
-

 
 
 
 
 
 

-
-\subsection{Minimizing Distortion with Syndrome-Treillis Codes}\label{sub:stc}
+\subsection{Minimizing distortion with syndrome-trellis codes}\label{sub:stc}

 \input{stc}
 
 
 \input{stc}
 
 


@@ -211,41 +205,49 @@ It  is further referred to as \emph{adaptive+STC} and is detailled in the nex se
 
 
 
 
 
 

-\subsection{Data Extraction}\label{sub:extract}
-The message extraction summarized in Fig.~\ref{fig:sch:ext} follows data embedding
+\subsection{Data extraction}\label{sub:extract}
+The message extraction summarized in Fig.~\ref{fig:sch:ext} 
+follows the data embedding approach 

 since there exists a reverse function for all its steps.
 since there exists a reverse function for all its steps.

-First of all, the same edge detection is applied (on the 7 first bits) to 
-get the set of LSBs,
-which is  sufficiently large with respect to the message size given as a key.  
-Then the STC reverse algorithm is applied to retrieve the encrypted message.
+
+More precisely, the same edge detection is applied on the $b$ first bits  to 
+produce the sequence $y$ of LSBs. 
+If the STC approach has been selected in embedding, the STC reverse
+algorithm is directly executed to retrieve the encrypted message. 
+This inverse function takes the $H$ matrix as a parameter.
+Otherwise, \textit{i.e.}, if the \emph{sample} strategy is retained,
+the same random bit selection than in the embedding step 
+is executed with the same seed, given as a key.

 Finally, the Blum-Goldwasser decryption function is executed and the original
 message is extracted.
 
 
 Finally, the Blum-Goldwasser decryption function is executed and the original
 message is extracted.
 
 

-\subsection{Running Example}\label{sub:xpl}
-In this example, the cover image is  Lena 
-which is a 512*512  image with 256 grayscale levels.
+\subsection{Running example}\label{sub:xpl}
+In this example, the cover image is  Lena, 
+which is a $512\times512$  image with 256 grayscale levels.

 The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667
 The message is the poem Ulalume (E. A. Poe), which is constituted by 104 lines, 667

-words, and 3754 characters, \textit{i.e.}  30032 bits.
-Lena and the the first verses are given in Fig.~\ref{fig:lena}.
+words, and 3,754 characters, \textit{i.e.},  30,032 bits.
+Lena and the first verses are given in Fig.~\ref{fig:lena}.

 
 \begin{figure}
 \begin{center}
 
 \begin{figure}
 \begin{center}

-\begin{minipage}{0.4\linewidth}
-\includegraphics[width=3cm]{Lena.eps}
+\begin{minipage}{0.49\linewidth}
+\begin{center}
+\includegraphics[scale=0.20]{Lena.eps}
+\end{center}

 \end{minipage}
 \end{minipage}

-\begin{minipage}{0.59\linewidth}
+\begin{minipage}{0.49\linewidth}

 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak
 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak

-$~$ The leaves they were crisped and sere—\linebreak
-$~$ The leaves they were withering and sere;\linebreak
+$\qquad$ The leaves they were crisped and sere—\linebreak
+$\qquad$ The leaves they were withering and sere;\linebreak

 It was night in the lonesome October\linebreak
 It was night in the lonesome October\linebreak

-$~$ Of my most immemorial year;\linebreak
+$\qquad$ Of my most immemorial year;\linebreak

 It was hard by the dim lake of Auber,\linebreak
 It was hard by the dim lake of Auber,\linebreak

-$~$ In the misty mid region of Weir—\linebreak
+$\qquad$ In the misty mid region of Weir—\linebreak

 It was down by the dank tarn of Auber,\linebreak
 It was down by the dank tarn of Auber,\linebreak

-$~$ In the ghoul-haunted woodland of Weir.
+$\qquad$ In the ghoul-haunted woodland of Weir.

 \end{scriptsize}
 \end{flushleft}
 \end{minipage}
 \end{scriptsize}
 \end{flushleft}
 \end{minipage}


@@ -253,9 +255,11 @@ $~$ In the ghoul-haunted woodland of Weir.
 \caption{Cover and message examples} \label{fig:lena}
 \end{figure}
 
 \caption{Cover and message examples} \label{fig:lena}
 \end{figure}
 

-The edge detection returns 18641 and 18455 pixels when $b$ is
-respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
-
+The edge detection returns 18,641 and 18,455 pixels when $b$ is
+respectively 7 and 6. These edges are represented in Figure~\ref{fig:edge}.
+When $b$ is 7, it remains one bit per pixel to build the cover vector.
+This configuration leads to a cover vector of size  18,641 if b is 7 
+and 36,910 if $b$ is 6.  

 
 \begin{figure}[t]
   \begin{center}
 
 \begin{figure}[t]
   \begin{center}


@@ -263,7 +267,7 @@ respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{edge7.eps}
+          \includegraphics[scale=0.20]{edge7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -272,21 +276,30 @@ respectively 7 and 6. These edges are represented in Fig.~\ref{fig:edge}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{edge6.eps}
+          \includegraphics[scale=0.20]{edge6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Edge Detection wrt $b$.}
+  \caption{Edge detection wrt $b$}

   \label{fig:edge}
 \end{figure}
 
 
 
   \label{fig:edge}
 \end{figure}
 
 
 

-In the former configuration, only 9320 bits are available 
-for embeding whereas in the latter we have 9227.
-In the both case, about the third part of the poem is hidden into the cover.
+The STC algorithm is optimized when the rate between message length and 
+cover vector length is lower than 1/2. 
+So, only 9,320 bits  are available for embedding 
+in the  configuration where $b$ is 7.
+
+When $b$ is 6, we could have considered 18,455 bits for the message.
+However, first experiments have shown that modifying this number of bits is too 
+easily detectable. 
+So, we choose to modify the same amount of bits (9,320) and keep STC optimizing
+which bits to change among  the 36,910 ones.
+
+In the two cases, about the third part of the poem is hidden into the cover. 

 Results with \emph{adaptive+STC} strategy are presented in 
 Fig.~\ref{fig:lenastego}.
 
 Results with \emph{adaptive+STC} strategy are presented in 
 Fig.~\ref{fig:lenastego}.
 


@@ -296,7 +309,7 @@ Fig.~\ref{fig:lenastego}.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{lena7.eps}
+          \includegraphics[scale=0.20]{lena7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -305,32 +318,31 @@ Fig.~\ref{fig:lenastego}.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{lena6.eps}
+          \includegraphics[scale=0.20]{lena6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Stego Images wrt $b$.}
+  \caption{Stego images wrt $b$}

   \label{fig:lenastego}
 \end{figure}
 
 
 Finally, differences between the original cover and the stego images  
   \label{fig:lenastego}
 \end{figure}
 
 
 Finally, differences between the original cover and the stego images  

-are presented in Fig.~\ref{fig:lenadiff}. For each pixel pair of picel $X_{ij}$  
-$Y_{ij}$, $X$ and $Y$ being the cover and the stego content respectively, 
-The pixel value $V_{ij}$ of the difference is defined with the following map
+are presented in Fig.~\ref{fig:lenadiff}. For each pair of pixel $X_{ij}$ and  $Y_{ij}$ ($X$ and $Y$ being the cover and the stego content respectively), 
+the pixel value $V_{ij}$ of the difference is defined with the following map

 $$
 V_{ij}= \left\{
 \begin{array}{rcl}
 0 & \textrm{if} &  X_{ij} = Y_{ij} \\
 $$
 V_{ij}= \left\{
 \begin{array}{rcl}
 0 & \textrm{if} &  X_{ij} = Y_{ij} \\

-75 & \textrm{if} &  \abs{ (X_{ij} - Y_{ij})} = 1 \\
-75 & \textrm{if} &  \abs{ (X_{ij} - Y_{ij})} = 2 \\
-225 & \textrm{if} &  \abs{ (X_{ij} - Y_{ij})} = 1 
+75 & \textrm{if} &  \vert X_{ij} - Y_{ij} \vert = 1 \\
+150 & \textrm{if} &  \vert X_{ij} - Y_{ij} \vert = 2 \\
+225 & \textrm{if} &  \vert X_{ij} - Y_{ij} \vert = 3 

 \end{array}
 \end{array}

-\right.
-$$.
-This function allows to emphase differences between content.
+\right..
+$$
+This function allows to emphasize differences between contents.

 
 \begin{figure}[t]
   \begin{center}
 
 \begin{figure}[t]
   \begin{center}


@@ -338,7 +350,7 @@ This function allows to emphase differences between content.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{emb.pdf}

-          \includegraphics[scale=0.15]{diff7.eps}
+          \includegraphics[scale=0.20]{diff7.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}
         \end{center}
       \end{minipage}
       %\label{fig:sch:emb}


@@ -347,12 +359,14 @@ This function allows to emphase differences between content.
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

-          \includegraphics[scale=0.15]{diff6.eps}
+          \includegraphics[scale=0.20]{diff6.eps}

         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}
         \end{center}
       \end{minipage}
       %\label{fig:sch:ext}
     }%\hfill
   \end{center}

-  \caption{Differences  with Lena's Cover  wrt $b$.}
+  \caption{Differences  with Lena's cover  wrt $b$}

   \label{fig:lenadiff}
 \end{figure}
   \label{fig:lenadiff}
 \end{figure}

+
+