quelques modifs

[canny.git] / ourapproach.tex

diff --git a/ourapproach.tex b/ourapproach.tex
index 5389d35484badac9c346b0c8fc799c513c473de6..36fefc31a4cba7ed397041b6eee2884ba28482bb 100644 (file)
--- a/ourapproach.tex
+++ b/ourapproach.tex
@@ -3,12 +3,13 @@ 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 then 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.

 
 
 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.
+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}).
 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}).


@@ -16,21 +17,20 @@ Let us first focus on the data embedding.
 
 \begin{figure*}%[t]
   \begin{center}
 
 \begin{figure*}%[t]
   \begin{center}

-    \subfloat[Data Embedding.]{
-      \begin{minipage}{0.49\textwidth}
+    \subfloat[Data Embedding]{
+      \begin{minipage}{0.4\textwidth}

         \begin{center}
         \begin{center}

-          %\includegraphics[width=5cm]{emb.pdf}
-          \includegraphics[scale=0.45]{emb.ps}
+            %\includegraphics[scale=0.45]{emb}
+            \includegraphics[scale=0.4]{emb}

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

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

       \begin{minipage}{0.49\textwidth}
         \begin{center}
       \begin{minipage}{0.49\textwidth}
         \begin{center}

-          %\includegraphics[width=5cm]{rec.pdf}
-          \includegraphics[scale=0.45]{rec.ps}
+            \includegraphics[scale=0.4]{dec}

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


@@ -48,13 +48,17 @@ Let us first focus on the data embedding.
 
 
 \subsection{Security considerations}\label{sub:bbs}
 
 
 \subsection{Security considerations}\label{sub:bbs}

-Among methods of the message encryption/decryption 
+To provide a self-contained article without any bias, we shor\-tly 
+present the selected encryption process.
+Among the methods of  message encryption/decryption 

 (see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)
 (see~\cite{DBLP:journals/ejisec/FontaineG07} for a survey)

-we implement the Blum-Goldwasser cryptosystem~\cite{Blum:1985:EPP:19478.19501}
+we implement the asymmetric 
+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.
 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 
+The main justification of this choice 
+is that 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 


@@ -89,7 +93,7 @@ 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.


@@ -97,30 +101,33 @@ As far as fuzzy edge methods are concerned, they are obviously based on fuzzy lo
 
 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
 
 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, smartphones,  desktop machines, and
+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}
+

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

-More precisely, only $b$ most 
-significant bits are concerned by this step, where 
-the parameter $b$ is practically set with $6$ or $7$. 
+More precisely, only $b$ most significant bits are concerned by this step, 
+where the parameter $b$ is practically set with $6$ or $7$. 
+Notice that 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.

 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.   
 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, 
-and the LSB of pixels if $b$ is 7.
-
-
-
+Moreover, to provide edge gradient value, 
+the Canny algorithm computes derivatives  
+in the two directions with respect to a mask of size $T$. 
+The higher $T$ is, the coarse the approach is. Practically, 
+$T$ is set with $3$, $5$, or $7$.
+In our flowcharts, this step is represented by ``Edge Detection(b, T, X)''.

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

-The next  section presents 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 
+with respect to the size
+of the message $m$ to embed and the size of the cover $x$.
+

 
 
  
 
 
  


@@ -129,61 +136,47 @@ adapts  when the size of $x$  is not sufficient for the message $m$ to embed.
 
 
 \subsection{Adaptive embedding rate}\label{sub:adaptive}
 
 
 \subsection{Adaptive embedding rate}\label{sub:adaptive}

-Two strategies have been developed in our scheme, 
-depending on the embedding rate that is either \emph{adaptive} or \emph{fixed}.
+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 
 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 a high threshold.
-The message length is thus defined to be less than 
+Canny algorithm with parameters $b=7$ and $T=3$.
+The message length is thus defined to be lesser than 

 half of this set cardinality.
 half of this set cardinality.

-If $x$ is then too short for $m$, the message is split 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. 
 
 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  
 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 
+a set of edge pixels related to increasing values of $T$ and 

 until its cardinality
 until its cardinality

-is sufficient. 
-
+is sufficient. Even in this situation, our scheme adapts
+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 changed. 

 The first one randomly chooses the subset of pixels to modify by 
 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 first one randomly chooses the subset of pixels to modify by 
 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 is a direct application of the 
-STC algorithm~\cite{DBLP:journals/tifs/FillerJF11}. 
+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.
 
 
 
 
 
 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-trellis codes}\label{sub:stc}
+\subsection{Minimizing distortion with Syndrome-Trellis Codes}\label{sub:stc}

 \input{stc}
 
 
 \input{stc}
 
 


@@ -205,14 +198,26 @@ It  is further referred to as \emph{STC} and is detailed in the next section.
 % but the whole approach can be updated to consider 
 % the fuzzy logic edge detector.   
 
 % but the whole approach can be updated to consider 
 % the fuzzy logic edge detector.   
 

-% Next, following~\cite{Luo:2010:EAI:1824719.1824720}, our scheme automatically
-% modifies Canny parameters to get a sufficiently large set of edge bits: this 
-% one is practically enlarged untill its size is at least twice as many larger 
-% than the size of embedded message.
-
-
-
-%%RAPH: paragraphe en double :-)
+For a given set of parameters, 
+the Canny algorithm returns a numerical value and
+states whether a given pixel is an edge or not.
+In this article, in the Adaptive strategy 
+we consider that all the edge pixels that 
+have been selected by this algorithm have the same
+distortion cost, \textit{i.e.}, $\rho_X$ is always 1 for these bits. 
+In the Fixed strategy, since pixels that are detected to be edge
+with small values of $T$ (e.g., when $T=3$) 
+are more accurate than these with higher values of $T$,
+we give to STC the following distortion map of the corresponding bits
+$$  
+\rho_X= \left\{ 
+\begin{array}{l}
+1 \textrm{ if an edge for $T=3$,} \\
+10 \textrm{ if an edge for $T=5$,} \\
+100 \textrm{ if an edge for $T=7$.}
+\end{array}
+\right.
+$$

 
 
 
 
 
 


@@ -222,11 +227,13 @@ 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.
 
 follows the data embedding approach 
 since there exists a reverse function for all its steps.
 

-More precisely, the same edge detection is applied on the $b$ first bits  to 
+More precisely,  let $b$ be the most significant bits and 
+$T$ be the size of the Canny mask, both be given as a key.
+Thus, the same edge detection is applied on a stego content $Y$ 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. 
 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.
+This inverse function takes the $\hat{H}$ 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.
 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.


@@ -245,21 +252,21 @@ Lena and the first verses are given in Fig.~\ref{fig:lena}.
 \begin{center}
 \begin{minipage}{0.49\linewidth}
 \begin{center}
 \begin{center}
 \begin{minipage}{0.49\linewidth}
 \begin{center}

-\includegraphics[scale=0.20]{Lena.eps}
+\includegraphics[scale=0.20]{lena512}

 \end{center}
 \end{minipage}
 \begin{minipage}{0.49\linewidth}
 \begin{flushleft}
 \begin{scriptsize}
 The skies they were ashen and sober;\linebreak
 \end{center}
 \end{minipage}
 \begin{minipage}{0.49\linewidth}
 \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}


@@ -268,8 +275,11 @@ $~$ In the ghoul-haunted woodland of Weir.
 \end{figure}
 
 The edge detection returns 18,641 and 18,455 pixels when $b$ is
 \end{figure}
 
 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}.
-
+respectively 7 and 6 and $T=3$.
+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}


@@ -277,7 +287,7 @@ respectively 7 and 6. These edges are represented in Figure~\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=0.20]{edge7.eps}
+          \includegraphics[scale=0.20]{edge7}

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


@@ -286,22 +296,31 @@ respectively 7 and 6. These edges are represented in Figure~\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=0.20]{edge6.eps}
+          \includegraphics[scale=0.20]{edge6}

         \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 $b$ with $T=3$}

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

-Only 9,320 bits (resp. 9,227 bits) are available for embedding 
-in the former configuration where $b$ is 7 (resp. where $b$ is 6).
-In both cases, about the third part of the poem is hidden into the cover.
-Results with \emph{adaptive+STC} strategy are presented in 
+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 {Adaptive} and {STC} strategies are presented in 

 Fig.~\ref{fig:lenastego}.
 
 \begin{figure}[t]
 Fig.~\ref{fig:lenastego}.
 
 \begin{figure}[t]


@@ -310,7 +329,7 @@ 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=0.20]{lena7.eps}
+          \includegraphics[scale=0.20]{lena7}

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


@@ -319,7 +338,7 @@ 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=0.20]{lena6.eps}
+          \includegraphics[scale=0.20]{lena6}

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


@@ -337,13 +356,19 @@ $$
 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 \\
-150 & \textrm{if} &  \abs{ X_{ij} - Y_{ij}} = 2 \\
-225 & \textrm{if} &  \abs{ X_{ij} - Y_{ij}} = 3 
+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..
+\right.

 $$
 This function allows to emphasize differences between contents.
 $$
 This function allows to emphasize differences between contents.

+Notice that since $b$ is 7 in Fig.~\ref{fig:diff7}, the embedding is binary 
+and this image only contains 0 and 75 values.
+Similarly, if $b$ is 6 as in Fig.~\ref{fig:diff6}, the embedding is ternary 
+and the image contains all the values in $\{0,75,150,225\}$.
+
+

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


@@ -351,21 +376,24 @@ This function allows to emphasize differences between contents.
       \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=0.20]{diff7.eps}
+          \includegraphics[scale=0.20]{diff7}

         \end{center}
       \end{minipage}
         \end{center}
       \end{minipage}

-      %\label{fig:sch:emb}
+      \label{fig:diff7}

     }%\hfill
     \subfloat[$b$ is 6.]{
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}
     }%\hfill
     \subfloat[$b$ is 6.]{
       \begin{minipage}{0.49\linewidth}
         \begin{center}
           %\includegraphics[width=5cm]{rec.pdf}

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

         \end{center}
       \end{minipage}
         \end{center}
       \end{minipage}

-      %\label{fig:sch:ext}
+      \label{fig:diff6}

     }%\hfill
   \end{center}
   \caption{Differences  with Lena's cover  wrt $b$}
   \label{fig:lenadiff}
 \end{figure}
     }%\hfill
   \end{center}
   \caption{Differences  with Lena's cover  wrt $b$}
   \label{fig:lenadiff}
 \end{figure}

+
+
+