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standard technique for estimating sparse signals is through the formulation of an inverse problem with the @xmath0 norm as convex proxy for sparsity . in particular , consider the problem of estimating a signal @xmath1 from a noisy observation @xmath2 , @xmath3where @xmath4 represents awgn . we assume the underlying signal to be sparse with respect to an overcomplete tight frame @xmath5 , @xmath6 , which satisfies the tight frame condition , i.e. , @xmath7using an analysis - prior , we formulate the signal denoising problem as @xmath8_i ; a_i \right)\biggr\rbrace,\end{aligned}\ ] ] where @xmath9 are the regularization parameters , and @xmath10 is a non - smooth sparsity inducing penalty function . the parameters @xmath11 control the non - convexity of @xmath12 in case it is non - convex . the analysis prior is used in image processing and computer vision applications @xcite . commonly , the @xmath0 norm is used to induce sparsity , i.e. , @xmath13 @xcite . in that case , problem is strictly convex and the global optimum can be reliably obtained . the @xmath0 norm is not the tightest envelope of sparsity @xcite . it under - estimates the non - zero values of the underlying signal @xcite . non - zero values can be more accurately estimated using suitable non - convex regularizers . non - convex regularization in an analysis model has been used for mri reconstruction @xcite , eeg signal reconstruction @xcite , and for computer vision problems @xcite . however , the use of non - convex regularizers comes at a price : the objective function is generally non - convex . consequently , several issues arise ( spurious local minima , a perturbation of the input data can change the solution unpredictably , convergence is guaranteed to the local minima only , etc . ) .    in order to maintain convexity of the objective function while using non - convex regularizers , we propose to restrict the parameter @xmath11 of the non - convex regularizer @xmath12 . by controlling the degree of non - convexity of the regularizer we guarantee that the total objective function @xmath14 is convex . this idea which dates to blake and zisserman @xcite and nikolova @xcite , has been applied to image restoration and reconstruction @xcite , total variation denoising @xcite , and wavelet denoising @xcite .    in this letter we provide a critical value of parameter @xmath15 to ensure @xmath14 in is strictly convex ( even though @xmath12 is non - convex ) . in contrast to the above works , we consider transform domain regularization and prove that admm @xcite applied to the problem converges to the global optimum . the convergence of admm is guaranteed , provided the augmented lagrangian parameter @xmath16 , satisfies @xmath17 . in order to induce sparsity more strongly than the @xmath0 norm , we use non - convex penalty functions @xmath10 parameterized by the parameter @xmath18 . we make the following assumption of such penalty functions . [ theorem::assumption 1 ] the non - convex penalty function @xmath19 satisfies the following    1 . @xmath12 is continuous on @xmath20 , twice differentiable on @xmath21 and symmetric , i.e. , @xmath22 2 . @xmath23 3 . @xmath24 4 . @xmath25 5 . @xmath26 6 . @xmath27 .     and the function @xmath28 , @xmath29 . ]    since @xmath27 , the @xmath0 norm is recovered as a special case of the penalty function @xmath12 . the parameter @xmath15 controls the degree of non - convexity of @xmath12 . note that the @xmath30 norm does not satisfy assumption [ theorem::assumption 1 ] . the rational penalty function @xcite , @xmath31the logarithmic , and the arctangent penalty functions @xcite are examples that satisfy assumption [ theorem::assumption 1 ] . the rational penalty @xmath12 for @xmath32 is shown in fig . [ fig::phi minus absolute ] . the proximity operator of @xmath12 @xcite , @xmath33 , is defined as @xmath34 for @xmath35 satisfying assumption [ theorem::assumption 1 ] , with @xmath36 , the proximity operator is a continuous non - linear threshold function with @xmath37 as the threshold value , i.e. , @xmath38 . the proximity operator of the absolute value function is the soft - thresholding function . there is a constant gap between the identity function and the soft - threshold function due to which the non - zero values are underestimated @xcite . on the other hand , non - convex penalty functions satisfying assumption [ theorem::assumption 1 ] are specifically designed so that the threshold function approaches identity asymptotically . these non - convex penalty functions do not underestimate large values .      in order to benefit from convex optimization principles in solving , we seek to ensure @xmath14 in is convex by controlling the parameter @xmath11 . for later , we note the following lemma . [ theorem::lemma 1 ] let @xmath10 satisfy assumption [ theorem::assumption 1 ] . the function @xmath39 defined as @xmath40is twice continuously differentiable and concave with @xmath41    since @xmath12 and the absolute value function are twice continuously differentiable on @xmath42 , we need only show @xmath43 and @xmath44 . from assumption [ theorem::assumption 1 ] , we have @xmath45 , hence @xmath46 . again by assumption [ theorem::assumption 1 ] we have @xmath47 , hence @xmath48 . further , @xmath49 and @xmath50 . thus the function @xmath51 is twice continuously differentiable . the function @xmath51 is concave since @xmath52 . using assumption [ theorem::assumption 1 ] it follows that @xmath53 .    figure  [ fig::phi minus absolute ] displays the function @xmath54 , which is twice continuously differentiable even though the penalty function @xmath12 is not differentiable . the following theorem states the critical value of parameter @xmath11 to ensure the convexity of @xmath14 in . [ theorem::theorem 1 ] let @xmath55 be a non - convex penalty function satisfying assumption [ theorem::assumption 1 ] and @xmath56 be a transform satisfying @xmath57 , @xmath58 . the function @xmath59 defined in is strictly convex if @xmath60    consider the function @xmath61 defined as @xmath62_i ; a_i ) . \end{aligned}\ ] ] since @xmath63 is twice continuously differentiable ( using lemma [ theorem::lemma 1 ] ) , the hessian of @xmath63 is given by @xmath64where @xmath65_i;a_i\right)$ ] . using , we write the hessian as @xmath66the transform @xmath56 has full column rank , from , hence @xmath67 is positive definite if @xmath68 thus , @xmath67 is positive definite if @xmath69_i;a_i ) > -\dfrac{1}{r\lambda_i}.\end{aligned}\ ] ] using lemma [ theorem::lemma 1 ] , we obtain the critical value of @xmath11 to ensure the convexity of @xmath63 , i.e. , @xmath70it is straightforward that @xmath71_i|.\end{aligned}\ ] ] thus , being a sum of a strictly convex function and a convex function , @xmath14 is strictly convex . note that if @xmath72 , then the function @xmath73 is not convex , as the hessian of @xmath73 is not positive definite . as a result , @xmath74 is the critical value of @xmath11 to ensure the convexity of the function @xmath14 . the following corollary provides a convexity condition for the situation where the same regularization parameter is applied to all coefficients . [ theorem::corollary 1 ] for @xmath75 , the function @xmath14 in is strictly convex if @xmath76 @xmath77    we illustrate the convexity condition using a simple example with @xmath78 . we set @xmath79 , \quad { \mathnormal{a}}^{t}{\mathnormal{a } } = 4{\mathnormal{i } } , \end{aligned}\]]and @xmath80 . theorem [ theorem::theorem 1 ] states that the function @xmath63 defined in is convex for @xmath81 and non - convex for @xmath82 . it can be seen in fig . [ fig::convexity condition ] that the function @xmath63 is convex for @xmath83 , even though the penalty function is not convex . however , when @xmath84 , the function @xmath63 ( hence @xmath14 ) is non - convex . a benefit of ensuring convexity of the objective function is that we can utilize convex optimization approaches to obtain the solution . in particular , for @xmath13 , the widely used methods for solving are proximal methods @xcite and admm @xcite . the convergence of admm to the optimum solution is guaranteed when the functions appearing in the objective function are convex @xcite . the following theorem states that admm can be used to solve with guaranteed convergence , provided the augmented lagrangian parameter @xmath16 is appropriately set . such a condition on @xmath16 was also given in @xcite . note that @xmath16 does not affect the solution to which admm converges , rather the speed at which it converges . .iterative algorithm for the solution to . [ cols= " < " , ]     [ table::algorithm ]    let @xmath12 satisfy assumption [ theorem::assumption 1 ] and the transform @xmath56 satisfy the parseval frame condition . let @xmath85 . the iterative algorithm [ table::algorithm ] converges to the global minimum of the function @xmath14 in if @xmath86    we re - write the problem using variable splitting @xcite as    [ eq::variable split ] @xmath87    the minimization is separable in @xmath88 and @xmath89 . applying admm to yields the following iterative procedure with the augmented lagrangian parameter @xmath16 . [ eq::iterative procedure ] @xmath90    the sub - problem for @xmath88 can be solved explicitly as @xmath91using . the sub - problem for @xmath92 can be solved using @xmath93 , provided the function @xmath94 is convex . consider the function @xmath95 defined as @xmath96 from lemma [ theorem::lemma 1 ] and the proof of theorem [ theorem::theorem 1 ] , @xmath97 is positive definite if @xmath98 since @xmath99 , it follows that @xmath100 is positive definite if @xmath17 . hence @xmath101 is strictly convex for @xmath17 . note that @xmath102 . hence , the function @xmath94 , being the sum of a convex and a strictly convex function , is strictly convex . as such , the minimization problem in is well - defined and its solution can be efficiently computed using the proximity operator of @xmath12 , i.e. , @xmath103_i ; \lambda_i/\mu_i , a_i \bigr).\end{aligned}\ ] ]    since @xmath56 has full column rank , admm converges to a stationary point of the objective function ( despite having a non - convex function in the objective ) @xcite ; see also @xcite . moreover , the function @xmath14 is strictly convex ( by theorem [ theorem::theorem 1 ] ) and the sub - problems of the admm are strictly convex for @xmath17 . as a result , the iterative procedure converges to the global minimum of @xmath14 .    a globally convergent algorithm based on a different splitting is presented in @xcite . in that approach , the objective function is split into two functions , both of which are convex regardless of the auxillary parameter value . hence , no parameter constraint is required to ensure convergence . we consider the problem of denoising a 1d signal that is sparse with respect to the undecimated wavelet transform ( udwt ) @xcite , which satisfies the condition with @xmath104 . in particular , we use a 4-scale udwt with three vanishing moments . the noisy signal is generated using wavelab ( http://www-stat.stanford.edu/%7ewavelab/ ) with awgn of @xmath105 . we set the regularization parameters @xmath106 . we use the same @xmath107 for all the coefficients in scale @xmath108 . the value of @xmath109 is chosen to obtain the lowest rmse for convex and non - convex regularization respectively . to maximally induce sparsity we set @xmath110 . for the 1d signal denoising example , we use the non - convex arctangent penalty and its corresponding threshold function @xcite . for comparison we use reweighted @xmath0 minimization @xcite , with @xmath109 chosen in order to obtain the lowest rmse .         for the 1d signal denoising example . ] figure  [ fig::1d example ] shows that the denoised signal obtained using non - convex regularization has the lowest rmse and preserves the discontinuities . further , the peaks are less attenuated using non - convex regularization in comparison with @xmath0 norm regularization .    for further comparison , we generate the noisy signal in fig . [ fig::1d example ] for @xmath111 , and denoise it with non - convex and convex regularization . we also denoise the noisy signal by direct non - linear thresholding of the noisy wavelet coefficients and by reweighted @xmath0 minimization . we use the same @xmath109 values as in fig . [ fig::1d example ] . the value of @xmath109 for direct non - linear thresholding is also chosen to obtain the lowest rmse . as seen in fig .  [ fig::rmse comparison ] , the non - convex regularization outperforms the three methods by giving the lowest rmse . the rmse values are obtained by averaging over 15 realizations for each @xmath112 . we consider the problem of denoising a 2d image corrupted with awgn . we use the 2d dual - tree complex wavelet transform ( dt - cwt ) @xcite , which is 4-times expansive and satisfies with @xmath113 . the noisy ` peppers ' image has peak signal - to - noise ratio ( psnr ) value of 14.6 db . we use the same @xmath37 for all the sub - bands . as in the previous example , we set the value of @xmath37 for each case ( convex and non - convex ) as a constant multiple of @xmath112 that gives the highest psnr . norm regularization . ]    . ( b ) psnr as a function of @xmath112 . ] figure  [ fig::2d example ] shows that the denoised image ( non - convex case ) contains fewer wavelet artifacts and has a higher psnr . [ fig::psnr values](a ) shows the psnr values ( convex and non - convex ) for different values of @xmath37 . to further assess the performance of tight - frame non - convex regularization , we realize several noisy ` peppers ' images with @xmath114 . as in the case of the 1d signal denoising , fig . 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this letter considers the problem of signal denoising using a sparse tight - frame analysis prior . 
 the @xmath0 norm has been extensively used as a regularizer to promote sparsity ; however , it tends to under - estimate non - zero values of the underlying signal . 
 to more accurately estimate non - zero values , we propose the use of a non - convex regularizer , chosen so as to ensure convexity of the objective function . 
 the convexity of the objective function is ensured by constraining the parameter of the non - convex penalty . 
 we use admm to obtain a solution and show how to guarantee that admm converges to the global optimum of the objective function . 
 we illustrate the proposed method for 1d and 2d signal denoising .