Jérôme Fehrenbach

Pierre Weiss

Corinne Lorenzo

VSNR: THE VARIATIONAL STATIONARY NOISE REMOVER 1 Variational algorithms to remove stationary noise. Applications to microscopy imaging

Keywords: Stripe removal, non linear filtering, fast algorithms, total variation, Gabor filter, SPIM microscope, scanning electron microscope, nanotomography, texture-geometry decompositions

A framework and an algorithm are presented in order to remove stationary noise from images. This algorithm is called VSNR (Variational Stationary Noise Remover). It can be interpreted both as a restoration method in a Bayesian framework and as a cartoon+texture decomposition method. In numerous denoising applications the white noise assumption fails: structured patterns (e.g. stripes) appear in the images. The model described here addresses these cases. Applications are presented with images acquired using different modalities: scanning electron microscope, FIB-nanotomography, and an emerging fluorescence microscopy technique called SPIM (Selective Plane Illumination Microscope).

I. INTRODUCTION

M Any imaging modalities tend to impair images with structured noise. For instance, stripes appear in images produced by recent microscopes such as Single Plane Illumination Microscope (SPIM) [START_REF] Huisken | Optical Sectioning Deep Inside Live Embryos by Selective Plane Illumination Microscopy[END_REF], Atomic Force Microscope (AFM), scanning electron microscopy (SEM) [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF], [START_REF] Kienberger | Improving the contrast of topographical AFM images by a simple averaging filter[END_REF], Synchrotron X-Ray Tomographic images, Focused Ion Beam nanotomography [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF], or in remote sensing imaging such as MODerate-resolution Imaging Spectroradiometer (MODIS) images or digital elevation models [START_REF] Chen | Destriping CMODIS Data by Power Filtering[END_REF], [START_REF] Chen | Oblique Striping Removal in Remote Sensing Imagery Based on Wavelet Transform[END_REF]. Figure 1 shows examples of stripes in images obtained using different modalities.

The main purpose of this work is to propose a simple noise model which describes accurately the kind of distortions appearing in Figure 1. This statistical modeling allows to formulate the restoration problem in a Bayesian framework. It leads to the resolution of a large scale convex programming problem which is solved using recent advances in numerical optimization. We call our algorithm VSNR which stands for Variational Stationary Noise Remover.

As far as we know, few works address this restoration problem. Recently, [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF] proposed a combined wavelet and Fourier method which allows to interpolate and restore wavelet coefficients impaired by stripes. In [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF], a Fourier filtering is proposed, where noisy Fourier coefficients are first detected using statistical tests. Contrarily to our work, these approaches do not rely on a precise statistical modeling of the noise and incorporate prior information on the images only implicitly.

The general image formation model we consider can be stated as follows:

u 0 = Hu + b
Jérôme Fehrenbach and Pierre Weiss are with Université de Toulouse, the IMT-UMR5219 and with ITAV-UMS3039, France.

Corinne Lorenzo is with Université de Toulouse and ITAV-UMS3039, France. where u 0 is the observed digital image, H is a deterministic linear operator that models the acquisition process (sampling, convolution, indirect measurements, ...) and b is a noise or a texture. In restoration applications, the objective is to retrieve u knowing the observation u 0 and the observation operator H. In texture+cartoon decomposition, H is the identity operator, and the objective is to retrieve both the cartoon component u and the texture component b.

A standard assumption in the literature is that the noise is white, meaning that the different components of b are independent random variables of finite variance. Though this noise assumption models accurately some real applications, it fails to do so in many scenarii. In the case of image restoration/denoising problems, it often leads to unsatisfactory results.

In this article we propose a noise model that suits more complex frameworks and that can be handled numerically in reasonable computing times. We replace the white noise assumption by a stationary noise assumption. A stationary noise b can be defined as a stochastic process having the following translation invariance property:

∀τ ∈ Z 2 , p(b) = p(T τ (b)) (1)
where p denotes the probability density function and T τ is the translation of vector τ with periodic boundary conditions on the image domain. This hypothesis is natural in many applications, since it amounts to assume that there is no a-priori knowledge about the location of any feature or noise pattern in the image. However, the sole hypothesis of stationarity appears to be hardly numerically tractable. In this work, we restrict to a subclass of stationary stochastic processes, that can be easily described in the frequency domain: these processes are generated by convolving white noise with a given kernel. The noise thus appears as "structured" in the sense that some pattern might be visible, see Figure 3.

The proposed model shares many similarities with the negative norm models of Y. Meyer [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF], its successors [START_REF] Aujol | Structure-Texture Image Decomposition -Modeling, Algorithms, and Parameter Selection[END_REF] and the decomposition algorithms proposed in [START_REF] Fadili | Image decomposition and separation using sparse representations: an overview[END_REF]. Meyer's idea is to decompose an image into a piecewise smooth component and an oscillatory component. We refer to [START_REF] Aujol | Structure-Texture Image Decomposition -Modeling, Algorithms, and Parameter Selection[END_REF] for a review of the principles, algorithms and results using this approach. An alternative way of decomposing images was proposed in [START_REF] Fadili | Image decomposition and separation using sparse representations: an overview[END_REF]. The idea is to seek for components that are sparse in given dictionaries. Different choices for the elementary atoms composing the dictionary allow to recover different kind of textures.

The main contributions of the present paper are: 1) We propose a simple class of random processes that describes real-life noises and textures. 2) Similarly to [START_REF] Fadili | Image decomposition and separation using sparse representations: an overview[END_REF], the texture is described through a dictionary. In our work each dictionary is composed of a single pattern shifted in space, ensuring translation invariance and fast computation in Fourier domain.

3) The proposed formalism and algorithms can be used when dealing with inverse problems such as deblurring or sampling with indirect measurements. 4) A Bayesian approach is provided in order to take into account the statistical nature of textures precisely. 5) The decomposition problem is recast into a convex optimization problem that is solved with a recent algorithm of A. Chambolle and T. Pock [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] allowing to obtain results in an interactive time. 6) We propose a C and Matlab implementation on our webpage in the spririt of reproducible research http: //www.math.univ-toulouse.fr/ ∼ weiss/PageCodes.html. The outline of this paper is as follows. In section II, we introduce the notation. Section III contains detailed definitions and some elementary properties of the noise model. Section IV presents a restoration or decomposition model based on the maximum a posteriori (MAP) estimator. Section V details an efficient numerical scheme based on [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] to solve the resulting convex programming problems. In section VI we present some applications and results on synthetic and real images.

II. NOTATION

In all the paper n = n x × n y will refer to the number of pixels of the degraded image. m stands for the number of filters used to describe the noise. Let u ∈ R nx×ny be an image.

u(x) denotes the pixel of coordinates x ∈ {1, 2, • • • , n x } × {1, 2, • • • , n y }. u p refers to the standard p -norm of u. Let (u 1 , u 2 ) ∈ R n × R n the dot product is denoted u 1 , u 2 = u T 1 u 2 . Let Λ = R n×m and Q = R n×2
. These spaces are endowed with inner products •, • Λ and •, • Q . We set q Q = q, q Q and λ Λ = λ, λ Λ . Let A : Λ → Q be a linear operator. The adjoint A * of A is defined by:

Aλ, q Q = A * q, λ Λ , ∀(λ, q) ∈ Λ × Q. ( 2
)
The norm of the operator A is defined by:

| A | = max λ Λ ≤1
Aλ Q (3) * is the convolution operator with periodic boundary conditions. F and F -1 respectively denote the discrete Fourier and inverse Fourier transforms. We will also use the notation û = Fu. Finally, ∇ denotes the discrete gradient operator. In this article we use the following discretization:

∇u = (∂ x u, ∂ y u)
where

(∂ x u)(i, j) = u(i + 1, j) -u(i, j) if i < n x 0 if i = n x (4)
(∂ y u)(i, j) = u(i, j + 1) -u(i, j) if j < n y 0 if j = n y . (5)
Let φ : Λ → R ∪ {+∞} be a convex, closed function with non-empty domain. The domain of φ is defined by dom(φ) = {λ ∈ Λ, φ(λ) < +∞}. φ * refers to the Fenchel conjugate of φ defined by:

φ * (η) = sup λ∈Λ η, λ Λ -φ(λ). (6)
The sub-differential of φ at λ 1 is the set defined by:

∂φ(λ 1 ) = {η ∈ Λ, φ(λ 2 ) ≥ φ(λ 1 )+ η, λ 2 -λ 1 Λ , ∀λ 2 ∈ Λ}.
A function φ is said to be strongly convex of parameter γ > 0, if the following inequality holds for all λ 1 , λ 2 ∈ dom(φ) and for all η ∈ ∂φ(λ 1 ):

φ(λ 2 ) ≥ φ(λ 1 ) + η, λ 2 -λ 1 Λ + γ 2 λ 2 -λ 1 2 Λ ,
Let Ξ ⊆ Λ be a set. The indicator function of Ξ is denoted χ Ξ and defined by:

χ Ξ (x) = 0 if x ∈ Ξ +∞ otherwise .
The resolvent or proximal operator of φ at point λ is defined by:

(Id + ∂φ) -1 (λ) = arg min λ ∈Λ φ(λ ) + 1 2 λ -λ 2 Λ .
It can be seen as a generalization of the projection operator since (Id + ∂χ Ξ ) -1 (λ) is the projection of λ on the set Ξ. We refer to [START_REF] Rockafellar | Convex Analysis[END_REF] for a detailed presentation of the above tools.

III. NOISE MODEL

A. A class of stationary noises

In this paper, we address the class of stationary noises that satisfy the following hypothesis.

Hypothesis 3.1: We assume that the noise b reads:

b = m i=1 λ i * ψ i , (7)
where:

• each ψ i ∈ R n is a known elementary pattern,

• The λ i 's are independent realizations of white noise processes with known probability density functions. Proposition 3.1: This class of noises is stationary -in the sense that it satisfies property [START_REF] Arrenberg | Optogenetic control of cardiac function[END_REF]. The proof of this proposition can be found in [START_REF] Shiryaev | Probability Second Edition[END_REF], page 404.

B. Rationale for the noise model

The images presented in Figure 1 are corrupted by parallel stripes. A noise removal algorithm should be able both to locate their position and to estimate their intensity. Such a perturbation b can be modeled by the following equation

b = λ * ψ, (8)
where ψ is an elementary pattern, similar to a stripe, and λ describes the pattern positions and intensities.

In realistic applications, the noise is seldom describable simply as in Equation [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. A more realistic noise description would be the sum of processes of type [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. For instance, in SPIM imaging in addition to stripes, Poisson or Gaussian white noise appears due to the imperfections of the imaging system. The Gaussian white noise is a special case of (8) where ψ is a Dirac function and λ is a sample of a Gaussian white noise process.

The common feature of the stripes is their orientation θ, as well as their dimensions (width and length) that belong to a restricted range. In this paper, we simply model a stripe as an anisotropic Gaussian function defined by :

ψ(x, y) = exp - x 2 σ 2 x - y 2 σ 2 y ,
where x = x cos θ + y sin θ and y = -x sin θ + y cos θ. In order to describe stripes of different width or length, several filters ψ 1 , . . . , ψ m with different geometrical parameters can be used. IV. RESTORATION/DECOMPOSITION MODEL Throughout this paper, we assume the following image formation model:

u 0 = Hu + b (9)
where H is a linear operator and b is independent of u and satisfies hypothesis 3.1. In the numerical results, we will concentrate on the case H = Id, however all the proposed algorithms extend to an arbitrary operator H = Id.

A. A MAP reconstruction approach

In this work we assume that the samples λ = {λ i } i=1..m are drawn from stochastic processes with known probability density functions. The prior probability on the space of images is denoted p(u). The maximum a posteriori (MAP) approach leads to maximize:

p(λ, u|u 0 ) = p(u 0 |λ, u)p(λ, u) p(u 0 ) = p(u 0 |λ, u)p(λ)p(u) p(u 0 ) .
Taking the log of the conditional probability leads to the following optimization problem:

Find λ * ∈ Arg min λ∈Λ -log p(u 0 |λ, u) -log p(λ) -log p(u).
(10) As we assumed independence of the λ i s,

-log p(λ) = m i=1 -log p(λ i ) and p(u 0 |λ, u) = p(Hu + m i=1 λ i * ψ i |λ, u) = δ Hu + m i=1 λ i ψ i -u 0 ,
where δ denotes the Dirac mass.

In order to specify problem [START_REF] Facchinei | Finite-dimensional variational inequalities and complementarity problems[END_REF], we need to define the densities p(u) (image prior) and p(λ) (noise prior).

Remark 4.1: In some applications the independence of u and λ is questionable and a multiplicative model could be considered. However our numerical experiments using multiplicative models did not improve the results sensibly compared to the additive models and required heavier computations.

B. Image prior

In this work, we use the standard assumption that images are smooth or piecewise smooth. This can be promoted by a p.d.f. of the form

p(u) ∝ exp(-∇u 1, ). (11)
where

• 1, : R n×2 → R q → q 1, = x φ ( q(x) 2 )
and

φ (t) = t 2 2 if |t| ≤ |t| -2 otherwise.
This function can be rewritten using duality as [START_REF] Weiss | Efficient schemes for total variation minimization under constraints in image processing[END_REF]:

||z|| 1, := max q ∞ ≤1 z, q - 2 q 2 2 .
The interest of this prior is twofold:

• by choosing = 0, • 1,0 corresponds to the standard isotropic 1 -norm R n×2 and lim

→+∞ • 1, = • 2 2
. This formalism thus captures the standard 1 (total variation) and 2 (Tikhonov) regularization.

• by setting = 0, the numerical scheme designed to solve [START_REF] Facchinei | Finite-dimensional variational inequalities and complementarity problems[END_REF] converges faster than for = 0 and may provide better results in some applications.

C. Noise priors

The p.d.f. p(λ i ) in equation ( 10) still needs to be defined in order to specify the optimization problem completely. The p.d.f. should be chosen depending on the noise nature. In the experiments of this paper we consider three cases:

• Uniform noise. It is obtained by setting p(λ i ) ∝ 1 if λ i ∞ ≤ α i 0 otherwise. .
This assumption allows to model bounded noises. See right images in Figure 3. • Gaussian noise. This hypothesis consists in setting

p(λ i ) ∝ exp(-α i λ i 2
2 ). The corresponding noise component b i = λ i * ψ i is then a colored Gaussian noise. Its power spectral density is given by the Fourier transform of ψ i . See middle images in Figure 3.

• Laplace noise. It is obtained by setting p(λ i ) ∝ exp(-α i λ i 1 )
. This distribution is known to be a convex approximation of Bernoulli processes [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF], [START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF]. See left images in Figure 3.

All these noise priors are generalized Gaussian distributions and their distribution function writes:

p(λ i ) ∝ exp(-φ i (λ i )). ( 12
)
Remark 4.2: The middle and right columns in Figure 3 look very similar. This can be explained using Lindeberg-Feller theorem (an extension of the central limit theorem). It asserts that under suitable hypotheses, a weighted sum of independent variables tends to a normal distribution. This remark has an important practical consequence: in most applications, it is convenient to approximate λ by a Gaussian process. Indeed, the numerical algorithms we develop have a convergence rate of order O(1/k 2 ) -where k is the iteration counter -for Gaussian priors, while it is only of order O(1/k) for the others (see proposition 5.3).

D. Resulting optimization problems

By replacing formulas ( 11) and ( 12) in ( 10) and denoting λ = {λ i } m i=1 , we obtain the following optimization problem:

Find (λ, u) ∈ arg min ∇u 1, + m i=1 φ i (λ i ) subject to u ∈ R n , λ ∈ Λ Hu + m i=1 λ i * ψ i = u 0 (13) In the case H = Id it simplifies to: Find λ ∈ Arg min λ∈Λ P (λ), (14)
where

P (λ) = ∇ u 0 - m i=1 λ i * ψ i 1, + m i=1 φ i (λ i ). ( 15
)
Remark 4.3: We show in [START_REF] Fehrenbach | Variational algorithms to remove stripes: a generalization of the negative norm models[END_REF] that the proposed model provides a Bayesian interpretation and generalizes the negative norm texture+cartoon decomposition models [START_REF] Aujol | Structure-Texture Image Decomposition -Modeling, Algorithms, and Parameter Selection[END_REF], [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF] in the discrete setting.

V. NUMERICAL ALGORITHMS

In the case of Tikhonov regularization (which corresponds to

φ i (λ) = αi 2 λ i 2 2 and J R (u) = ∇u 2 2 )
, Problem ( 14) can be solved exactly in O(m 3 n log n) operations using Fourier transforms and inversion of small m × m linear systems. In the more general case, it is impossible to get an exact solution. The objective of this section is to design iterative methods that lead to approximate solutions of [START_REF] Keller | Fast, high-contrast imaging of animal development with scanned light sheet-based structured-illumination microscopy[END_REF].

A. Problem relaxation

Since problem ( 14) cannot be solved exactly, we are interested in finding an -solution defined by: Find λ ∈ Λ such that P (λ ) -P (λ * ) ≤ .

(

) 16
An iterative optimization algorithm defines a sequence (λ k ) k∈N that converges to a solution of [START_REF] Keller | Fast, high-contrast imaging of animal development with scanned light sheet-based structured-illumination microscopy[END_REF]. The resolution of [START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF] amounts to define an appropriate stopping criterion. Since P (λ * ) is unknown, P (λ k ) -P (λ * ) cannot be evaluated, but this quantity can be bounded by the duality gap defined in equation ( 23) [START_REF] Hiriart-Urruty | Convex Analysis and Minimization Algorithms[END_REF]. In order to prevent the duality gap to be infinite (corresponding to an infeasible dual variable), we add a box constraint to problem [START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF]. This extra constraint also allows to obtain convergence rates, see Proposition 5.3.

To summarize, the constrained problem we consider reads :

Find λ ∈ Ξ such that P (λ ) -P (λ * ) ≤ .

where

Ξ = λ ∈ R n×m , λ ∞ ≤ C
and where λ * is one exact solution of ( 14) and C is large enough (e.g. C ≥ λ * ∞ ). Under the latter condition, the extra box-constraint is inactive and the solutions of ( 17) are the same as those of ( 16).

B. Reformulation as a saddle-point problem

This section is devoted to the resolution of [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF]. The extension to problems (13) is straightforward. In order to apply the ideas presented in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], we reformulate (17) as a saddlepoint problem.

To simplify the reading, let us first introduce some notation. A is the following linear operator:

A : Λ → Q λ → ∇ ( m i=1 λ i * ψ i ) (18)
We detail a procedure to compute the highest singular value L of A in appendix B. By denoting

F (g) = ∇u 0 -g 1, (19)
and

G(λ) = m i=1 φ i (λ i ) + χ Ξ (λ), (20)
problem ( 17) can be recast as the following convex-concave saddle-point problem:

min λ∈Ξ max q∈Q Aλ, q Q -F * (q) + G(λ). (21)
The interest of this saddle-point reformulation is twofold:

• It allows the use of primal-dual algorithms which are known as being robust and efficient. • It will allow to define a duality gap, which will provide a reliable stopping criterion for the iterative algorithm.

C. Elementary properties of the problem

By inverting the minimum and the maximum in ( 21) we obtain the following dual problem :

max q∈Q min λ∈Ξ λ, A * q Q -F * (q) + G(λ). (22)
Let us denote:

P (λ) = max q∈Q λ, A * q Λ -F * (q) + G(λ) = F (Aλ) + G(λ), D(q) = min λ∈Λ Aλ, q Q -F * (q) + G(λ) = -F * (q) -G * (-A * q) and ∆(λ, q) = P (λ) -D(q). (23)
The letter P stands for Primal and D stands for Dual. By using standard results of convex programming, we prove that:

Theorem 5.1 (Characterization of problem (17) solutions):

1) Problem ( 17) admits a convex, non-empty set of solutions. 2) Problem ( 21) admits a non-empty set of saddle points.

3) Let (λ * , q * ) be a saddle point of [START_REF] Nesterov | Introductory Lectures on Convex Optimization: A Basic Course[END_REF]. It satisfies ∆(λ * , q * ) = 0 and λ * is a solution of problem [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF]. 4) Finally, for any (λ, q) ∈ Ξ × R n×2 , ∆(λ, q) ≥ P (λ) -P (λ * ).

The last item in this theorem indicates that the duality gap might be used as a stopping criterion for an iterative algorithm: it is an upper bound of the difference between the objective function and the minimum.

Proof: Points (1) and ( 2) result from boundedness of the set Ξ × dom(F * ) and convexity of the functionals. Points (3) and ( 4) are standard results of convex analysis [START_REF] Rockafellar | Convex Analysis[END_REF].

The following propositions will be useful in order to determine the algorithm step sizes.

Proposition 5.1: In the standard Euclidean metric, if > 0, then F * is strongly convex with parameter .

Proposition 5.2: In the standard Euclidean metric, if

φ i (λ i ) = αi 2 λ i 2
2 for all i ∈ {1..m}, then G is strongly convex with parameter:

γ = min i∈{1..m} α i (24)

D. A primal-dual algorithm

Indeed, Nesterov paved the way to the development of new efficient algorithms in ( [START_REF] Nemirovski | Information-based complexity of linear operator equations[END_REF], [START_REF] Nesterov | Smooth minimization of non-smooth functions[END_REF]). From a theoretical point of view, these algorithms are shown to be optimal in the class of first order methods and they outperform second order algorithms -like interior point methods -for large scale problems and moderate accuracy. Among all the papers published recently on this topic, [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] is probably the most versatile and we decided to present and implement this strategy.

The algorithm proposed by Chambolle and Pock in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] applied to problem [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF], can be written as follows:

Algorithm 1: Primal-Dual algorithm [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] Input: : the desired precision; ( λ0 , q 0 ): a starting point; σ 0 , τ 0 such that σ 0 τ 0 L 2 < 1; Output: λ : a solution to problem [START_REF] Meyer | Oscillating Patterns in Image Processing and Nonlinear Evolution Equations[END_REF]. begin k=0; while ∆(λ k , q k ) > ∆(λ 0 , q 0 ) do

q k+1 = (Id + σ k ∂F * ) -1 (q k + σ k A λk ) λ k+1 = (Id + τ k ∂G) -1 (λ k -τ k A * q k+1 ) (τ k+1 , σ k+1 , θ k ) =Update(τ k , σ k , , γ, L) λk+1 = λ k+1 + θ k (λ k+1 -λ k ) k = k + 1; end end
The calculation of the resolvents of F * and G are described in the appendix. Following [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], the Update function should be implemented according to the following rules:

Algorithm 2:

Step size update rules Input: ε: parameter of strong convexity of F * γ: parameter of strong convexity of G (see eq. ( 24)) L: highest singular value of G σ k and τ k : step sizes at iteration k Output:

θ k , σ k+1 and τ k+1 . begin if φ i (•) = αi 2 • 2 2 for all i ∈ {1..m} then if ε > 0 then
The sequence ((τ k , σ k )) k∈N is constant:

µ = 2 √ γε/L τ k+1 = µ 2 * γ σ k+1 = µ 2 * ε θ k = 1 1+µ else θ k = 1 √ 1+2γτ τ k+1 = θ k τ k σ k+1 = σ k θ k end else
The sequence ((τ k , σ k )) k∈N is constant:

θ k = 1 τ k+1 = τ k σ k+1 = σ k end end
The convergence properties of this algorithm are summarized below.

Proposition 5.3: The sequence (λ k , q k ) k∈N converges to a saddle-point of [START_REF] Nesterov | Introductory Lectures on Convex Optimization: A Basic Course[END_REF]. Moreover it guarantees the following convergence rates:

• If = 0 and a function φ i is non quadratic then:

∆(λ k , q k ) = O L k
• If = 0 and all functions φ i are squared 2 -norms:

∆(λ k , q k ) = O L k 2 and λ k -λ * 2 2 = O L 2 k 2
• If > 0 and all functions φ i are 2 -norms, the convergence is at least linear:

∆(λ k , q k ) = O ω k/2 and λ k -λ * 2 2 = O ω k/2 with ω = 1 1+ √ γ L .
The proof of this proposition is a straightforward application of results in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF]. These convergence rates can be shown to be optimal for first order methods [START_REF] Nemirovski | Information-based complexity of linear operator equations[END_REF], [START_REF] Nesterov | Introductory Lectures on Convex Optimization: A Basic Course[END_REF].

Remark 5.1: As Chambolle-Pock's method is first-order, the algorithm depends on the choice of the inner products on the primal and dual spaces. It is possible to define different inner products •, • Λ and •, • Q , leading to different iterative methods. This remark can be used to precondition the problem and obtain faster convergence rates, see [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF] and paragraph VI-C for more details.

VI. RESULTS

In this section we first present experimental results on images impaired by synthetic noise. We perform comparisons between the proposed algorithm and two recent works dedicated to stripes removal [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF], [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. We also present an example where the noise is synthetized as the sum of m = 2 stationary processes. This illustrates our algorithm's ability to cope with more complicated noises. We then present results on real images acquired using different modalities: FIB-nanotomography, SEM and SPIM. We finish the section by showing the experimental convergence behavior of the algorithm.

A. Synthetic noise examples

In order to evaluate the performance of the proposed VSNR algorithm, we synthetise noise according to model [START_REF] Chen | Oblique Striping Removal in Remote Sensing Imagery Based on Wavelet Transform[END_REF] and add it to Figure 4. In the first experiment, we consider m = 1 noise component, the filter ψ 1 is a straight vertical line running over the entire vertical direction and λ 1 is the realization of a Gaussian white noise process with different variances. The noisy images are then restored using three algorithms: VSNR, the DeStripe algorithm available on http://biodev.cea.fr/destripe/ [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF], and the wavelet-FFT algorithm [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. The wavelet-FFT algorithm implementation was kindly provided by Dr. Beat Münch as a Fiji plugin.

We compare the results of the different restoration algorithms using a rescaled SNR measure (denoted SNRr) defined as follows: . This measure provides the best SNR among all possible affine transforms of the grey levels of u 0 . We chose this measure since the wavelet-FFT algorithm tends to decrease the contrasts.

SN Rr(u 0 , u) = SN R( u 0 , u) = 10 log 10 u 2 2 u 0 -u 2
The results are shown in Table I and in Figure 5. VSNR outperforms the two other methods by a few dBs. Note that when the noise level is low (first line of Table I), both the Destripe and wavelet-FFT methods tend to degrade images. This is due to the fact that the noise description is too rough and this affects signal recovery. In the second experiment, we consider m = 1 noise component, the filter ψ 1 is an elongated Gaussian and λ 1 is the realization of a Gaussian white noise process with different variances. The results are shown in table II and in Figure 6. Once again, VSNR clearly outperforms the two other methods. Figure 6 shows that the algorithm recovers most of the signal even when the noise level is higher than the signal level. In the third experiment, we consider m = 2 noise components, the first filter ψ 1 is a radial sinc function, the second filter ψ 2 is an elongated Gaussian in the vertical direction, λ 1 is a sample of a white Gaussian process and λ 2 is a sample of a white Bernoulli process with low probability. The noisy image is shown in Figure 7 and the decomposition result is shown in Figure 8. [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. Bottom-right: denoised using Wavelet-FFT [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. This experiment corresponds to the last line of Table I. The noise pattern is a vertical line. The Destripe and Wavelet-FFT algorithms do not completely remove low frequencies while VSNR leads to near perfect results.

B. Real images

In order to assess the efficiency of VSNR algorithm, we perform tests on images coming from different modalities:

• FIB-nanotomography. This image was used in [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. Results are shown in Figure 9. • SEM imaging. This image was used in [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. Results are shown in Figure 10. • SPIM imaging. Results are shown in Figure 11 and12.

The images come from a microscope prototype developed at ITAV-UMS3039 (CNRS). In all experiments, algorithm 1 is applied with m = 2 filters ψ 1 and ψ 2 . The first filter ψ 1 is a Dirac (which allows the recovery of white noise), and the second filter ψ 2 is an anisotropic Gabor filter with principal axis directed by the stripes (this orientation is determined by the user). Different parameters were tested for filter ψ 2 and led to similar results, outlining the robustness of this approach with respect to the filter choice. Since no ground truth is available for these images, only visual inspection can help assessing the algorithm's performance. Let us discuss the results:

• Figure 9 and 10 show that VSNR is able to preserve the small image details and to remove the stripes and most of the white noise. The output of the wavelet-FFT and DeStripe algorithms are still impaired with low frequency striping and white noise. • In Figure 11 only the cells contour is stained. The images are significantly denoised and small details are preserved. This is a particularly hard example as the images are not piecewise constant (i.e. the perfect BV images).

Fig. 6. Top-left: noisy pirate. Top-right: denoised using VSNR. Bottom-left: denoised using DeStripe [START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. Bottom-right: denoised using Wavelet-FFT [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. This experiment illustrates the third line of Table II. In that example, the noise pattern is an elongated Gaussian in the vertical direction. The noise level is higher that the signal level. The VSNR algorithm retrieves most of the signal while the other two algorithms do not.

C. Numerical behavior

In this section, we briefly describe some aspects of the algorithm and of its numerical performance. The convergence rates depend on the parameters in algorithm 1. There are at least 4 parameters that should be tuned:

1) First, it is important to have a tight estimate of L, the highest singular value of A, in order to choose σ and τ as large as possible such that στ L 2 < 1. An overestimation of L will slow down the scheme and an underestimation will make the scheme diverge. In the case of total variation, the calculation of L can be done explicitly, see appendix B. 2) Second, the relationship στ L 2 < 1 still leaves a degree of freedom in the choice of σ and τ . It is uneasy to choose these step sizes and it seems that only experimental tuning is available today. 3) Third, as stated in remark 5.1, choosing different inner products in the primal space Λ and dual space Q will lead to different algorithms and can change the numerical performances of algorithm 1 drastically. This question is related to the previous one and we are currently conducting theoretical investigations in order to provide analytical solutions for these choices. A first answer was provided in [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF]. 4) Finally, it is important to have a reliable stopping criterion in order to automatize the algorithm. In all our experiments, choosing = 10 -3 as a stopping criterion in algorithm 1 led to solutions that were accurate enough for the visual system (no visible difference with the true solution). We believe that this is a very nice property of the proposed scheme as no user input is necessary. b) Analytical and empirical complexity: The scheme's analytical complexity A is given by:

A = (iterations number) × 2 × m × (FFT(n) + O(n) op.) +m × FFT(n) ( 25
)
where n is the pixels number, m is the number of filters used in the model, FFT(n) indicates an FFT applied to a size n vector and op. means operations. This theoretical convergence rate indicates that the scheme is adapted to large scale problems: the dependence in n is just O(n log(n)).

Our experiments showed that when the parameters in algorithm 1 are chosen correctly (σ, τ and the metrics), the scheme requires less than 50 iterations in order to decrease the initial duality gap by a factor = 10 -3 . The overall cost is thus around 100 × m FFT computations. The computational time is around 5 seconds for a 1024 × 1024 image, on a 1.2GHz laptop, using m = 1 filter. This can be accelerated using parallel computing.

A more formal complexity result could be obtained using the worst case convergence rates of the algorithm, but this rate seems to be always outperformed for the class of problems considered in this paper. c) A heuristic choice of the metrics: All the tools used in algorithm 1 (Adjoints, singular values, Fenchel transforms, subgradients, proximal operators,...) depend on the choice of the inner products •, • Λ and •, • Q . In [START_REF] Pock | Diagonal preconditioning for first order primal-dual algorithms in convex optimization[END_REF] some choices of the metrics are proposed in order to improve the convergence rate. In the present work, we simply rescaled the filters ψ i and weights α i . This normalization is equivalent to a block-wise constant diagonal preconditionner.

VII. CONCLUSION

We proposed a variational approach to denoise images impaired by a class of stationary noises. It takes advantage of recent advances in convex optimization, leading to interactive computing times. Applications to synthetic images, and to images issued from different microscopy imaging modalities were presented, leading to clear improvements of the image quality compared to existing methods.

APPENDIX A EXPLICIT EXPRESSIONS OF THE RESOLVENTS

In this section, we assume that •, • Λ and •, • Q are the standard dot products. The Fenchel conjugates F * and G * can be computed using Equation (6):

F * (q) = 2 q 2 2 -∇u 0 , q Q if q ∞ ≤ 1 +∞ otherwise and G * (η) = m i=1 (φ i + χ [-C,C] n ) * (η i ) .
The adjoint of A is A * = A T , where:

A T : R n×2 → R m×n q → ∇ T q * ψ1 , • • • , ∇ T q * ψm and ψi = F -1 ψ i .
The different Fenchel transforms useful for the computation of G * are:

• If φ i (λ) = α i λ 1 : (φ i + χ [-C,C] n ) * (η) = C max(0, |η| -α i ) 1 . • If φ i (λ) = αi 2 λ 2 2 : (φ i + χ [-C,C] n ) * (η) = α min η α , C η α - 1 2 min λ α , C 2 1 • If φ i (λ) = χ [-αi,αi] (λ): (φ i + χ [-C,C] n ) * (η) = min(α i , C) η 1
1) Resolvent of F * : Let us detail the calculation of:

q * = (Id + σ∂F * ) -1 (q n ) = arg min q∈Q, q ∞≤1 σ 2 q 2 2 -σ∇u 0 , q + 1 2 q -q n 2 Q . (26)
Let qn = q n + σ∇u 0 By writing the Karush-Kuhn-Tucker optimality conditions, we obtain the following solution to problem (26):

q * (x) = qn (x) max(|q n (x)|, σ + 1)
2) Resolvent of G: In the second step of the algorithm, we must compute

λ * = (I + τ ∂G) -1 (λ n ) = arg min λ∈Λ τ G(λ) + 1 2 λ -λ n 2 Λ = arg min λ∈Ξ m i=1 φ i (λ i ) + 1 2τ λ i -λ n,i 2 Λi
As all the functions φ i are separable, this problem reduces to m × n unidimensional problems of form:

arg min λ∈R,|λ|≤C τ f (λ) + 1 2 |λ -λ n | 2 (27)
where f is a convex function. In this work we focus on the cases:

• f (λ) = α|λ|.
The solution of ( 27) is given by:

λ * = max(|λ n | -τ α, 0) • sign(λ n ) max(1, max(|λ n | -τ α, 0)/C) . • f (λ) = α λ 2 2 .
The solution of ( 27) is given by:

λ * = λn τ α+1 max(1, | λn τ α+1 |/C) . • f (λ) = 0 if |λ| ≤ α +∞ otherwise
Let δ = min(C, α). The solution of ( 27) is given by:

λ * = λ n max(1, |λ n |/δ)
.

APPENDIX B COMPUTATION OF THE OPERATOR NORM L

In order to Algorithm 1 as fast as possible, it is important to use a tight estimation of L, the highest singular value of A defined in equation [START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. In the case of total variation, L can be computed exactly if periodic boundary conditions are used to define the discrete gradient operator.

With these boundary conditions, the discrete gradient operator ∇ = ∂ 1 ∂ 2 can be rewritten using convolution products and is thus diagonalized by the discrete Fourier tranform (DFT):

∂ 1 u = d 1 * u = F -1 diag( d1 )Fu
AA T = F -1 0 0 F -1 • |D 1 | 2 Σ D 1 ΣD * 2 D 2 ΣD * 1 |D 2 | 2 Σ • F 0 0 F .
The eigenvalues of AA T are the same as those of

|D 1 | 2 Σ D 1 ΣD * 2 D 2 ΣD * 1 |D 2 | 2 Σ .
This matrix is symmetric, positive, semi-definite and is constituted of four diagonal blocks. L is thus given by the maximum largest eigenvalue of the following n, 2 × 2 matrices:

M (k) =   | d1 (k)| 2 Σ(k) d1 (

Fig. 1 .

 1 Fig. 1. Stripes for various imaging modalities. Top-left: cement paste image using Ion beam nanotomography. Top-right: sintered specimen of CeO2 image using Scanning Electron Microscope (SEM). Bottom: medaka fish embryo image using Selective Plane Illumination Microscope (SPIM)

Fig. 2 .

 2 Fig. 2. Left: a Gaussian function in space domain -Right: the same Gaussian function in Fourier domain.

Fig. 3 .

 3 Fig. 3. Examples of stationary processes obtained by convolving white noises with different marginals with a filter. Top row : convolution with a Gabor filter. Bottom row: convolution with a fish pattern. From left to right : convolution with a white Bernoulli process, a Gaussian white noise and a uniform white noise.

where u 0

 0 = au 0 -b and (a, b) = arg min (a,b)∈R 2 au 0 -b -u 2 2

Fig. 4 .

 4 Fig. 4. Original pirate image.

Fig. 5 .

 5 Fig.5. Top-left: noisy pirate. Top-right: denoised using VSNR. Bottom-left: denoised using DeStripe[START_REF] Chen | DeStripe: frequency-based algorithm for removing stripe noises from AFM images[END_REF]. Bottom-right: denoised using Wavelet-FFT[START_REF] Münch | Stripe and ring artifact removal with combined wavelet-Fourier filtering[END_REF]. This experiment corresponds to the last line of TableI. The noise pattern is a vertical line. The Destripe and Wavelet-FFT algorithms do not completely remove low frequencies while VSNR leads to near perfect results.

Fig. 7 .

 7 Fig. 7. Synthetic image used for the toy example. PSNR = 21.5dB.

Fig. 8 .

 8 Fig. 8. Toy example. Left column: real components; right column: estimated components using our algorithm. (a,d): Baboon image. PSNR = 27dB. -(b,e): Colored Gaussian noise -(c,f): Impulse like noise

and ∂ 2 u

 2 = d 2 * u = F -1 diag( d2 )Fuwhere d 1 and d 2 are finite difference filters in the horizontal and vertical directions.Let H i be the matrix associated to the convolution product with ψ i . It satisfies H i λ i = ψ i * λ i and is thus diagonalized by the DFT:H i = F -1 diag ψi F. (28)Let us denoteD i = diag di , |D i | 2 = Di D i where Di is the complex conjugate of D i and Σ = m i=1 diag | ψi | 2 .Elementary calculus then leads to:

  k) d2 (k)Σ(k) d1 (k) d2 (k)Σ(k) | d2 (k)| 2 Σ(k)   ,where k belongs to the frequency domain andΣ(k) = m i=1 diag | ψi (k)| 2 .This computation is achieved in O(n log n) arithmetic operations.

Fig. 9 .

 9 Fig. 9. Destriping of a single section from a 3D volume of unhydrated particles of cement paste accessed by FIB-nanotomography. (a) original image is severely affected by the waterfall effect. (b) restored image using VSNR. (c) restored image using wavelet-FFT. (d) restored image using DeStripe.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. SEM imaging on a sintered specimen of CeO2 (cerium oxide) at a sintering temperature of 1400 C for 2 hours and 50 minutes [8]. The image is composed of 512x512 pixels with each side measuring 70.55 µm; the intensity unit is 0.1 µm.

Fig. 12 .

 12 Fig. 12. Isosurface rendering (in red) of a three-dimensional stack of 42 planes of a Multicellular Tumor Spheroid expressing a fluorescent nuclear protein, H2B-HcRed. SPIM images were acquired with an objective 10X NA 0.25, an excitation wavelength of 595 nm and a detection using a 593 nm long pass filter (a) raw data (b) denoised data. The voxel size is 0.645*0.645*1µm.

TABLE I SNRR

 I OF THE IMAGES DENOISED USING DIFFERENT ALGORITHMS. THE NOISE IS OBTAINED BY CONVOLVING WHITE GAUSSIAN NOISE WITH A STRAIGHT LINE. THE THIRD LINE CORRESPONDS TO FIGURE 5.

	Initial SNRr	DeStripe	Wavelet -FFT	VSNR
	33.85	33.70	22.61	43.00
	19.73	20.52	22.03	37.54
	8.63	16.60	15.15	25.32

TABLE II SNRR

 II OF THE IMAGES DENOISED USING DIFFERENT ALGORITHMS. THE NOISE IS OBTAINED BY CONVOLVING WHITE GAUSSIAN NOISE WITH AN ELONGATED GAUSSIAN. THE THIRD LINE CORRESPONDS TO FIGURE 6.

	Initial SNRr	DeStripe	Wavelet-FFT	VSNR
	32.68	30.64	23.63	34.42
	21.80	22.53	18.98	26.31
	8.14	8.15	8.29	13.31

ACKNOWLEDGMENT

The authors would like to thank Bernard Ducommun, Raphaël Jorand and Valérie Lobjois from the IP3D team in Toulouse cancéropole for their tireless support during this work and for all SPIM related questions. They also thank Franc ¸ois De Vieilleville for fruitful discussions. They thank Beat Münch for providing a Fiji implementation of the Wavelet-FFT algorithm and test images and Jean-Luc Pellequer for providing us images. Jérôme Fehrenbach was partly supported by a grant from CNRS.


Jamal Ibrahim Haidar
email: jhaidar@worldbank.org

The Impact of Business Regulatory Reforms on Economic Growth

Keywords: Growth, Reform, Development, Regulations, Doing Business, and Institutions JEL Codes: O12, O17, O50, P48

I investigate the link between business regulatory reforms and economic growth in 172 countries.

I create a five year dataset on business regulatory reforms from the World Bank's Doing Business reports. Then, I test the hypothesis that business regulatory reforms increase economic growth, using data on micro-economic reforms. These data do not suffer the endogeneity issues associated with other datasets on changes in economic institutions. The results provide a robust support for the claim that business regulatory reforms are good for economic growth. The paper establishes that, on average, each business regulatory reform is associated with a 0.15 percent increase in growth rate of GDP. J'ai examiné le lien entre les réformes réglementaires et la croissance économique dans 172 pays. J'ai créé une base de donnée pour 5 années sur les réformes réglementaires, à partir des rapports « Doing Business » de la banque mondiale. J'ai ensuite testé l'hypothèse que les réformes réglementaires favorisent (augmentent) la croissance économique, en utilisant des données sur les réformes micro-économiques. Ces données ne souffrent pas des questions d'endogénéité associées avec d'autres bases de données sur les changements dans les institutions économiques. Les résultats fournissent un support robuste pour la réclamation que les réformes réglementaires sont bonnes pour la croissance économique. L'article établit que, en moyenne, chaque réforme réglementaire est associée à une augmentation de 0,15 pour cent en taux de croissance du PIB.

Introduction

The World Bank has been publishing the annual Doing Business reports since 2004 to investigate the scope and manner of regulations that enhance business activity and those that constrain it. These reports compare countries2 on the basis of quantitative indicators of business regulations. A fundamental premise of business regulations is that economic activity requires good rules -rules that establish and clarify property rights and reduce the cost of resolving disputes; rules that increase the predictability of economic interactions and provide contractual partners with certainty and protection against abuse [START_REF] Acemoglu | The Colonial Origins of Comparative Development: an Empirical Investigation[END_REF][START_REF] Djankov | The Regulation of Entry[END_REF]). The objective is regulations designed to be efficient, accessible to all and simple in their implementation. [START_REF] Djankov | Regulation and growth[END_REF] and [START_REF] Haidar | Investor protections and economic growth[END_REF] show in cross country regressions that burdensome business regulatory procedures are negatively correlated with GDP growth. This paper is different in 2 different ways. First, while [START_REF] Djankov | Regulation and growth[END_REF] and [START_REF] Haidar | Investor protections and economic growth[END_REF] focus on one year, much less countries, a lower number of indicators, and a narrower geographic scope, this paper looks at a five-year period, a larger regulatory scope, and a broader set of countries. Second, the main independent variable in [START_REF] Djankov | Regulation and growth[END_REF] and [START_REF] Haidar | Investor protections and economic growth[END_REF] is different. The authors look at the regulatory status at one year but this paper considers a more interesting and important variable, regulatory reform (not status) at a given year and over time.

They consider that growth is a function of the existing regulatory framework. I argue that the level of income can be a function of existing regulatory framework but a change in income level is more of a function of how regulatory framework changes and improves. This paper uses the World Bank Doing Business indicators as proxies of business regulations, identifies business regulatory reforms by Doing Business reforms, and goes further towards answering the question about whether regulatory reforms enhance economic growth by studying a major determinant of economic growth: regulatory reforms governing business activity.

A vibrant private sector -with firms making investments, creating jobs, and improving productivity -promotes growth and expands opportunities for poor people (OECD and World Bank ( 2006)). To strengthen private sector, governments around the world have implemented wide-ranging reforms, including macro-stabilization programs, price liberalization, privatization, and trade-barrier reductions. In many countries, however, entrepreneurial activity remains limited, poverty stays high, and growth is not significantly far from stagnant. And other countries have spurned orthodox macro reforms and done well.

Although macro policies are unquestionably important, there is a growing consensus that the quality of business regulation and the institutions that enforce it are a major determinant of prosperity. Hong Kong (China)'s economic success, Botswana's stellar growth performance, and Hungary's smooth transition experience have all been stimulated by a good business regulatory environment 3 . However, little research has measured specific aspects of business regulation and analyzed their impact on economic outcomes such as growth, productivity, investment, informality, corruption, unemployment, and poverty. The lack of systematic knowledge prevents policymakers from assessing how good legal and regulatory systems are and determining what to reform.

The World Bank Doing Business regulatory indicators have four key goals. First, they aim to motivate reforms through country benchmarking. Second, they try to inform the design of reforms by highlighting specifically what needs to be changed. Third, the dataset enriches international initiatives on development effectiveness. Fourth, the dataset tries to inform theory by producing new indicators that quantify various aspects of regulation, facilitating tests of existing theories, and contributing to the empirical foundation for new theoretical work on the relation between regulation and development.

Governments around the world reported 1140 business regulatory reforms over the five years up to 2010 4 . Against the backdrop of the global financial and economic crisis, policy makers around the world continue to reform business regulations at the level of the firm, in some areas at an even faster pace than before. Most reforms were nested in broader programs of investment climate reform aimed at enhancing economic competitiveness, as in Colombia, Kenya, and Liberia. In structuring their reform programs for the business environment, governments use multiple data sources and indicators. And, reformers respond to many stakeholders and interest groups, all of whom bring important issues and concerns to the debate. World Bank Group dialogue with governments on the investment climate is designed to encourage critical use of the data, sharpen judgment, avoid a narrow focus on improving rankings, and encourage broadbased reforms that enhance the investment climate. These continued efforts prompt questions:

What is the impact of business regulatory reforms on economic growth? This paper aims to present new findings toward answering this question.

The paper proceeds as follows. Section 2 reviews literature. Section 3 describes data. Section 4 presents main empirical results. Section 5 provides robustness checks, and section 6 concludes.

Literature review:

Hall and Jones (1999), [START_REF] Acemoglu | The Colonial Origins of Comparative Development: an Empirical Investigation[END_REF], [START_REF] Djankov | The Regulation of Entry[END_REF][START_REF] Djankov | Courts[END_REF][START_REF] Botero | The Regulation of Labor[END_REF][START_REF] Djankov | Regulation and growth[END_REF], among others 5 , show that institutions are a major determinant of wealth and long-term growth.

Countries that had better political and economic institutions in the past are richer today. I add to the literature on institutions and growth by studying a new measure of institutional reforms. The analysis focuses on a particular type of institutional reforms: business regulations. I use a new country-level data set to establish the impact of business regulatory reforms on economic growth.

The relationship between regulations and business activity has been under investigation in many academic circles in many articles over the last two decades. [START_REF] Winston | U.S. Industry Adjustment to Economic Deregulation[END_REF] provides a literature review, finds that business regulations affect large and most concentrated industries and are sector-specific. However, much fewer studies look at the impact of business regulatory reforms on economic growth, partly due to lack of data availability. This study tries to fill a research gap by addressing business regulatory reforms impact on economic growth.

Various empirical studies look at business regulations trends across countries over the last decade. [START_REF] Djankov | The Regulation of Entry[END_REF] presents new data on the regulation of entry of start-up firms in 85 countries. Countries with heavier regulation of entry have higher corruption and larger unofficial economies, but not better quality of public or private goods. Countries with more democratic and limited governments have lighter regulation of entry. [START_REF] Djankov | Courts[END_REF] finds that formalism is systematically greater in civil than in common law countries, and is associated with higher expected duration of judicial proceedings, less consistency, less honesty, less fairness in judicial decisions, and more corruption. These results suggest that legal transplantation may have led to an inefficiently high level of procedural formalism, particularly in developing countries. [START_REF] Botero | The Regulation of Labor[END_REF] investigates the regulation of labor markets through employment, collective relations, and social security laws in 85 countries. They find that heavier regulation of labor is associated with lower labor force participation and higher unemployment, especially of the young.

On a related front, Conway, Janod and Nicoletti (2005) shows that within some countries product market policies have become more consistent across different regulatory provisions, although relatively restrictive countries still tend to have a more heterogeneous approach to competition. In general, domestic barriers to competition tend to be higher in countries that have higher barriers to foreign trade and investment, and high levels of state control and barriers to competition. Also, Djankov, Freund, and Pham (2010) established the impact of time delays on international trade. They estimated a difference gravity equation that controls for remoteness, and find that each additional day that a product is delayed prior to being shipped reduces trade by more than one percent. The results that I establish in this paper (in the main regressions and in Appendix Table A8) are related to the latter paper. Appendix Table A8 shows that each positive reform in trading-across-borders regulations (i.e. time, costs, and procedures needed to export or import a cargo) is associated with a 0.88% increase in average economic growth rate. Thus, this paper also highlights the importance of reducing trade costs (as opposed to tariff barriers) to stimulate economic growth.

Starting early 2000s, articles focusing on the effect of regulations on economic fundamentals appeared. On the business entry regulations front, Desai, Gompers, and Lerner (2003) finds cross-country correlations between entry regulations and firm entry rates. They explore the impact of the institutional environment on the nature of entrepreneurial activity Documents de Travail du Centre d'Economie de la Sorbonne -2012.44 across Europe. Greater fairness and greater protection of property rights increase entry rates, reduce exit rates, and lower average firm size. Moreover, Klapper, Laevan and Rajan (2004) uses a comprehensive database of firms in Western and Eastern Europe to study how the business environment in a country drives the creation of new firms. They find entry regulations hamper entry, especially in industries that naturally should have high entry. Also, value-added per employee in naturally "high entry" industries grow more slowly in countries with onerous regulations on entry. And, [START_REF] Viviano | Entry Regulations and Labour Market Outcomes: Evidence from the Italian Retail Trade Sector[END_REF] exploits reforms to regional entry regulations in the Italian retail trade sector, finding that entry barriers have a negative impact on employment growth and on the efficiency of small firms.

On the labor regulations front, [START_REF] Scarpetta | The Role of Policy and Institutions for Productivity and Firm Dynamics: Evidence from Micro and Industry Data[END_REF] use firm-level survey data from OECD countries to analyze firm entry and exit, finding that higher product market and labor regulations are negatively correlated with the entry of small and medium sized firms. In addition, Hasan, Mitra and Ramaswamy (2007) find that labor demand elasticities in Indian manufacturing industries are higher for Indian states with more flexible labor regulations. And, [START_REF] Besley | Can Labor Regulation Hinder Economic Performance? Evidence from India[END_REF] find that Indian states which imposed tighter labor regulations experienced reduced manufacturing output, employment, investment and productivity in formal sector manufacturing and increased output in informal manufacturing.

Data description

The sample consists of 172 countries for which information on the main variables is available.

The time period covered by the study is 2006-2010. In the analysis, I utilize several sources of data including the World Bank's Doing Business annual reports, World Development Indicators, Polity IV, and Freedom House. Table 1 provides a definition of all variables and their sources. In addition to the main independent variable, Reform, I ran 10 separate regressions using the main independent variable as the total number of reforms in each category of regulatory reforms. The results hold. I show the ten separate regressions in the Appendix (A1-A10).

Information on changes in the quality of the business environment is also available from alternative sources such as Heritage Foundation's Freedom of the World Index or Fraser Institute's Economic Freedom of the World. One could use annual changes in these data to construct a measure of reform similar to the ones described above. However, I use the Doing Business data as I consider that it offers two advantages. First, unlike other data sources that are based in part on experts' perceptions, the Doing Business data are based on actual reforms.

Second, and more important, since the Doing Business data cover a specific set of policy reforms, reverse causality from the dependent variable to reform is unlikely. It is difficult, for example, to imagine that the enactment of higher growth rates leads to more efficient bankruptcy law and, hence, would influence the societal bend towards reform. In contrast, other available The main empirical specifications below are relatively sparse, given the use of withincountry variation, the limited quantity of data, and the desire to maximize the number of usable observations. Because the main threat to econometric identification comes from unobserved economy -boosting public sector actions taken with similar timing to the observed regulatory reforms -the other control variables are chosen to capture these factors. government expenditures is 65.13. Moreover, the Trade variable helps us control for the impact of trade openness on growth as well.

Third, to capture the different implications of changes in population, foreign direct investment, and fixed capital formation over this period, every regression equation includes these explanatory variables. In a range of robustness checks, macroeconomic variables like inflation rates, real interest rates, and current account deficits were tested as controls as well. Their weak correlations with reform timing, however, led to a focus on sparser specifications. Moreover, I also control for initial status of business regulatory environment, initial level of income, and dummy variables for each geographic region 6 .

Main empirical results

Figure 1 illustrates the main empirical finding visually. The figure is a scatter plot showing the relationship between Reform and Growth on average over the sample period, and controlling for GDP per capita and country fixed effects. This regression is of average values of growth on total business regulatory reforms and other controls taken over the sample period (2006-2010). There is suggestive slope, corresponding to a positive relationship between reforms and increases in GDP growth rates. In addition, the confidence intervals are not large and do not expand rapidly for big reforms. Overall, the existing data can support certain assertions one way or the other about the aggregate impacts of the micro regulatory reforms on economic growth regulations.

The main empirical results are provided in Table 4. The estimated coefficient values and their significance levels are obtained using an ordinary least squares specification with Huber-White robust standard errors clustered at the country level. Without any additional controls, the estimated coefficient of Reform is positive equaling 0.150, significant at the 1% level (column 1). In words, on average, each business regulatory reform is associated, on average, with a 0.15 percent increase in economic growth. Moving from the country with the least number of reforms (0) to the highest number of reforms (23) increases the growth rate by 3.45 percentage points. This is a large effect given that the mean value of the dependent variable is only 3.93.

The estimated coefficient of the Reform variable remains large and statistically significant when I control for various proxies of institutions and economic variables (columns 2-10). The coefficient value is lowest when I control for corruption, equaling 0.114 significant at the 5% level (column 10). Controlling for foreign direct investment and fixed capital formation growth lowers the estimated coefficient of reform but keeps it statistically significant at the 1% level (columns 2 and 3). In addition, the estimated coefficient value of Reform remains positive, economically large and statistically significant at less than the 5 percent level even after controlling for population size and trade openness (columns 4 and 5). However, the coefficient value does decline in magnitude from 0.150 (column 1) to 0.128 (column 5). This decline is almost entirely due to the control for foreign direct investment and trade. These two measures are negatively correlated with Growth.

Given that controlling for trade and investment levels had a fairly large effect on the estimated coefficient value of Reform, controlling for government expenditures and financial freedom, additional measures of overall financial development becomes more important.

Columns [START_REF] Conway | Product Market Regulation in OECD Countries, 1998 to 2003[END_REF][START_REF] Djankov | The Regulation of Entry[END_REF] expecting that countries with higher levels of financial freedom would enjoy higher growth rates.

Nevertheless, the sample period includes a financial crises time span, which affected more sharply countries with higher levels of financial freedom and more developed systems. Sure, other factors caused these countries (i.e. US, EU) to have lower growth rates too. This is why I do not consider the result as surprising in this particular context. More importantly, the result does not contradict my expectations given the main independent variable does not capture financial freedom.

Reasonably, countries of more political stability, rule of law, and control of corruption need less business regulatory reforms, as also exhibited in Table 3. The largest impact on the estimated coefficient of Reform occurs when I control for corruption. It decreases from 0.150 (column1) to 0.114 (significant at the 1% level) due to the control of corruption (column 10).

Importantly, the estimated coefficient value of Reform holds in terms of economic magnitude and remains statistically significant after controlling for each of the above economic variables.

In sum, growth is associated with micro-economic reforms, and this association remains strong when I control for various measures of institutional quality and economic variables. One concern with the results discussed above could be statistical significance. That is, while the magnitude of the estimated coefficient of Reform is not much affected by the various controls, its statistical significance level goes down to from 1 to 5 percent once I control for population, trade, and corruption. Does this mean that the results for the Growth-Reform relationship are somewhat weak? It is not necessarily the case because a 5 percent significance level is not necessarily weak. The next section shows that the results hold even after I allow a lagged impact of reform on growth. Hence, the stated weakness appears to be due to a specification bias. This matter makes the focus on the lagged impact much more important.

Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

Robustness

I repeat the regression exercise using lagged reforms as well as country fixed effects estimators.

The relationship between reforms and growth holds (Table 5). Unlike above, it is stronger in economic magnitude and still significant at the 5 and 10 % levels when I control for FDI, population, and corruption. Based on the estimated coefficients, one additional business regulatory reform in a given country during 2006-08 is associated, on average, with a 0.18% increase in its economic growth during 2009-10.

Unlike other studies that do not go beyond whether a country reformed/not reformed or beyond counting the regulatory reforms, I consider the timing of reforms at the country-year level for two reasons. First, the availability of sequential reform information at the country level allows me to control for idiosyncratic patterns in the economic growth variable at the country level. Then, I do not worry about estimation bias due to unobserved controls that may drive differences in growth rates and are invariant during the sample period. Second, it helps me to partial out unobserved non-varying heterogeneity. For example, the fixed effects estimators that I use are valid in cases where economies with less flexible regulatory frameworks intervene more in their markets via the channel of state owned enterprises conditional on that such trends are time invariant.

Another worrisome zone is the co-movement of macroeconomic variables. If the timing of reforms is correlated with the business cycle, then the main regression in Table 5 will deliver biased coefficient estimates. And, in fact, such correlation can happen as output rates follow persistent, cyclical processes with substantial autocorrelation. One example can be when reforms take places during periods of prosperity. Then, the macroeconomic indicators move downward with the business cycles, causing the coefficients on reforms in during periods of macroeconomics downturns. Then, as a result, improvements in the economic framework of a country are expected regardless of whether reforms have any impact, causing the coefficients on reforms in Table 5 to be upward-biased.

This section relates changes in economic growth rates to changes in lagged values of regulatory indicators and control variables, conditional on the dynamics of the macroeconomic process of economic growth. A central difficulty in measuring the impact of a specific business regulatory reform on economic growth is inability to capture all other simultaneous actions. A fixed-effects regression exploiting variation within countries over time is capable of measuring impact of reform if the government actions are distributed over time. But, there is an omitted variables problem in the case where policy makers implement simultaneous reforms. For example, one government may improve the business regulatory framework during the same period that it provides more loans to small and medium enterprises. For this reason, specifically in order not to affiliate the impact of the unobserved policy (i.e. loans to SMEs in this case), I control for other institutional variables to address this potential problem. The hypothesis that I propose in the introductionthat business regulatory reforms have a positive impact on economic growthcannot be rejected. The analysis shows that the p-values on business regulatory reform coefficients are statistically significant at conventional levels, and the confidence intervals widen substantially after including controls. I tried various ways to increase power by reducing multi-collinearityi.e. by measuring the impact of reforms in each separate business regulatory area as well as by dropping controlsbut the positive impact of business regulatory reforms holds.

Conclusion

With business regulatory reforms expanding significantly within countries and are being used as benchmark measures for success of policy makers, the link between the level of reforms and growth is receiving renewed interest. This study expands the previous literature based on anecdotal evidence and case studies of countries by using new and improved data. The findings confirm that an expansion of micro-economic reforms is likely to increase growth.

This paper studies the impact of business regulatory reforms on economic growth rates over the period 2006-10. It uses cross-country differences in business regulatory reforms as well as variables that capture macroeconomic dynamics. The key empirical finding that we establish in this paper is: over the period 2006-10, there is statistically significant evidence, across 172 countries, for economic growth response to business regulatory reforms. There is fairly robust evidence of positive impacts of regulatory reforms and these estimated impacts are sizeable and plausibly large. Each additional reform during 2006-10 is associated, on average, with a 0.15% increase in economic growth.

In addition, the onset of the global economic crisis has led to a slump in global economic growth. However, the extent to which economic growth has decreased differed among countries

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

First, I include

  yearly averages of two measures of the quality of government policies and institutions from the World Governance Indicators (WGI). I aggregate the WGI's indices of rule of law, political stability, and control of corruption into average indices. These three indicators contain aggregate indicators of two dimensions of governance. The indicators are constructed using an unobserved components methodology. The three governance indicators are measured in units ranging from about -2.5 to 2.5, with higher values corresponding to better governance outcomes. Significant changes in the attitudes or policies of governments towards the private sector and changes in the risk environment facing investors should be picked up in these indices, helping to address potential biases associated with correlation between such changes and the observed regulatory reforms. Second, I utilize Freedom House data on financial freedom and government expenditures to capture changes in countries' political environment which might have an effect on the confidence of the private sector and, thus, on economic growth. These measures range from 0 (not free) to 100 (free). Financial freedom ranges from 10 to 90 in the sample, and average Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

  Figure 1

Table 2

 2 provides summary statistics, and Table3provides correlations between the main variables. An example is reducing the number of days to get an industry license, eliminating the minimum capital requirement for start-ups, or increasing the legal rights of creditors and minority shareholders.Using this dataset, I define the main independent variable, Reform, as the total number of reforms happening in a country during a certain period of time (i.e. 2006-2010 in Table4 and2006-2008 in Table5). Each individual reform is coded as a dummy variable equal to 1 if a positive reform occurred in one or more of the ten indicators in a given year and 0 otherwise.The mean value of the variable equals 6.51 and the standard deviation is 4.67 (Table2). For example, between 2006 and 2010, Trinidad and Tobago implemented reforms in paying taxes and access to credit implying a score of 2 for the independent variable.

	Dependent variable
	This study primarily focuses on one dependent variable: annual percentage growth rate of
	income per capita, available from the World Bank's World Development Indicators (WDI). I use
	the WDI because it covers a larger set of countries than the Penn World Tables used by Hall and
	Jones (1999) and Sala-i-Martin et al. (2004). By way of context, 2006-10 was a period of
	relatively mixed (strong and weak) economic performance across the world. For developed
	countries, 2006 marked recovery after the US slump of 2001-05. Among developing countries,

some experienced sharp downturnsnotably Argentina and Zimbabwebut most have enjoyed growth rates in excess of those of the 1980s and 1990s. GDP growth rates averaged 3.92, and this average growth rate ranged between -6.3% (Zimbabwe) and 19.9% (Azerbaijan), as shown in Table

2

.

Explanatory variables

I

deduce the main independent variable from World Bank's Doing Business annual reports. As part of its annual exercise, Doing Business compiles ten sets of indicators covering various aspects of the business climate including starting a business, paying taxes, obtaining licenses, getting credit, protecting investors, employing workers, international trade, property registration, closing a business and enforcement of private contracts. The annual Doing Business report includes information on important reforms on each of these indicators. I code this information as a dummy variable which equals 1 if a country implemented a positive reform during the year on a given indicator and 0 otherwise. A positive reform, as defined in Doing Business reports, is one that makes it faster, cheaper or administratively easier for local businesses to start and run operations; or a reform that defines and increases the protection of Documents de Travail du Centre d'Economie de la Sorbonne -2012.44 property rights.

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.44 indicators of the business environment are aggregate or macro level measures. Glaeser et al (2006), among others, highlight a plausible feedback from macro level changes in the business environment to growth rates.

  show that controlling for government expenditures and financial freedom has little effect on the estimated coefficient value of Reform, and the statistical level of the coefficient stays significant at 1%. Unlike government spending, financial freedom shows a negative correlation with Reform less business climate reforms among the countries of relatively more financial freedom. Regarding the negative and significant coefficients on financial freedom

variable, it is a result that initially causes a puzzle but does not contradict expectations. I am Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

  Table 5 to be downward-biased. Another example can be when business regulatory reforms are motivated by and implemented

Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

Table 5

 5 presents the robustness results. It includes total number of reforms and average data for main explanatory variables available for the period 2006-08 as well as data on average growth rates between 2009 and 2010. I regress average economic growth on lagged reforms. The results for economic growth rates in all columns hold positive and statistically significant and validate the main results in Table4. Without any other controls, on average, each additional

	reform during 2006-08 is associated with a 0.182% increase in economic growth during 2009-10
	(column 1,

Table 5 )

 5 . I controlled for all the variables reported in Table4simultaneously. The relationship between growth and reform remains robust to all these checks.

Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.44which reformed at least one area during the 3 years that preceded the recent financial crisis to those which did not. Using a unique micro, business regulatory reforms data from the World Bank Doing Business project, this study signals that reforms, which improved business and investment climate, may have helped to mitigate the effects of the 2008 global slump in economic growth. Countries with more business regulatory reforms enjoyed higher economic growth rates. Further research can look at whether countries that reformed more before the recent economic and financial crisis were less affected by the crisis.

Table 2 : Summary statistics

 2

			Std.			Observations
	Variable	Mean	deviation Minimum	Maximum	(countries)
	Economic Growth	3.93	3.12	-6.3	19.91	172
	Regulatory Reforms	6.51	4.67	0	23	175
	Foreign Direct
	Investment	6.05	6.22	-8.51	27.80	166
	Fixed Capital
	Formation Growth	6.67	9.058	-18.99	63.12	117
	Population	15.71	2.015	9.91	21.01	174
	Trade	93.50	52.12	25.26	422.02
	Government Spending	65.13	22.55	0	94.73	164
	Financial Freedom	49.62	18.01	10	90	164
	Political Stability	-0.09	0.97	-2.88	1.21	160
	Rule of Law	-0.09	0.94	-1.89	1.89	171
	Control of Corruption	-0.06	0.95	-1.627	2.33	173

Table 3 : Correlation between main variables

 3

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9) (10) (11)
	Dependent variable: Economic
	Growth	1
	Regulatory Reforms	0.27	1
	Foreign Direct Investment	-0.05	0.05	1
	Fixed Capital Formation Growth	0.45	0.10	0.16	1
	Population	0.24	0.12	-0.14	0.05	1
	Trade	-0.05	0.05	0.59	-0.05	-0.19	1
	Government Spending	0.47	0.04	-0.18	0.23	0.13	-0.14	1
	Financial Freedom	-0.40	-0.05	0.26	-0.30	-0.18	0.29	-0.32	1
	Political Stability	-0.37	-0.18	0.14	-0.22	-0.16	0.26	-0.57	0.42	1
	Rule of Law	-0.45	-0.13	0.13	-0.41	-0.04	0.23	-0.61	0.66	0.75	1
	Control of Corruption	-0.48	-0.20	0.12	-0.38	-0.08	0.20	-0.63	0.63	0.72	0.97	1

Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

Table 4 :

 4

				Regression results
		(1)	(2)		(4)	(5)	(6)	(7)		(9)	(10)
	Dependent variable: Economic Growth
	Regulatory Reforms	0.150***	0.129***	0.145***	0.130**	0.128**	0.131***	0.132***	0.133***	0.134***	0.114**
		[0.003]	[0.009]	[0.008]	[0.015]	[0.017]	[0.007]	[0.006]	[0.008]	[0.008]	[0.036]
	Foreign Direct Investment		-0.033	-0.067	-0.051	-0.074	-0.036	-0.018	-0.021	-0.019	-0.018
			[0.381]	[0.164]	[0.286]	[0.214]	[0.510]	[0.735]	[0.712]	[0.738]	[0.726]
	Fixed Capital Formation Growth			0.151***	0.147***	0.150***	0.121***	0.103***	0.102***	0.106***	0.110***
				[0.000]	[0.000]	[0.000]	[0.000]	[0.000]	[0.000]	[0.000]	[0.000]
	Population				0.000**	0.000**	0.000**	0.000*	0.000*	0.000*	0.000
					[0.027]	[0.023]	[0.034]	[0.058]	[0.063]	[0.097]	[0.164]
	Trade					0.004	0.005	0.006	0.006	0.006	0.006
						[0.512]	[0.384]	[0.251]	[0.299]	[0.307]	[0.317]
	Government Spending						0.049***	0.043***	0.043***	0.045***	0.046***
							[0.000]	[0.000]	[0.001]	[0.001]	[0.002]
	Financial Freedom							-0.034**	-0.032**	-0.037**	-0.032
								[0.013]	[0.029]	[0.042]	[0.029]
	Political Stability								-0.024	-0.135	-0.208
									[0.943]	[0.751]	[0.626]
	Rule of Law									0.223	1.519
										[0.657]	[0.183]
	Control of Corruption										-1.24
											[0.205]
	Initial income per capita										-0.721
											[0.346]
	Initial ease of business regulations										-0.128
											[0.108]
	East Asia and Pacific										0.812*
											[0.095]
	Europe and Central Asia										0.314
											[0.647]
	OECD										-1.634
											[0.278]
	Middle East and North Africa										0.124
											[0.176]
	Latin America and Carribean										0.957
											[0.234]
	South Asia										0.879
											[0.768]
	Observations	172	166	117	117	117	116	116	113	113	113
	R-squared	0.051	0.047	0.265	0.296	0.299	0.422	0.454	0.449	0.454	0.462

The Doing Business 2011 dataset covers 183 countries. Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

See the World Bank Doing Business 2004 report for details. Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

We deduce these reforms from the annual Doing Business reports. The paper looks at these

specific years for (i) data consistency and (ii) higher country coverage. Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

i.e.[START_REF] Amin | Trade Facilitations and Country Size[END_REF],[START_REF] Haidar | Investor protections and economic growth[END_REF], and[START_REF] Amin | The Cost of Registering Property: Does Legal Origin Matter?[END_REF].[START_REF] Haidar | Investor protections and economic growth[END_REF] looks at how the state of investors protections affects income level and growth at one given year. However, this paper looks at how changes in investor protections, among 9 other regulatory aspects, in a given country affect its income growth over time.Documents deTravail du Centre d'Economie de la Sorbonne -2012.44

Specifically, I used (i) data on GDP per capita in the initial year to control for initial level of income as typically higher income countries tend to grow slower; and (ii) dummy variables for each geographic region. Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

p-values in brackets. All standard errors used are Huber-White robust and clustered on the country. All regressions use a constant term (not shown). Significance level is denoted by ***(1 percent or less), **(5 percent or less) and *(10 percent or less). Documents de Travail du Centre d'Economie de la Sorbonne -2012.44

Rule of Law

The Rule of Law Index is a measure of "the extent to which agents have confidence in and abide by the rules of society." The degree to which a society's atmosphere is conducive to regular, orderly social and economic activity and the protection of private property is an important measure of government effectiveness.