Siddhartha Chandra
email: siddhartha.chandra@inria.friasonas.kokkinos@ecp.fr

Iasonas Kokkinos

Fast, Exact and Multi-Scale Inference for Semantic Image Segmentation with Deep Gaussian CRFs

In this work we propose a structured prediction technique that combines the virtues of Gaussian Conditional Random Fields (G-CRF) with Deep Learning: (a) our structured prediction task has a unique global optimum that is obtained exactly from the solution of a linear system (b) the gradients of our model parameters are analytically computed using closed form expressions, in contrast to the memory-demanding contemporary deep structured prediction approaches [1,2] that rely on back-propagation-through-time, (c) our pairwise terms do not have to be simple hand-crafted expressions, as in the line of works building on the DenseCRF [1,3], but can rather be 'discovered' from data through deep architectures, and (d) out system can trained in an end-to-end manner. Building on standard tools from numerical analysis we develop very efficient algorithms for inference and learning, as well as a customized technique adapted to the semantic segmentation task. This efficiency allows us to explore more sophisticated architectures for structured prediction in deep learning: we introduce multi-resolution architectures to couple information across scales in a joint optimization framework, yielding systematic improvements. We demonstrate the utility of our approach on the challenging VOC PASCAL 2012 image segmentation benchmark, showing substantial improvements over strong baselines. We make all of our code and experiments available at https://github.com/ siddharthachandra/gcrf.

Introduction

Over the last few years deep learning has resulted in dramatic progress in the task of semantic image segmentation. Early works on using CNNs as feature extractors [START_REF] Farabet | Learning hierarchical features for scene labeling[END_REF][START_REF] Mostajabi | Feedforward semantic segmentation with zoom-out features[END_REF][START_REF] Hariharan | Hypercolumns for object segmentation and fine-grained localization[END_REF] and combining them with standard superpixel-based front-ends gave substantial improvements over well-engineered approaches that used hand-crafted features. The currently mainstream approach is relying on 'Fully' Convolutional Networks (FCNs) [START_REF] Long | Fully convolutional networks for semantic segmentation[END_REF][START_REF] Farabet | Scene parsing with multiscale feature learning, purity trees, and optimal covers[END_REF], where CNNs are trained to provide fields of outputs used for pixelwise labeling.

A dominant research direction for improving semantic segmentation with deep learning is the combination of the powerful classification capabilities of FCNs with structured prediction [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF][START_REF] Vemulapalli | Gaussian conditional random field network for semantic segmentation[END_REF][START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF][START_REF] Chen | Learning Deep Structured Models[END_REF][START_REF] Vemulapalli | Deep gaussian conditional random field network: A model-based deep network for discriminative denoising[END_REF][START_REF] Ionescu | Matrix backpropagation for deep networks with structured layers[END_REF], which aims at improving classification by capturing interactions between predicted labels. One of the first works in the direction of combining deep networks with structured prediction was [START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF] which advocated the use of denselyconnected conditional random fields (DenseCRF) [START_REF] Krähenbühl | Efficient inference in fully connected crfs with gaussian edge potentials[END_REF] to post-process an FCNN output so as to obtain a sharper segmentation the preserves image boundaries. This was then used by Zheng et al. [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF] who combined DenseCRF with a CNN into a single Recurrent Neural Network (RNN), accommodating the DenseCRF post processing in an end-toend training procedure.

Most approaches for semantic segmentation perform structured prediction using approximate inference and learning [START_REF] Chen | Learning Deep Structured Models[END_REF][START_REF] Couprie | Multi-label energy minimization for object class segmentation[END_REF]. For instance the techniques of [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF][START_REF] Vemulapalli | Gaussian conditional random field network for semantic segmentation[END_REF][START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF][START_REF] Vemulapalli | Deep gaussian conditional random field network: A model-based deep network for discriminative denoising[END_REF] perform mean-field inference for a fixed number of 10 iterations. Going for higher accuracy with more iterations could mean longer computation and eventually also memory bottlenecks: back-propagation-through-time operates on the intermediate 'unrolled inference' results that have to be stored in (limited) GPU memory. Furthermore, the non-convexity of the mean field objective means more iterations would only guarantee convergence to a local minimum. The authors in [START_REF] Lin | Efficient piecewise training of deep structured models for semantic segmentation[END_REF] use piecewise training with The network populates the unary terms (c), and horizontal and vertical pairwise terms. The G-CRF module collects the unary and pairwise terms from the network and proposes an image hypothesis, i.e. scores (d) after inference. These scores are finally converted to probabilities using the Softmax function (e), which are then thresholded to obtain the segmentation. It can be seen that while the unary scores in (c) miss part of the torso because it is occluded behind the hand. The flow of information from the neighbouring region in the image, via the pairwise terms, encourages pixels in the occluded region to take the same label as the rest of the torso (d). Further it can be seen that the person boundaries are more pronounced in the output (d) due to pairwise constraints between pixels corresponding to the person and background classes.

CNN-based pairwise potentials and three iterations of inference, while those in [START_REF] Liu | Semantic image segmentation via deep parsing network[END_REF] use highly-sophisticated modules, effectively learning to approximate mean-field inference. In these two works a more pragmatic approach to inference is taken, considering it as a sequence of operations that need to be learned [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF]. These 'inferning'-based approaches of combining learning and inference may be liberating, in the sense that one acknowledges and accommodates the approximations in the inference through end-toend training. We show however here that exact inference and learning is feasible, while not making compromises in the model's expressive power.

Motivated by [START_REF] Tappen | Learning gaussian conditional random fields for low-level vision[END_REF][START_REF] Jancsary | Regression tree fields -an efficient, nonparametric approach to image labeling problems[END_REF], our starting point in this work is the observation that a particular type of graphical model, the Gaussian Conditional Random Field (G-CRF), allows us to perform exact and efficient Maximum-A-Posteriori (MAP) inference. Even though Gaussian Random Fields are unimodal and as such less expressive, Gaussian Conditional Random Fields are unimodal conditioned on the data, effectively reflecting the fact that given the image one solution dominates the posterior distribution. The G-CRF model thus allows us to construct rich expressive structured prediction models that still lend themselves to efficient inference. In particular, the log-likelihood of the G-CRF posterior has the form of a quadratic energy function which captures unary and pairwise interactions between random variables. There are two advantages to using a quadratic function: (a) unlike the energy of general graphical models, a quadratic function has a unique global minimum if the system matrix is positive definite, and (b) this unique minimum can be efficiently found by solving a system of linear equations. We can actually discard the probabilistic underpinning of the G-CRF and understand G-CRF inference as an energy-based model, casting structured prediction as quadratic optimization (QO).

G-CRFs were exploited for instance in the regression tree fields model of Jancsary et al. [START_REF] Jancsary | Regression tree fields -an efficient, nonparametric approach to image labeling problems[END_REF] where decision trees were used to construct G-CRF's and address a host of vision tasks, including inpainting, segmentation and pose estimation. In independent work [START_REF] Vemulapalli | Gaussian conditional random field network for semantic segmentation[END_REF] proposed a similar approach for the task of image segmentation with CNNs, where as in [START_REF] Lin | Efficient piecewise training of deep structured models for semantic segmentation[END_REF][START_REF] Liu | Semantic image segmentation via deep parsing network[END_REF][START_REF] Vu | Context-aware cnns for person head detection[END_REF] FCNs are augmented with discriminatively trained convolutional layers that model and enforce pairwise consistencies between neighbouring regions.

One major difference to [START_REF] Vemulapalli | Gaussian conditional random field network for semantic segmentation[END_REF], as well as other prior works [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF][START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF][START_REF] Vemulapalli | Deep gaussian conditional random field network: A model-based deep network for discriminative denoising[END_REF][START_REF] Lin | Efficient piecewise training of deep structured models for semantic segmentation[END_REF][START_REF] Liu | Semantic image segmentation via deep parsing network[END_REF], is that we use exact inference and do not use back-propagation-through-time during training. In particular building on the insights of [START_REF] Tappen | Learning gaussian conditional random fields for low-level vision[END_REF][START_REF] Jancsary | Regression tree fields -an efficient, nonparametric approach to image labeling problems[END_REF], we observe that the MAP solution, as well as the gradient of our objective with respect to the inputs of our structured prediction module can be obtained through the solution of linear systems. Casting the learning and inference tasks in terms of linear systems allows us to exploit the wealth of tools from numerical analysis. As we show in Sec. 3, for Gaussian CRFs sequential/parallel meanfield inference amounts to solving a linear system using the classic Gauss-Seidel/Jacobi algorithms respectively. Instead of these under-performing methods we use conjugate gradients which allow us to perform exact inference and back-propagation in a small number (typically 10) iterations, with a negligible cost (0.02s for the general case in Sec. 2, and 0.003s for the simplified formulation in Sec. 2.5) when implemented on the GPU.

Secondly, building further on the connection between MAP inference and linear system solutions, we propose memory-and time-efficient algorithms for weight-sharing (Sec. 2.5) and multi-scale inference (Sec. 3.2). In particular, in Section 2.5 we show that one can further reduce the memory footprint and computation demands of our method by introducing a Potts-type structure in the pairwise term. This results in multifold accelerations, while delivering results that are competitive to the ones obtained with the unconstrained pairwise term. In Sec. 3.2 we show that our approach allows us to work with arbitrary neighbourhoods that go beyond the common 4-connected neighbourhoods. In particular we explore the merit of using multi-scale networks, where variables computed from different image scales interact with each other. This gives rise to a flow of information across different-sized neighborhoods. We show experimentally that this yields substantially improved results over single-scale baselines.

In Sec. 2 we describe our approach in detail, and derive the expressions for weight update rules for parameter learning that are used to train our networks in an end-to-end manner. In Sec. [START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF] we analyze the efficiency of the linear system solvers and present our multi-resolution structured prediction algorithm. In Sec. 4 we report consistent improvements over well-known baselines and state-of-the-art results on the VOC PAS-CAL test set.

Quadratic Optimization Formulation

We now describe our approach. Consider an image I containing P pixels. Each pixel p ∈ {p 1 , . . . , p P } can take a label l ∈ {1, . . . , L}. Although our objective is to assign discrete labels to the pixels, we phrase our problem as a continuous inference task. Rather than performing a discrete inference task that delivers one label per variable, we use a continuous function of the form x(p, l) which gives a score for each pairing of a pixel to a label. This score can be intuitively understood as being proportional to the log-odds for the pixel p taking the label l, if a 'softmax' unit is used to post-process x.

We denote the pixel-level ground-truth labeling by a discrete valued vector y ∈ Y P where Y ∈ {1, . . . , L}, and the inferred hypothesis by a real valued vector x ∈ R N , where N = P × L. Our formulation is posed as an energy minimization problem. In the following subsections, we describe the form of the energy function, the inference procedure, and the parameter learning approach, followed by some technical details pertinent to using our framework in a fully convolutional neural network. Finally, we describe a simpler formulation with pairwise weight sharing which achieves competitive performance while being substantially faster. Even though our inspiration was from the probabilistic approach to structured prediction (G-CRF), from now on we treat our structured prediction technique as a Quadratic Optimization (QO) module, and will refer to it as QO henceforth.

Energy of a hypothesis

We define the energy of a hypothesis in terms of a function of the following form:

E(x) = 1 2 x T (A + λI)x -Bx (1)
where A denotes the symmetric N × N matrix of pairwise terms, and B denotes the N × 1 vector of unary terms. In our case, as shown in Fig. 1, the pairwise terms A and the unary terms B are learned from the data using a fully convolutional network. In particular and as illustrated in Fig. 1, A and B are the outputs of the pairwise and unary streams of our network, computed by a forward pass on the input image. These unary and pairwise terms are then combined by the QO module to give the final per-class scores for each pixel in the image. As we show below, during training we can easily obtain the gradients of the output with respect to the A and B terms, allowing us to train the whole network end-to-end. Eq. 1 is a standard way of expressing the energy of a system with unary and pairwise interactions among the random variables [START_REF] Jancsary | Regression tree fields -an efficient, nonparametric approach to image labeling problems[END_REF] in a vector labeling task. We chose this function primarily because it has a unique global minimum and allows for exact inference, alleviating the need for approximate inference. Note that in order to make the matrix A strictly positive definite, we add to it λ times the Identity Matrix I, where λ is a design parameter set empirically in the experiments.

Inference

Given A and B, inference involves solving for the value of x that minimizes the energy function in Eq. 1. If (A + λI) is symmetric positive definite, then E(x) has a unique global minimum [START_REF] Shewchuk | An introduction to the conjugate gradient method without the agonizing pain[END_REF] at:

(A + λI)x = B. (2)
As such, inference is exact and efficient, only involving a system of linear equations.

Learning A and B

Our model parameters A and B are learned in an end-to-end fashion via the backpropagation method. In the back-propagation training paradigm each module or layer in the network receives the derivative of the final loss L with respect to its output x, denoted by ∂L ∂x , from the layer above. ∂L ∂x is also referred to as the gradient of x. The module then computes the gradients of its inputs and propagates them down through the network to the layer below.

To learn the parameters A and B via back-propagation, we require the expressions of gradients of A and B, i.e. ∂L ∂A and ∂L ∂B respectively. We now derive these expressions.

Derivative of Loss with respect to B To compute the derivative of the loss with respect to B, we use the chain rule of differentiation: ∂L ∂x = ∂L ∂B ∂B ∂x . Application of the chain rule yields the following closed form expression, which is a system of linear equations:

(A + λI) ∂L ∂B = ∂L ∂x . ( 3
)
When training a deep network, the right hand side ∂L ∂B is delivered by the layer above, and the derivative on the left hand side is sent to the unary layer below.

Derivative of Loss with respect to A The expression for the gradient of A is derived by using the chain rule of differentiation again:

∂L ∂A = ∂L ∂x ∂x ∂A . Using the expression ∂x ∂A = ∂ ∂A (A+λI) -1 B, substituting ∂ ∂A (A+λI) -1 = -(A+ λI) -T ⊗ (A + λI) -1
, and simplifying the right hand side, we arrive at the following expression:

∂L ∂A = - ∂L ∂B ⊗ x, (4)
where ⊗ denotes the kronecker product. Thus, the gradient of A is given by the negative of the kronecker product of the output x and the gradient of B.

Softmax Cross-Entropy Loss

Please note that while in this work we use the QO module as the penultimate layer of the network, followed by the softmax cross-entropy loss, it can be used at any stage in a network and not only as the final classifier. We now give the expressions for the softmax cross-entropy loss and its derivative for sake of completeness. The image hypothesis is a scoring function of the form x(p, l). For brevity, we denote the hypothesis concerning a single pixel by x(l). The softmax probabilities for the labels are then given by p l = e x(l) L e x(l) . These probabilities are penalized by the cross-entropy loss defined as L =l y l log p l , where y l is the ground truth indicator function for the ground truth label l * , i.e. y l = 0 if l = l * , and y l = 1 otherwise. Finally the derivative of the softmax-loss with respect to the input is given by: ∂L ∂x(l) = p l -y l .

Quadratic Optimization with Shared Pairwise Terms

We now describe a simplified QO formulation with shared pairwise terms which is significantly faster in practice than the one described above. We denote by A pi,pj (l i , l j ) the pairwise energy term for pixel p i taking the label l i , and pixel p j taking the label l j . In this section, we propose a Potts-type pairwise model, described by the following equation:

A pi,pj (l i , l j ) = 0 l i = l j A pi,pj l i = l j . (5)
In simpler terms, unlike in the general setting, the pairwise terms here depend on whether the pixels take the same label or not, and not on the particular labels they take. Thus, the pairwise terms are shared by different pairs of classes. While in the general setting we learn P L × P L pairwise terms, here we learn only P × P terms. To derive the inference and gradient equations after this simplification, we rewrite our inference equation

(A + λI) x = B as,      λI Â • • • Â Â λI • • • Â . . . Â Â • • • λI      ×      x 1 x 2 . . . x L      =      b 1 b 2 . . . b L      (6)
where x k , denotes the vector of scores for all the pixels for the class k ∈ {1, • • • , L}.

The per-class unaries are denoted by b k , and the pairwise terms  are shared between each pair of classes. The equations that follow are derived by specializing the general inference (Eq. 2) and gradient equations (Eq. 3,4) to this particular setting. Following simple manipulations, the inference procedure becomes a two step process where we first compute the sum of our scores i x i , followed by x k , i.e. the scores for the class k as:

λI + (L -1) Â i xi = i bi, (7)
(λI -Â)x k = b k -Â i xi. (8)
Derivatives of the unary terms with respect to the loss are obtained by solving:

λI + (L -1) Â i ∂L ∂bi = i ∂L ∂xi , ( 9
) (λI -Â) ∂L ∂b k = ∂L ∂x k -Â i ∂L ∂bi . (10)
Finally, the gradients of  are computed as

∂L ∂ Â = k ∂L ∂b k ⊗ i =k x i . (11)
Thus, rather than solving a system with A ∈ R P L×P L , we solve L + 1 systems with  ∈ R P ×P . In our case, where L = 21 for 20 object classes and 1 background class, this simplification empirically reduces the inference time by a factor of 6, and the overall training time by a factor of 3. We expect even larger acceleration for the MS-COCO dataset which has 80 semantic classes. Despite this simplification, the results are competitive to the general setting as shown in Sec. 4.

Linear Systems for Efficient and Effective Structured Prediction

Having identified that both the inference problem in Eq. 2 and computation of pairwise gradients in Eq. 3 require the solution of a linear system of equations, we now discuss methods for accelerated inference that rely on standard numerical analysis techniques for linear systems [START_REF] Press | Numerical Recipes in C, 2nd Edition[END_REF][START_REF] Golub | Matrix computations[END_REF]. Our main contributions consist in (a) using fast linear system solvers that exhibit fast convergence (Sec. 3.1) and (b) performing inference on multiscale graphs by constructing block-structured linear systems (Sec. 3.2).

Our contributions in (a) indicate that standard conjugate gradient based linear system solvers can be up to 2.5 faster than the solutions one could get by a naive application of parallel mean-field when implemented on the GPU. Our contribution in (b) aims at accuracy rather than efficiency, and is experimentally validated in Sec. 4

Fast Linear System Solvers

The computational cost of solving the linear system of equations in Eq. 2 and Eq. 3 depends on the size of the matrix A, i.e. N × N , and its sparsity pattern. In our experiments, while N ∼ 10 5 , the matrix A is quite sparse, since we deal with small Fig. 2: The table in (a) shows the average number of iterations required by various algorithms, namely Jacobi, Gauss Seidel, Conjugate Gradient, and Generalized Minimal Residual (GMRES) iterative methods to converge to a residual of tolerance 10 -6 . Figure (b) shows a plot demonstrating the convergence of these iterative solvers. The conjugate gradient method outperforms the other competitors in terms of number of iterations taken to converge.

4-connected, 8-connected and 12-connected neighbourhoods. While a number of direct linear system solver methods exist, the sheer size of the system matrix A renders them prohibitive, because of large memory requirements. For large problems, a number of iterative methods exist, which require less memory, come with convergence (to a certain tolerance) guarantees under certain conditions, and can be faster than direct methods. In this work, we considered the Jacobi, Gauss-Seidel, Conjugate Gradient, and Generalized Minimal Residual (GMRES) methods [START_REF] Press | Numerical Recipes in C, 2nd Edition[END_REF], as candidates for iterative solvers. The table in Fig. 2 (a) shows the average number of iterations required by the aforementioned methods for solving the inference problem in Eq. 2. We used 25 images in this analysis, and a tolerance of 10 -6 . Fig. 2 shows the convergence of these methods for one of these images. Conjugate gradients clearly stand out as being the fastest of these methods, so our following results use the conjugate gradient method. Our findings are consistent with those of Grady in [START_REF] Grady | Random walks for image segmentation[END_REF].

As we show below, mean-field inference for the Gaussian CRF can be understood as solving the linear system of Eq. 2, namely parallel mean-field amounts to using the Jacobi algorithm while sequential mean-field amounts to using the Gauss-Seidel algorithm, which are the two weakest baselines in our comparisons. This indicates that by resorting to tools for solving linear systems we have introduced faster alternatives to those suggested by mean field.

In particular the Jacobi and Gauss-Seidel methods solve a system of linear equations Ax = B by generating a sequence of approximate solutions x (k) , where the current solution x (k) determines the next solution x (k+1) .

The update equation for the Jacobi method [START_REF] Golub | Matrix computations[END_REF] is given by

x (k+1) i ← 1 a ii    b i - j =i a ij x (k) j    . ( 12
)
The updates in Eq. 12 only use the previous solution x (k) , ignoring the most recently available information. For instance, x

(k) 1 is used in the calculation of x (k+1) 2 , even though x (k+1) 1
is known. This allows for parallel updates for x. In contrast, the Gauss-Seidel [START_REF] Golub | Matrix computations[END_REF] method always uses the most current estimate of x i as given by:

x (k+1) i ← 1 a ii    b i - j<i a ij x (k+1) j - j>i a ij x (k) j    . ( 13
)
As in [START_REF] Rue | Gaussian Markov Random Fields: Theory and Applications[END_REF], the Gaussian Markov Random Field (GMRF) in its canonical form is expressed as π(x) ∝ exp 1 2 x T Θx + θ T x , where θ and Θ are called the canonical parameters associated with the multivariate Gaussian distribution π(x). The update equation corresponding to mean-field inference is given by [START_REF] Wainwright | Graphical models, exponential families, and variational inference[END_REF],

µ i ← - 1 Θ ii    θ i + j =i Θ ij µ j    , ( 14
)
The expression in Eq. 14 is exactly the expression for the Jacobi iteration (Eq. 12), or the Gauss-Seidel iteration in Eq. 13 for solving the linear system µ = -Θ -1 θ, depending on whether we use sequential or parallel updates. One can thus understand sequential and parallel mean-field inference and learning algorithms as relying on weaker system solvers than the conjugate gradient-based ones we propose here. The connection is accurate for Gaussian CRFs, as in our work and [START_REF] Vemulapalli | Gaussian conditional random field network for semantic segmentation[END_REF], and only intuitive for Discrete CRFs used in [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF][START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF].

Multiresolution graph architecture

We now turn to incorporating computation from multiple scales in a single system. Even though CNNs are designed to be largely scale-invariant, it has been repeatedly reported [START_REF] Chen | Attention to scale: Scale-aware semantic image segmentation[END_REF][START_REF] Kokkinos | Pushing the Boundaries of Boundary Detection using Deep Learning[END_REF] that fusing information from a CNN operating at multiple scales can improve image labeling performance. These results have been obtained for feedforward CNNswe consider how these could be extended to CNNs with lateral connections, as in our case. A simple way of achieving this would be to use multiple image resolutions, construct one structured prediction module per resolution, train these as disjoint networks, and average the final results. This amounts to solving three decoupled systems which by itself yields a certain improvement as reported in Sec. [START_REF] Farabet | Learning hierarchical features for scene labeling[END_REF] We advocate however a richer connectivity that couples the scale-specific systems, allowing information to flow across scales. As illustrated in Fig. 3 the resulting linear system captures the following multi-resolution interactions simultaneously: (a) pairwise constraints between pixels at each resolution, and (b) pairwise constraints between the same image region at two different resolutions. These inter-resolution pairwise terms connect a pixel in the image at one resolution, to the pixel it would spatially correspond to at another resolution. The inter-resolution connections help enforce a different kind of pairwise consistency: rather than encouraging pixels in a neighbourhood to have the same/different label, these encourage image regions to have the same/different labels across resolutions. This is experimentally validated in Sec. 4 to outperform the simpler multi-resolution architecture outlined above. In this example, we have the input image at 2 resolutions. The pairwise matrix A contains two kinds of pairwise interactions: (a) neighbourhood interactions between pixels at the same resolution (these interactions are shown as the blue and green squares), and (b) interactions between the same image region at two resolutions (these interactions are shown as red rectangles). While interactions of type (a) encourage the pixels in a neighbourhood to take the same or different label, the interactions of type (b) encourage the same image region to take the same labels at different resolutions.

Implementation Details and Computational Efficiency

Our implementation is fully GPU based, and implemented using the Caffe library. Our network processes input images of size 865 × 673, and delivers results at a resolution that is 8 times smaller, as in [START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF]. The input to our QO modules is thus a feature map of size 109 × 85. While the testing time per image for our methods is between 0.4 -0.7s per image, our inference procedure only takes ∼ 0.02s for the general setting in Sec. 2, and 0.003s for the simplified formulation (Sec. 2.5). This is significantly faster than dense CRF postprocessing, which takes 2.4s for a 375 × 500 image on a CPU and the 0.24s on a GPU. Our implementation uses the highly optimized cuBlas and cuSparse libraries for linear algebra on large sparse matrices. The cuSparse library requires the matrices to be in the compressed-storage-row (CSR) format in order to fully optimize linear algebra for sparse matrices. Our implementation caches the indices of the CSR matrices, and as such their computation time is not taken into account in the calculations above, since their computation time is zero for streaming applications, or if the images get warped to a canonical size. In applications where images may be coming at different dimensions, considering that the indexes have been precomputed for the changing dimensions, an additional overhead of ∼ 0.1s per image is incurred to read the binary files containing the cached indexes from the hard disk (using an SSD drive could further reduce this). Our code and experiments are publicly available at https://github.com/siddharthachandra/gcrf.

Experiments

In this section, we describe our experimental setup, network architecture and results.

Dataset. We evaluate our methods on the VOC PASCAL 2012 image segmentation benchmark. This benchmark uses the VOC PASCAL 2012 dataset, which consists of 1464 training and 1449 validation images with manually annotated pixel-level labels for 20 foreground object classes, and 1 background class. In addition, we exploit the additional pixel-level annotations provided by [START_REF] Hariharan | Hypercolumns for object segmentation and fine-grained localization[END_REF], obtaining 10582 training images in total. The test set has 1456 unannotated images. The evaluation criterion is the pixel intersection-over-union (IOU) metric, averaged across the 21 classes.

Baseline network (basenet). Our basenet is based on the Deeplab-LargeFOV network from [START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF]. As in [START_REF] Kokkinos | Pushing the Boundaries of Boundary Detection using Deep Learning[END_REF], we extend it to get a multi-resolution network, which operates at three resolutions with tied weights. More precisely, our network downsamples the input image by factors of 2 and 3 and later fuses the downsampled activations with the original resolution via concatenation followed by convolution. The layers at three resolutions share weights. This acts like a strong baseline for a purely feedforward network. Our basenet has 49 convolutional layers, 20 pooling layers, and was pretrained on the MS-COCO 2014 trainval dataset [START_REF]Microsoft coco: Common objects in context[END_REF]. The initial learning rate was set to 0.01 and decreased by a factor of 10 at 5K iterations. It was trained for 10K iterations.

QO network. We extend our basenet to accommodate the binary stream of our network. Fig. 1 shows a rough schematic diagram of our network. The basenet forms the unary stream of our QO network, while the pairwise stream is composed by concatenating the 3 rd pooling layers of the three resolutions followed by batch normalization and two convolutional layers. Thus, in Fig. 1, layers C 1 -C 3 are shared by the unary and pairwise streams in our experiments. Like our basenet, the QO networks were trained for 10K iterations; The initial learning rate was set to 0.01 which was decreased by a factor of 10 at 5K iterations. We consider three main types of QO networks: plain (QO), shared weights (QO s ) and multi-resolution (QO mres ).

Experiments on train+aug -val data

In this set of experiments we train our methods on the train+aug images, and evaluate them on the val images. All our images were upscaled to an input resolution of 865 × 673. The hyper-parameter λ was set to 10 to ensure positive definiteness. We first study the effect of having larger neighbourhoods among image regions, thus allowing richer connectivity. More precisely, we study three kinds of connectivities: (a) 4-connected (QO 4 ), where each pixel is connected to its left, right, top, and bottom neighbours, (b) 8-connected (QO 8 ), where each pixel is additionally connected to the 4 diagonally adjacent neighbours, and (c) 12-connected (QO 12 ), where each pixel is connected to 2 left, right, top, bottom neighbours besides the diagonally adjacent ones. Table 1 demonstrates that while there are improvements in performance upon increasing connectivities, these are not substantial. Given that we obtain diminishing returns, rather than trying even larger neighbourhoods to improve performance, we focus on increasing the richness of the representation by incorporating information from various scales. As described in Sec. 3.2, there are two ways to incorporate information from multiple scales; the simplest is to have one QO unit per resolution (QO res ), thereby enforcing pairwise consistencies individually at each resolution before fusing them, while the more sophisticated one is to have information flow both within and across scales, amounting to a joint multi-scale CRF inference task, illustrated in Fig. 3. In Table 2 It can be seen that our weight sharing simplification, while being significantly faster, also gives better results than QO. Finally, the multi-resolution framework outperforms the other variants, indicating that having information flow both within and across scales is desirable, and a unified multi-resolution framework is better than merely averaging QO scores from different image resolutions.

Experiments on train+aug+val -test data

In this set of experiments, we train our methods on the train+aug+val images, and evaluate them on the test images. The image resolutions and λ values are the same as those in Sec. 4.1. In these experiments, we also use the Dense CRF post processing as in [START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF][START_REF] Chen | Weakly-and semi-supervised learning of a deep convolutional network for semantic image segmentation[END_REF]. Our results are tabulated in Tables 3 and4. We first compare our methods QO, QO s and QO mres with the basenet, where the relative improvements can be most clearly demonstrated. Our multi-resolution network outperforms the basenet and other QO networks. We achieve a further boost in performance upon using the Dense CRF post processing strategy, consistently for all methods. We observe that our method yields an improvement that is entirely complementary to the improvement obtained by combining with Dense-CRF. We also compare our results to previously published benchmarks in Table 4. When benchmarking against directly comparable techniques, we observe that even though we do not use end-to-end training for the CRF module stacked on top of our QO network, our method outperforms the previous state of the art CRF-RNN system of [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF] by a margin of 0.8%. We anticipate further improvements by integrating end-to-end CRF training with our QO. In Table 4, we compare our methods to previously published, directly comparable methods, namely those that use a variant of the VGG [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] network, are trained in an end-to-end fashion, and use structured prediction in a fully-convolutional framework.

Method

mean IoU (%) Deeplab-Cross-Joint [START_REF] Chen | Weakly-and semi-supervised learning of a deep convolutional network for semantic image segmentation[END_REF] 73.9 CRFRNN [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF] 74.

Experiments with Deeplab-V2 Resnet-101

In this section we use our Potts-type model alongside the deeplab-v2 [START_REF] Chen | Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs[END_REF] Resnet-101 network. This network is a 3 branch multi-resolution version of the Resnet-101 network from [START_REF] He | Deep residual learning for image recognition[END_REF]. It processes the input image at 3 resolutions, with scaling factors of 0.5, 0.75, and 1.0, and then combines the network responses at the different resolutions by upsampling the responses at the lower scales to the original scale, and taking an element-wise maximum of the responses corresponding to each pixel. We learn Potts type shared pairwise terms, and these pairwise terms are drawn from a parallel Resnet-101 network which has layers through conv-1 to res5c, and processes the input image at the original scale. Table 5 reports quantitative results on the PASCAL VOC 2012 test set. We show some qualitative results in Fig. 4. It can be seen that our method refines the object boundaries, leading to a better segmentation performance.

Method mean IoU (%) Deeplab-v2 + CRF [START_REF] Chen | Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs[END_REF] 79.7 QO s 79.5 QO s + CRF 80.2

Conclusions and Future Work

In this work we propose a quadratic optimization method for deep networks which can be used for predicting continuous vector-valued variables. The inference is efficient and exact and can be solved in 0.02 seconds on the GPU for each image in the general setting, and 0.003 seconds for the Potts-type pairwise case using the conjugate gradient method. We propose a deep-learning framework which learns features and model parameters simultaneously in an end-to-end FCN training algorithm. Our implementation is fully GPU based, and implemented using the Caffe library. Our experimental results indicate that using pairwise terms boosts performance of the network on the task of image segmentation, and our results are competitive with the state of the art methods on the VOC 2012 benchmark, while being substantially simpler. While in this work we focused on simple 4 -12 connected neighbourhoods, we would like to experiment with fully connected graphical models. Secondly, while we empirically verified that setting a constant λ parameter brought about positive-definiteness, we are now exploring approaches to ensure this constraint in a general case. We intend to exploit our approach for solving other regression and classification tasks as in [START_REF] Eigen | Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture[END_REF][START_REF] Kokkinos | Ubernet: A universal cnn for the joint treatment of low-, mid-, and high-level vision problems[END_REF].

  Schematic of a fully convolutional neural network with a G-CRF module 1(b) Input Image 1(c) Person unary 1(d) QO Output 1(e) Person Probability

Fig. 1 :

 1 Fig. 1: (a) shows a detailed schematic representation of our fully convolutional neural network with a G-CRF module. The G-CRF module is shown as the box outlined by dotted lines. The factor graph inside the G-CRF module shows a 4-connected neighbourhood. The white blobs represent pixels, red blobs represent unary factors, the green and blue squares represent vertical and horizontal connectivity factors. The input image is shown in (b).The network populates the unary terms (c), and horizontal and vertical pairwise terms. The G-CRF module collects the unary and pairwise terms from the network and proposes an image hypothesis, i.e. scores (d) after inference. These scores are finally converted to probabilities using the Softmax function (e), which are then thresholded to obtain the segmentation. It can be seen that while the unary scores in (c) miss part of the torso because it is occluded behind the hand. The flow of information from the neighbouring region in the image, via the pairwise terms, encourages pixels in the occluded region to take the same label as the rest of the torso (d). Further it can be seen that the person boundaries are more pronounced in the output (d) due to pairwise constraints between pixels corresponding to the person and background classes.

1 Fig. 3 :

 13 Fig. 3: Schematic diagram of matrix A for the multi-resolution formulation in Sec. 3.2.In this example, we have the input image at 2 resolutions. The pairwise matrix A contains two kinds of pairwise interactions: (a) neighbourhood interactions between pixels at the same resolution (these interactions are shown as the blue and green squares), and (b) interactions between the same image region at two resolutions (these interactions are shown as red rectangles). While interactions of type (a) encourage the pixels in a neighbourhood to take the same or different label, the interactions of type (b) encourage the same image region to take the same labels at different resolutions.

Fig. 4 :

 4 Fig. 4: Qualitative results when our Potts type pairwise terms are used in combination with the deeplab-V2 Resnet-101 network. Column (a) shows the input image, (b) shows the heatmap of the unary scores, (c) shows the heatmap of the scores after inference, and (d) shows the softmax probabilities. We notice that the object boundaries are significantly finer after incorporating cues from the pairwise terms.

Fig. 5 :

 5 Fig. 5: Visual results on the VOC PASCAL 2012 test set. The first column shows the colour image, the second column shows the basenet predicted segmentation, the third column shows the basenet output after Dense CRF post processing. The fourth column shows the QO mres predicted segmentation, and the final column shows the QO mres output after Dense CRF post processing. It can be seen that our multi-resolution network captures the finer details better than the basenet: the tail of the airplane in the first image, the person's body in the second image, the aircraft fan in the third image, the road between the car's tail in the fourth image, and the wings of the aircraft in the final image, all indicate this. While Dense CRF post-processing quantitatively improves performance, it tends to miss very fine details.

Table 1 :

 1 , we Connectivity Method QO QO s QO res QO mres

	Method QO4 QO8 QO12
	IoU	76.36 76.40 76.42	IoU	76.36 76.59 76.69	76.93

Table 2 :

 2 Comparison of 4 variants of our G-CRF network.

	Method IoU IoU after Dense CRF
	Basenet 72.72	73.78
	QO	73.41	75.13
	QO s	73.20	75.41
	QO mres 73.86	75.46

Table 3 :

 3 Performance of our methods on the VOC PASCAL 2012 Image Segmentation Benchmark. Our baseline network (Basenet) is a variant of Deeplab-LargeFOV[START_REF] Chen | Semantic image segmentation with deep convolutional nets and fully connected crfs[END_REF] network. In this table, we demonstrate systematic improvements in performance upon the introduction of our Quadratic Optimization (QO), and multi-resolution (QO mres ) approaches. DenseCRF post-processing gives a consistent boost in performance.

compare 4 variants of our QO network: (a) QO (Sec. 2), (b) QO with shared weights (Sec. 2.5), (c) three QO units, one per image resolution, and (d) multi-resolution QO (Sec. 3.2).

Table 4 :

 4 Comparison of our method with directly comparable previously published approaches on the VOC PASCAL 2012 image segmentation benchmark.

		7
	Basenet	73.8
	QO	75.1
	QO s	75.4
	QO mres	75.5

Table 5 :

 5 Performance of our Potts type pairwise terms on the VOC PASCAL 2012 test set with the deeplab-v2 Resnet-101 network.

Acknowledgements This work has been funded by the EU Projects MOBOT FP7-ICT-2011-600796 and I-SUPPORT 643666 #2020.


Florence Danglade
email: florence.danglade@ensam.eu

Philippe Veron
email: philippe.veron@ensam.eu

Jean-Philippe Pernot
email: philippe.pernot@ensam.eu

Estimation of CAD model simplification impact on CFD analysis using machine learning techniques

Keywords: CAD/FEA transfer, CAD model simplification, Machine learning

  L'archive ouverte pluridisciplinaire

Introduction:

In the field of transfer from Computer-Aided Design (CAD) to Finite Element Analysis (FEA), preparation processes based on CAD model simplification ensure the quality and the reliability of analysis results. For convective heat transfer analysis, the analysis computation is based on a mesh of a fluid volume wrapping the simplified CAD model (Fig. 1 c). A huge number of elements are necessary to mesh all the local details. Without simplification, the CAD model of the fluid volume and its meshing are often impossible to obtain or the computing time is too high. Existing methods and tools for CAD model simplification allow preparing CAD model to produce an appropriate heat transfer analysis model. Thakur and al. [START_REF] Thakur | A survey of CAD model simplification techniques for physics-based simulation applications[END_REF] propose a classification of simplification technologies based on surface entity operators (SE class), volume entity operators (VE class), explicit features operators (EF class) or dimension reduction operators. We can add to this list, operations based on the simplification of assembly trees (AT class). Among all methods of simplification, we have selected appropriate operations according to our case and used them in an industrial context with ready-to-use software tools: Depending on the use case, these seven methods can be combined in different orders to produce a simplified model. Moreover, for each operation, setting parameters must be adjusted (e.g. kinds of details to be removed by defeaturing, accuracy for decimation, and so on). Thus, a simplification process is defined by the choice of a sequence of simplification operations, and their setting parameters. For a component, a very large number of simplification processes can be driven. Currently, numerical simulation experts have to test different processes without necessarily achieving the optimal process. In practice, the choice of setting parameters is often empirical. Knowing the a priori best simplification process, will reduce design/simulation iterations. Although much research has been devoted to a posteriori evaluation of simplification impact on analysis [START_REF] Foucault | Adaptation of CAD model topology for finite element analysis[END_REF], little works have been led about a priori evaluation. The impact of local simplification [START_REF] Gopalakrishnan | A formal theory for estimating defeaturing-induced engineering analysis errors[END_REF] and defeaturing ( [START_REF] Tang | Evaluating defeaturing-induced impact on model analysis[END_REF], [START_REF] Danglade | On the use of Machine Learning to Defeature CAD Models for Simulation[END_REF]) processes on analysis results have aroused great interest. In the case of simplification process for heat transfer analysis, the level of simplification is very high. The characteristics of the CAD model are strongly impacted. In case of high level of simplification, it appears the accuracy of estimation is low with such application. So it is difficult to implement these methods. Only few attentions have been given to the impact of global simplification methods like convex hull modeling or substitution. The challenge addressed by this work is to estimate the impact on convective FEA results due to the simplification process on a model (the volume of fluid) although it was not created.

Machine learning [START_REF] Mitchell | Machine Learning[END_REF] techniques, like neural networks, support vector machine or decision tree, are able to imitate and accurately predict behaviour from a given set of examples. The use of these methods requires to precisely define input and output variables and choose carefully the set of examples for learning. The translation of raw data from CAD models to learning or testing database is a crucial point of this study. Examples for learning should be as representative as possible so that all the "level of simplification / cost of simplification" spectrum was covered. The new unknown cases must be in the field of learning cases. In the following section, an approach is proposed to build a database of representative examples in order to estimate the impact of simplification for a CAD model using machine learning techniques.

Estimation of the simplification impact on convective heat transfer analysis Cost of simplification and analysis result error criteria define the impact of simplification on analysis results. For a new unknown case, classifiers will estimate these output variables from the initial CAD model of the component and a proposed simplification process. Classifiers and their parameters were obtained previously by learning on examples of known cases. The first step of the proposed approach (Fig. 2 step1) consists in building a database with examples for various component configurations. For this, we extracted key data from initial CAD model of products (e.g. surface area, volume, relationship with boundary conditions, number of faces) and from simplified CAD model previously computed (benefits between initial and prepared CAD model, Hausdorff distance). Others key data are also extracted about the analysis case (boundaries conditions, analysis results, costs) and about simplification process performed (methods used for the simplification, setting parameters, operations sequencing).

Learning

Config. N

Config. 1

Config. 2

3.Use on new case

Simplification processus Prepared CAD models Initial CAD models New unknown case: Initial CAD model + simplifcation processus

Pre-processing

Cost of simplification Analysis result error

Fig. 2: general workflow for estimating the impact of simplification on analysis results.

All these data are compiled in a matrix whose rows are component configurations and the columns are vectors of input and output variables. In the rows, each CAD model configuration is defined by a specific simplification process. In order to cover the whole spectrum "level / cost of simplification", the database contains the configurations corresponding to the initial model, to the most simplified model, to the reference model (lowest simplified model can be simulated), and to other intermediate models (Fig. 3) with various level of simplification (LOS). The database contains all range of LOS. In the columns, vectors of variables include two main output variables (analysis result error and simplification cost) and 20 input variables defined below. Data must be adapted to machine learning technique like Neural Network. For that, variables are corrected in order to normalize them (from -1 to +1), to remove aberrant values and to discretize them into classes of variables which must be representative of the variable distributions. Determinant input variables for each output variable are identified by ranking variables according to their impact on the classification accuracy. A backward removing method is used. Useful variables for analysis error prediction are constituted of five variables relative to the simplification process and three variables relative to the level of simplification (benefits) (Tab. 1). Useful variables for simplification cost prediction are constituted of twelve variables relative to the simplification process, three variables relative to the CAD model and five variables relative to the level of simplification (benefits). Fig. 4 shows examples of CAD model simplification.

Simplification processes

Tab. 1. Useful variables for predictions. Examples of simplification processes.

Variables relative to the level of simplification are unknown variables for a new case. These intermediate output variables are themselves estimated from their own input variables. In a second step, machine learning techniques are used for carrying out classifiers for the prediction of output variables and intermediates variables. Classifiers and their parameters are selected by mean of three criteria, which are the average quadratic error, the percentage of classification errors and the receiver operating characteristic curve (curve giving the true-positive rate against the false-positive rate for varied thresholds).

The classifiers with best prediction accuracy are neural networks with 3 layers using Tan-Sigmoid transfer function and back-propagation method. The final step consists in estimating impact of simplification on analysis for a new case. In this step, only the characteristics of initial CAD model, the boundaries conditions are known. A set of simplification processes is built. This set contains several sequences of operations pre-selected by the user and all combinations of driving parameters. The impact of simplification process is estimated for each planned simplification process. Among this set, the best simplification process is the process with the lowest analysis error and the lowest simplification cost.

Real variables Predicted variables

Simplification processes for new unknow case

Level of simplification Prediction error

Tab. 2. Examples of predictions results

Conclusion

These results show that machine learning techniques based on neural networks can be used to predict the impact of simplification processes on CAD model for heat transfer FEA purposes. A great set of processes can be tested without embedding simplifications and simulations. The proposed method provides us to perform time-consuming simulations.

The best simplification process is identified among a set of candidate processes, limited and proposed by the analysts. All simplification processes with all operations and setting parameters may be tested. Further studies should therefore implement an optimization loop in order to be able to suggest the optimal simplification process to the analysts. The proposed approach could be easily applied to the prediction of simplification impact for more than one component and to the global decision making. At the end, the proposed approach and tools should reduce significantly the number and duration of design/simulation iterations.

Fig. 1 :

 1 Fig. 1: Example of CAD model representations.

Fig. 3 :

 3 Fig. 3: Examples of representative configurations of CAD model simplification for learning.

Fig. 4 :

 4 Fig. 4: Examples of CAD model simplification

  EF class) by removing features likes holes, pockets, protrusions, fillets or chamfers ; 5. substitution (EF class) by removing parts and modeling new basic parts (cylinders or parallelepipeds) ; 6. filtering (AT class) for removing parts from an assembly ; 7. merging (AT class) of several parts into a single component.

1. decimation (SE class) of edges, faces or vertices of the model ; 2. convex hull modeling (SE class) by wrapping of the model ; 3. voxels based simplification (VE class) by discretization and re-building of the model ; 4. defeaturing (

Table 1

 1 gives examples of useful variables for simplification cost and analysis error prediction.

	Proceedings of CAD'15, London, UK, June 22-25, 2015, aaa-bbb
	© 2015 CAD Solutions, LLC, http://www.cadconferences.com

1) Initial CAD model § Faces #: 2984 § Simplification cost: 0 § Analysis result error: 0 5) Defeaturing (small details) § Faces #:140 § Simplification cost: 18 § Analysis result error: -8,8%

  ResultsExamples for learning and test were a set of single parts and products in STEP214 files format. CAD models were simplified with CATIAV5, GPURE or NX5 SIEMENS software. Examples for learning were distributed in two sets: training set (66%) and a test set (33%) statistically equivalent. Meshes were carried out by ANSYS Meshing software. Meshes characteristics were the same for all examples (triangular volume mesh, medium size, without adaptation). The mesh quality is the same for all examples. Analysis simulations were carried out by ANSYS Fluent software. Classifiers and theirs parameters were selected and performed using the Weka platform. Tab. 2 shows an example of a new case for which eleven simplification processes were tested. With a maximum prediction error of 11% and no unacceptable error, the confidence rates of classifiers are satisfactory. The best simplification process for this case is the process number # 2.

			Simplification	cost	Analysis result	error	Simplification	cost	Analysis result	error	Simplification	cost	Analysis result	error
	None simplification	1	0	0	Failed		0	Failed		0%	0%
	Substitution (cylinder)	2	0.49	33		0		35		0		6%	0%
	Filering + decimation	8	0.01	63		0.054		70		0.05		11%	8%
	Convex hull	3	1.0	14		45.8		15		50		7%	9%
			Proceedings of CAD'15, London, UK, June 22-25, 2015, aaa-bbb
				© 2015 CAD Solutions, LLC, http://www.cadconferences.com

Proceedings of CAD'15, London, UK, June 22-25, 2015, aaa-bbb © 2015 CAD Solutions, LLC, http://www.cadconferences.com


Borís Burle

Sonia Allain

Franck Vidal

Thierry Hasbroucq

M M Botvinick

T S Braver

C S Carter

D M Barch

J D Cohen

Boris Burle
email: boris.burle@univ-amu.fr

Boris Burle

Sequential Compatibility Effects and Cognitive Control: Does Conflict Really Matter?

Keywords: sequential adjustments, Ne/ERN, conflict-loop theory, partial errors

, who proposed the conflict-loop theory. This theory has been successful in accounting for the reduction of compatibility effects after an incompatible trial: The level of conflict being, on average, higher during an incompatible trial, more control occurs after such a trial. The authors have tested this prediction by sorting the trials on the basis of amount of conflict (quantified by the electromyographic activity) they presented. A reduction of the compatibility effect was observed after incompatible trials, but it was independent of the level of conflict on previous trials, suggesting that the conflict does not trigger changes in executive control. Consequences for the conflict monitoring model are discussed.

(see, e.g., [START_REF] Cohen | A parallel distributed processing approach to automaticity[END_REF]) such as the tasks of [START_REF] Stroop | Studies of interference in serial verbal reactions[END_REF], Eriksen [START_REF] Eriksen | Effects of noise letters upon the identification of target letter in a non-search task[END_REF], and Simon [START_REF] Craft | Processing symbolic information from a visual display: Interference from irrelevant directional cue[END_REF]. In those tasks, the stimulus is made of two dimensions: a relevant one (e.g., the color of the stimulus in the Simon task) and an irrelevant one (e.g., the position of the stimulus), which, more or less automatically, activates the response strongly associated with this dimension (e.g., a left-side stimulus activates a left response). When the two dimensions correspond (compatible trials), the reaction time (RT) is short and error rate is low, whereas if they do not correspond (incompatible trials), RT is longer and error rate increases. This is termed the compatibility effect.

Two important aspects of connectionist models of compatibility are (a) that the responses are in competition (i.e., as soon as one response is activated, it sends an inhibition proportional to its own activation to the alternative responses) and (b) that an "attentional" module implements the instructions given to the participant (e.g., respond to the stimulus color in the Simon task) by biasing the connections between units involved in the processing of the relevant dimension. The compatibility effects are explained by the fact that, in incompatible trials, the incorrect response is activated and inhibits the correct response, hence delaying its execution. The core idea of the conflict-loop theory is that a conflict monitoring module, supposedly located in the anterior cingulate cortex (ACC), measures online the degree of conflict. The conflict is defined as the sum of the product of the activation of the possible responses weighted by the negative connection strengths between them [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF]. If only one response is activated, there is no conflict. On the contrary, as soon as at least two responses are activated, conflict occurs. According to this model, cognitive control is recruited if a conflict is detected. Given the simplicity of the core assumptions, the conflict-loop theory aims at providing a general 831 account of control mechanisms in a large variety of tasks [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF] and situations [START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF]. In a first series of simulations, [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF] evaluated the degree of agreement between conflict measures in the model and ACC activations obtained in various experiments. Notably, they related the amount of conflict and its timing to a specific electroencephalographic component called error negativity (Ne; [START_REF] Falkenstein | Effects of cross-modal divided attention on late ERP components: II. Error processing in choice reaction tasks[END_REF] or error related negativity (ERN; [START_REF] Gehring | A neural system for error detection and compensation[END_REF]. When participants commit errors in an RT task, this component appears; it starts around response time (or shortly after incorrect electromyography [EMG] onset; [START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF], peaks about 70 ms after the overt response, and is maximum at fronto-central site [START_REF] Falkenstein | Effects of cross-modal divided attention on late ERP components: II. Error processing in choice reaction tasks[END_REF][START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF]. It is thought to originate from either the ACC or the Supplementary Motor Area, or from both [START_REF] Dehaene | Localization of a neural system for error detection and compensation[END_REF]. Although this component was first interpreted as reflecting error detection [START_REF] Coles | Why is there an ERN/Ne on correct trials? Response representations, stimulus-related components, and the theory of errorprocessing[END_REF][START_REF] Falkenstein | Effects of cross-modal divided attention on late ERP components: II. Error processing in choice reaction tasks[END_REF][START_REF] Gehring | A neural system for error detection and compensation[END_REF], it has been reinterpreted as reflecting the amount of conflict as defined in the model [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF][START_REF] Van Veen | The timing of action-monitoring processes in the anterior cingulate cortex[END_REF][START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF]. The simulations revealed reasonable agreement between the model's predictions and the observed amplitude [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF] and timing [START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF] of the Ne. In the second set of simulations, [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF] addressed the impact of conflict measurement on control. Notably, they applied the conflict theory to an Eriksen task to account for sequential adjustment effects. In this task, the compatibility effect depends largely on the nature of the previous trial: It is much reduced, or even disappears, when the previous trial was incompatible compared with when the preceding trial was compatible [START_REF] Gratton | Pre-and poststimulus activation of response channels: A psychophysiological analysis[END_REF]. Such a sequential effect is also present in other compatibility tasks, like Simon's [START_REF] Stürmer | Control over location-based response activation in the Simon task: Behavioral and electrophysiological evidence[END_REF]. Although some authors have argued that those sequential effects are due merely to stimulus transition effects [START_REF] Hommel | Event files: Evidence for automatic integration of stimulus-response episodes[END_REF][START_REF] Hommel | A featureintegration account of sequential effects in the Simon task[END_REF][START_REF] Mayr | Conflict adaptation effects in the absence of executive[END_REF], they are largely considered as reflecting cognitive control [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF][START_REF] Ridderinkhof | Activation and suppression in conflict tasks: Empirical clarification through distributional analyses[END_REF][START_REF] Stürmer | Control over location-based response activation in the Simon task: Behavioral and electrophysiological evidence[END_REF] 1 . The conflict interpretation of this reduction is as follows: The incorrect response being more often activated in incompatible trials than in compatible ones, the conflict is higher. Following a high-conflict trial, the attentional module increases its control. As a consequence, interference is reduced on the next trial. It is important to note that, strictly speaking, conflict is independent of compatibility: It depends only on the coactivation of the two responses. Now, the two responses can be activated both in compatible and in incompatible trials. However, the incorrect response being more often activated in incompatible trials, the conflict will be, on average, higher in the incompatible situation than in the compatible one. As a consequence, postconflict adjustments will be, on average, higher after an incompatible trial than after a compatible one. But, if the conflict triggers sequential adjustments, equating the preceding trials in terms of conflict should lead to equivalent adjustments whatever the nature (compatible vs. incompatible) of the preceding trials.

Incorrect Electromyographic Activation As an Index of Conflict

One major problem is to estimate the level of conflict on a trialby-trial basis. Indeed, conflict is a covert process, not visible on behavior, occurring at the level of response activation. Electrophysiological indices have proven to be powerful in revealing response activation (see [START_REF] Coles | Modern mind-brain reading: Psychophysiology, physiology, and cognition[END_REF], for a review). Among those tools, the Lateralized Readiness Potential has attracted a lot of interest. However, this index is not very useful here, as one can only estimate it after averaging, and it is not possible to estimate response activation on individual trials. Another useful index is the electromyographic (EMG) activity of the muscles involved in the responses [START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF][START_REF] Coles | A psychophysiological investigation of the continuous flow of human information processing[END_REF][START_REF] Hasbroucq | Effect of the irrelevant location of the response signal on choice reaction time: An electromyographic study in humans[END_REF][START_REF] Smid | Selective response activation can begin before stimulus recognition is complete: A psychophysiological and error analysis of the continuous flow[END_REF]. Recording EMG activity has revealed, on a proportion of correct trials, the existence of small EMG activities in muscles associated with the incorrect response (see Figure 1) before the correct response is given ("incorrect-correct" trials)2 . It is important to note that for our purpose, these subthreshold activities can be detected on a trialby-trial basis. The question is, of course, whether those incorrect EMGs are a good index of the presence of conflict. First, EMG has explicitly been associated with response activation in the conflictloop model (Yeung et al., 2004, Note 2, p. 937). Second, the RT of incorrect-correct trials is longer than the trials without incorrect EMG [START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF][START_REF] Hasbroucq | Effect of the irrelevant location of the response signal on choice reaction time: An electromyographic study in humans[END_REF][START_REF] Smid | Selective response activation can begin before stimulus recognition is complete: A psychophysiological and error analysis of the continuous flow[END_REF], suggesting that the correct response is indeed inhibited by the activation of the incorrect one. Therefore, the incorrect EMG really reflects the activation of the incorrect response, and hence, as two responses are activated, conflict should occur. Within the conflict monitoring framework, a more direct argument for the presence of conflict after an incorrect EMG is the presence of a clear Ne timelocked to the incorrect EMG onset on incorrect-correct trials [START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF]. Such incorrect activations, although more numerous in incompatible situations, are also observed in compatible trials. They offer, therefore, the possibility to decorrelate compatibility and conflict and hence to test whether the reduction of the compatibility effect is due to the level of conflict on the preceding trial to the compatibility per se. We evaluated these two possibilities in a visual version of the Simon task. We sorted the "pure-correct" trials (see the Method section) as a function of both the compatibility and the presence of an incorrect EMG on the previous trial. This led to the 8 categories of trials: 2 compatibility levels on Trial n (compatible vs. incompatible) × 2 compatibility level on Trial n -1 (compatible vs. incompatible) × 2 types of trials on n -1 (pure- correct vs. incorrect-correct). It is important to note that, although the presence of an incorrect EMG evidences the presence of conflict, the opposite is not true. Indeed, conflict can occur at a more central level, without peripheral correlate. This is especially true for incompatible pure-correct trials, in which conflict likely also occurs. However, it seems reasonably safe to consider that on compatible pure-correct trials, the level of conflict, if any, must be low and lower than on compatible incorrect-correct trials. Therefore, if the sequential adjustments depend on conflict on the previous trial, the compatibility effect should be much reduced after compatible incorrect-correct trials than after compatible pure-correct trials. As the conflict is very high whenever an incorrect EMG occurs, sequential effects should be similar after incorrect-correct compatible and incorrect-correct incompatible trials. Note that such an outcome would also invalidate sequential effect explanation in terms of features-integration [START_REF] Hommel | A featureintegration account of sequential effects in the Simon task[END_REF] or stimulus-specific priming [START_REF] Mayr | Conflict adaptation effects in the absence of executive[END_REF], as the stimulus transitions are independent of conflict. Furthermore, as executive control is known to depend on the emphasis put on accuracy [START_REF] Gehring | A neural system for error detection and compensation[END_REF], the speed-accuracy trade-off was manipulated through the task instruction. Such manipulation allows one to check whether the sequential adjustments (compatibility reduction after incompatible trials) are affected by the response strategy of the participants.

Method Participants

Twelve participants (2 women, 10 men, ages 23-52 years, M = 35) volunteered for the experiment. They all had normal (or corrected-to-normal) vision and gave their informed consent.

Apparatus and task

The participants were seated in a chair, facing a computer screen (distance 2.1 m). The stimuli were colored (yellow or blue)-filled circles, 2.5 cm in diameter, appearing either on the right or the left of a fixation point (Stim system, Neuroscan, El Paso, TX; total visual angle: 0.68 • , eccentricity: 1.64 • ). Depending on the color of the stimulus, participants had to flex either the right or the left thumb so as to depress the key of a response pad (Neuroscan). Half the participants had to give a right response when the yellow disk was presented (and a left response for the blue disk), whereas the other half had to perform the opposite mapping. When the stimulus appeared on the side ipsilateral to the required response, the trial was termed compatible; it was termed incompatible in the alternative case. The two colors and the two sides of stimulation were equiprobable. EMG was recorded from the flexor pollicis brevis of each hand, by paired surface Ag/AgCl electrodes, amplified (5,000 times), filtered (bandwidth: 10 Hz-1 kHz), full wave rectified, integrated (integration window: 5 ms), and then digitized (sampling rate: 256 Hz). The electroencephalogram (EEG) was recorded from 15 Ag/AgCl scalp electrodes. The reference and ground were on the right and left mastoids, respectively. Impedances were kept below 5 kΩ. To estimate the time course of the surface Laplacian (Babiloni, Cincotti, Carducci, Rossini, & [START_REF] Babiloni | Spatial enhancement of EEG data by surface Laplacian estimation: The use of magnetic resonance imaging-based head models[END_REF]) by a modified version [START_REF] Mackay | On-line source density computation with a minimum of electrodes[END_REF] of the source derivation method [START_REF] Hjorth | An on-line transformation of EEG scalp potentials into orthogonal source derivations[END_REF], we used an electrode configuration that partly differs from the standard 10-20 electrode placement system. This configuration permitted the Laplacian to be estimated at 7 electrodes, called nodal electrodes (see [START_REF] Tandonnet | Spatial enhancement of EEG traces by surface Laplacian estimation: Comparison between local and global methods[END_REF][START_REF] Tandonnet | Spatial enhancement of EEG traces by surface Laplacian estimation: Comparison between local and global methods[END_REF]Vidal et al., 2000, for more details). The distance between a nodal electrode and its three surrounding electrodes was 1/20th of the inion-nasion plus tragus-tragus distance (i.e., 3.7 cm, on average). Only the Laplacian obtained over Cz (where the Ne was maximal) is presented here. The baseline was chosen between 20 ms and 50 ms after EMG onset to facilitate the comparison of the sizes and shapes of the Ne. The vertical electro-oculogram (EOG) was recorded bipolarly between two electrodes, one above and one below the right eye; the horizontal EOG was also recorded bipolarly between two electrodes, at the outer canthus of each eye. EEG and EOG signals were fed into Nicolet amplifiers (Nicolet, Madison, WI), amplified (30,000 times), filtered, and digitized (bandwidth: 0.016-100 Hz, 12 dB/octave; sampling rate: 256 Hz).

Procedure

The participants first ran 4 training blocks of 64 trials and then 16 blocks of 128 trials each. The first-order sequential effects were equilibrated; that is, each stimulus was followed by each other stimuli the same number of times, and the maximum number of stimulus repetitions (color and loca-tion) was set to three. Following the response signal (RS), participants had to press, as soon as possible, one of the buttons of the response pad. This turned off the stimulus, and 500 ms later, the next RS was delivered. RTs shorter than 130 ms or longer than 815 ms were discarded: They were considered as anticipations and omissions, respectively. When participants failed to respond within 815 ms, the RS was turned off, and the next RS was delivered 500 ms later (i.e., 1,315 ms after the previously omitted RS). In half of the blocks, the participants received an instruction emphasizing accuracy at the cost of slower RT (accurate condi-tion), whereas, in the other half, speed was emphasized at the cost of errors (speed instruction). The order of these speed-accuracy conditions was balanced across participants.

Results

Classification of Trials

All proportions were arcsine transformed [START_REF] Winer | Statistical principles in experimental design[END_REF]) before being analyzed. Participants made more errors in the incompatible situation (16.46%) than in the compatible situation (9.28%), F(1, 11) = 38.05, p < .01, and under the speed condition (18.7%) than under the accurate condition (7.27%), F(1, 11) = 62.12, p < .0001. These two factors interacted significantly, F(1, 11) = 11.57, p < .006 (see Table 1). Correct trials were sorted into two categories depending on whether an EMG activity occurred in the wrong response hand. These categories were labeled pure-correct (single EMG activity) and incorrect-correct (dual-activation trials, with the incorrect activation preceding the correct activation; see [START_REF] Hasbroucq | Effect of the irrelevant location of the response signal on choice reaction time: An electromyographic study in humans[END_REF]. Incorrect-correct trials were more numerous in incompatible (9.9%) than in compatible trials (6.5%), F(1, 11) = 21.83, p < .001. There was no effect of instruction (F < 1) or interaction between instruction and compatibility, F(1, 11) = 2.38, p = .15. Erroneous trials and correct trials following errors were excluded.

RT Results

Concerning RT, on pure-correct trials, there was a main effect of compatibility, F(1, 11) = 73.57, p < .0001; a main effect of instruction, F(1, 11) = 47.33, p < .001; and a trend toward an interaction between those effects, F(1, 11) = 3.84, p < .08 (see Table 1). As already reported [START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF][START_REF] Hasbroucq | Effect of the irrelevant location of the response signal on choice reaction time: An electromyographic study in humans[END_REF], the RT for incorrect-correct trials was longer (434 ms) than that for purecorrect trials (350 ms), F(1, 11) = 118, p < .0001; there was a main effect of compatibility, F(1, 11) = 5.13, p < .05, and there was a trend toward an interaction between the nature of the trial and the compatibility, F(1, 11) = 3.49, p < .09. This interaction reveals that the compatibility effect disappears for the RT of the incorrectcorrect trials (F < 1; see [START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF], for more details). There was a main effect of speed instruction, F(1, 11) = 22.45, p < .001. This factor did not interact with any other one (all Fs < 1). Concerning the latencies of the incorrect EMG of the incorrect-correct trials (i.e., the time between the stimulus and the onset of the incorrect EMG activity), there was no compatibility effect (compatible, 223 ms; incompatible, 219 ms; F < 1), but a speed instruction effect occurred (accurate, 230 ms; speed, 212 ms; F[1, 11] = 5.62, p < .05). These two factors did not interact (F < 1). The within-trial difference in latency between the incorrect and correct EMG activities (called correction time by [START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF] was not affected by compatibility (compatible, 141 ms; incompatible, 148 ms; F[1, 11] = 1.16, p = .30)3 .

Ne Triggered by Incorrect EMG

Figure 2 presents the grand-averaged Laplacian obtained over Cz, time locked to the onset of the incorrect EMG of the incorrectcorrect trials. One can see that, shortly after the onset of the incorrect EMG, a negative wave develops, reaching its maximum around 100 ms after EMG onset. This negative wave, already described for incorrect-correct trials (Scheffers, Coles, Bernstein, Gehring, & [START_REF] Scheffers | Event-related brain potentials and error-related processing: An analysis of incorrect responses to go and no-go stimuli[END_REF][START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF] and for overt errors [START_REF] Falkenstein | Effects of cross-modal divided attention on late ERP components: II. Error processing in choice reaction tasks[END_REF][START_REF] Gehring | A neural system for error detection and compensation[END_REF], has been interpreted as reflecting conflict [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF][START_REF] Van Veen | The timing of action-monitoring processes in the anterior cingulate cortex[END_REF][START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF]. It is important to note that the Ne can be seen for both incompatible and compatible trials, confirming that incorrect EMG on compatible trials also elicits conflict as defined by [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF]. However, as a negativity very similar to the Ne has been observed on pure-correct trials [START_REF] Allain | The modulation of the Ne-like wave on correct responses foreshadows errors[END_REF][START_REF] Vidal | Error negativity on correct trials: A reexamination of available data[END_REF][START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF], one may wonder whether the negativity reported in Figure 2 is not the one elicited by the correct EMG. The question as to whether the Ne observed after the incor-rect EMG of incorrect-correct trials is the one seen on pure-correct trials has been addressed by [START_REF] Allain | Electromyographic activities and the Ne[END_REF]: They averaged the EEG signal time locked to the incorrect and to the correct EMG on incorrect-correct trials. Allain, Carbonnell, Burle, et al. (2004, Figure 1) clearly observed two independent negativities, one time locked to the incorrect EMG and one smaller, time locked to the correct one. This is also the case in the present study: The Ne peaks around 100 ms after the incorrect EMG, whereas the mean asynchrony between the incorrect and the correct EMG is 145 ms. Therefore, the peak occurs before the correct EMG onset. The idea that the Ne elicited by the incorrect EMG on incorrect-correct trials is strengthened by the amplitude analysis. We performed an analysis of variance (ANOVA), with trial type as a three-level factor (pure-correct, incorrect-correct, and errors) on the surface under the curve in a time window from -20 to +20 centered on the latency of the peak of the negativity detected on the grand average (see [START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF], for more details). The ANOVA revealed a main effect of trial type, F(2, 22) = 14.32, p < .001. Contrast analysis confirmed that Ne amplitude was smaller on pure-correct (-0.33 µV/cm 2 ) trials than on incorrect-correct trials, -1.55 µV/cm 2 , F(1, 11) = 15.52, p < .01. The difference between incorrect-correct and error (-1.85 µV/cm 2 ) was, however, not significant, F(1, 11) = 3.74, p = .08, Cohen's d = 0.56.

Sequential Effects

The analysis involved speed instruction, compatibility on Trial n, category of Trial n -1, and compatibility of Trial n -1 as withinsubject factors. The results are presented in Table 2 for the accurate and speed instruction separately.

Reaction time. The analysis of RT on Trial n revealed a main effect of speed instruction, F(1, 11) = 31.66, p < .001; of compatibility on Trial n, F(1, 11) = 63.64, p < .001; and of the category of the previous trial (pure-correct vs. incorrect-correct), F(1, 11) = 12.40, p < .005, indicating a lengthening of RT on Trial n when the previous trials contained an incorrect EMG activation. This analysis revealed no effect of compatibility of the previous trial, F(1, 11) < 1. There was an interaction between compatibility on Trial n and on Trial n -1 (F(1, 11) = 38.22, p < .0001, replicating the fact that the compatibility effect largely depends on the compatibility on previous trials. The interaction between the compatibility on Trial n and the category of the previous trial was significant, F(1, 11) = 6.02, p < .05. However, this interaction reflects an increased compatibility effect after an incorrect-correct trial, not a decrease (see Figure 3). No other interaction was significant (all ps > .10), except the second-order interaction between compatibility on Trial n, on Trial n -1, and the speed instruction, which was marginally significant, F(1, 11) = 3.82, p = .08. This interaction may reflect that sequential adjustments are reduced under the speed instruction.

Error rate. The data obtained on error rate essentially replicated the pattern observed on RT. There was a main effect of instruction, F(1, 11) = 32.83, p < .001, of compatibility on Trial n, F(1, 11) = 24.90, p < .001, an effect of the category of the previous trial, F(1, 11) = 10.81, p < .01, and no effect of compatibility of Trial n -1 (F < 1). The interaction between compatibility on Trials n and n -1 was significant, F(1, 11) = 79.40, p < .001. The interaction between compatibility on Trial n and the category of the previous trial was far from significant (F < 1; see Figure 3). No other interaction was significant (all ps > .10), except the secondorder interaction between compatibility on Trial n, on Trial n -1, and the speed instruction, F(1, 11) = 5.80, p < .05. Therefore, both RT and error rate show a reduction of the compatibility effect after an incompatible trial but show no sign of such a reduction after compatible incorrect-correct trials 4 .

Discussion

In the present study, using a Simon task, we replicated the finding that after an incompatible trial, the compatibility effect is largely reduced or even disappears [START_REF] Stürmer | Control over location-based response activation in the Simon task: Behavioral and electrophysiological evidence[END_REF]. The conflict-loop theory explains such a reduction by executive control adjustments following high conflict trials. After a high conflict trial, control is thought to increase to prevent a new conflict. As conflict occurs more often on incompatible trials, adjustments also occur more often after those trials. To test this hypothesis, we dissociated conflict from compatibility by looking at trials on which an incorrect EMG activity occurred, on both compatible and incompatible trials. This dissociation allowed us to show that the reduction of the compatibility effect on Trial n occurs only after incompatible trials (on Trial n -1), independent of the presence of any incorrect EMG. Therefore, it seems that sequential adjustments are not triggered by the level of conflict on the previous trial. One can, however, argue that our measure of conflict might not correspond exactly to the conflict defined in the theory. Indeed, as pointed out by [START_REF] Coles | Modern mind-brain reading: Psychophysiology, physiology, and cognition[END_REF], incorrect response activation, and hence conflict, can occur without any peripheral (EMG) signs. More specifically, one may argue that the presence of an incorrect EMG on a compatible trial (incorrect-correct compatible) does not prove the presence of conflict. However, according to the conflict monitoring model, the conflict is reflected in the amplitude of the Ne [START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF], which was similar for compatible and incompatible in the present study. Therefore, following the model, conflict clearly occurred on compatible incorrect-correct trials. Now, despite the presence of conflict (as indicated by the Ne) on those trials, no reduction of the Simon effect was obtained on the next trial. The conflict-loop theory is based on two main assumptions: (a) Conflict is reflected by the ACC activity, whose electrophysiological correlate is the Ne, and (b) conflict triggers sequential adjustments. The present data show that at least one of the two is incorrect. One possibility to accommodate the conflict model with the present data is that sequential effects are not triggered by conflict. Note, however, that in this case, several arguments in favor of the conflict-loop theory [START_REF] Botvinick | Conflict monitoring and cognitive control[END_REF][START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF] need to be reevaluated. More important, if conflict detection is not used to set up adaptive adjustments, the function of such a conflict detection loses most of its interest. Another possibility could be that sequential adjustments 4 A similar pattern of results was also obtained on the percentage of incorrect-correct trials following either a pure-correct or an incorrect-correct trial. However, there were too few incorrectcorrect trials following an incorrect-correct trial for a reliable analysis (too many cells were at zero). If we restrict the analysis when Trial n -1 was pure-correct, we replicate the interaction between compatibility on Trials n and n -1, F(1, 11) = 27.75, p < .001, indicating that a compatibility effect was observed when n -1 was compatible, F(1, 11) = 43.73, p < .0001, but not when n -1 was incompatible (F < 1). are triggered by a "conflict-like" detection but that such a "conflict" is not reflected in the amplitude of the Ne. Here again, an important consequence would be that several strong arguments in favor of the conflict monitoring model based on the fit between the Ne and conflict [START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF] vanish. From a functional point of view, if the Ne does not index conflict, it could reflect the activity of a response evaluation process, independent of response competition, whose activity could change as a function of the degree of mismatch between an optimum response and the actual performed response [START_REF] Allain | Response monitoring without sensory feedback[END_REF][START_REF] Holroyd | A note on the N200 and the feedback ERN[END_REF]. Such an interpretation fits nicely with the idea that the Ne is not restricted to erroneous activities [START_REF] Allain | The modulation of the Ne-like wave on correct responses foreshadows errors[END_REF]Allain, Hasbroucq, et al., 2004;[START_REF] Vidal | Is the "error negativity" specific to errors?[END_REF] but is also observed on correct trials. In conclusion, it appears that, although the conflict model has been very successful in accounting for both the Ne data and sequential effects separately, it has difficulty in accounting for both at the same time.

Figure 1 .

 1 Figure 1. Example of a representative incorrect-correct trial recorded in this task. Rectified electromyographic activity of the muscles involved (correct activation) and noninvolved (incorrect EMG) in the correct response as a function of poststimulus time (in ms). The vertical dashed lines indicate the moment of the mechanical response. This trace comes from an incompatible trial in the speed condition. EMG = electromyography.

Figure 2 .

 2 Figure 2. Amplitude of the Ne, observed over Cz, evoked by partial errors (in ordinate: µV/cm 2 ) as a function of time (in abscissa: ms) for each experimental condition; the zero shown on the abcissa corresponds to incorrect electromyography (EMG) onset. The data shown are the Laplacian-transformed traces, and negativity is up.

Table 1

 1 Mean RT in Milliseconds and Percentage of Error as a Function of Compatibility and Speed Instruction

			Accurate	Speed
	Trial	M	% of Error M	% of Error
	Compatible	357	5.46 321	13.10
	Incompatible 375	9.09 346	23.84
	Note: RT= reaction time.

Table 2

 2 Mean RT in Milliseconds and Percentage of Error as a Function of Compatibility and Speed Instruction RT= reaction time; PC= pure-correct trials; IC= Incorrect-Correct trials; EMG= electromyography.

			Accurate			Speed
		Compatible	Incompatible	Compatible	Incompatible
	Trial n -1	M	% of Error	M	% of Error	M	% of Error	M	% of Error
	Compatible PC	339	2.95	383	13.49	299	8.96	340	33.97
	Compatible IC	342	2.40	398	12.12	313	5.74	354	26.79
	Incompatible PC 365	8.07	357	3.85	321	18.35	319	15.80
	Incompatible IC 360	7.35	362	5.93	333	20.62	342	11.26
	Note:

It is important to dissociate the mechanisms triggering control adjustments from the mechanisms by which such sequential effects occur (e.g., gating of the direct route, better suppression of the incorrect response activation, stimulus-response binding). Here we are interested in mechanisms triggering sequential adjustments, not in the one responsible for them, as data in this study do not shed much light on this issue.

2 Incorrect EMGs can also occur after the correct response. They are, however, less frequent and are not sensitive to compatibility[START_REF] Smid | Selective response activation can begin before stimulus recognition is complete: A psychophysiological and error analysis of the continuous flow[END_REF]. Furthermore, no conflict seems to occur for those late EMG activities, as they do not generate any Ne[START_REF] Allain | Electromyographic activities and the Ne[END_REF].

This difference is important as conflict depends on both the strength of the incorrect activation and its overlap with the correct response activation. The numerical difference obtained here, although not significant, is in agreement with previous studies[START_REF] Burle | Executive control in the Simon effect: An electromyographic and distributional analysis[END_REF] and goes in the direction of a larger overlap for compatible trials, suggesting that the conflict should be even larger for compatible trials. Note that such an increase of conflict for congruent trials is the argument advanced by[START_REF] Yeung | The neural basis of error detection: Conflict monitoring and the error-related negativity[END_REF] to account for the larger Ne on compatible errors than on incompatible errors.

This work was supported by French Ministry of Research Grant ACT54B and by doctoral Grant Direction Générale de l'Armement -CNRS to