Student Member, IEEE Sampriti Soor

Senior Member, IEEE B S Daya Sagar

Segmentation of Multi-band Images using Watershed Arcs

Keywords: Watershed Transformation, Watershed Arcs, Image Segmentation, Region Merging. ✦

Watershed transformation on grey-scale images has been one of the most reliable methods for image segmentation for years. To address the challenges of over-segmentation lies in watershed transformation, strategies like waterfall and P-algorithm came in the scene. Another new study involving watershed arcs, that divides an image into non-connected segments by generating a maximal vertex-cut in-between them, also was successful to overcome this challenge, where at each level some of the existing arcs are removed to merge the corresponding neighbor basins. Selecting the set of arcs to be removed in a level being determined from a arc-graph using only the existing arcs makes the run-time extremely low. In this study, we incorporate the concept of region merging using watershed arcs for multi-band images. We used a stricter criterion to determine which arcs to be removed in a level by imposing weight on an arc's dissimilarity from its neighboring arcs, a factor that was not there in the initial concept. The performance of the proposed method is evaluated on the Wiezmann dataset of color images in comparison to some of the existing methods in the literature.

INTRODUCTION

I MAGE segmentation is the process of dividing a digital image into separate partitions that are alike in intensity, color, or texture, to achieve a simpler representation of the image for easier analysis. Image segmentation is the fundamental step of various applications such as object detection, medical imagery, pattern recognition and target-tracking [START_REF] Morel | Variational methods in image segmentation: with seven image processing experiments[END_REF]. One of the earliest methods for image segmentation for many years has been region growing/merging, where from some seed pixels in the image homogeneous tiny regions, that are disjoint sets of connected pixels, are formed which are iteratively combined to generate bigger regions based on some specified criteria [START_REF] Ning | Interactive image segmentation by maximal similarity based region merging[END_REF], [START_REF] Peng | Automatic image segmentation by dynamic region merging[END_REF], [START_REF] Nock | Statistical region merging[END_REF]. A powerful and effective method for image segmentation in the field of mathematical morphology, watershed transformation, was first developed in the 1980s with the objective to produce continuous contours from digital elevation models [START_REF] Beucher | Use of watersheds in contour detection. int[END_REF]. An algorithmic definition of the watershed transform as region-growing approach following simulated immersion was given by Vincent and Soille [START_REF] Vincent | Watersheds in digital spaces: an efficient algorithm based on immersion simulations[END_REF].

A general problem occurs in the classical watershed transformation that yields over-segmentation. A new method presented in [START_REF] Soor | Watershed arcs in hierarchical image segmentation[END_REF] involving watershed arcs, that constitutes of the watershed points between distinct pairs of neighbor regions termed as basins, uses a modified definition of watershed transformation to construct connected watershed lines [START_REF] Meyer | Topographic distance and watershed lines[END_REF], [START_REF] Najman | Watershed of a continuous function[END_REF] in the discrete image domain. The neighbor basins are thematically merged in a hierarchical process by removing some of the arcs in each level by implementing the same watershed transformation on a spatial line-graph [START_REF] Trémeau | Regions adjacency graph applied to color image segmentation[END_REF], termed as arc-graph, created from the existing arcs. As only the constructed arc-graphs are processed in each level and their cardinality gets significantly smaller, in comparison to algorithms like waterfall [START_REF] Beucher | Watershed, hierarchical segmentation and waterfall algorithm[END_REF], which were introduced with a similar objective, the resulting segmentation can be acquired in much less time.

The Image Foresting Transform (IFT), presented in [START_REF] Falcao | The image foresting transform: theory, algorithms, and applications[END_REF], follows the region growing process considering the image as an edge-weighted graph, and assigns labels to each pixel of an image with the same label as one of the predefined seed pixels, based on the minimum-cost paths. In [START_REF] Lotufo | IFT-watershed from gray-scale marker[END_REF] IFT was applied to produce the watershed of an image via reconstruction from a marker image. In this study, we extend the idea of hierarchical partitioning based on watershed arcs removal, which up until now has been applied to singleband images considering the image as a node-weighted graph, to edge-weighted graphs in order to perform similar segmentation in multi-band images. The initial watershed arcs are obtained following the region growing process alike IFT. Each node in the arc-graph at a subsequent level is assigned by a real value derived from the features of its neighboring basins. When applying the watershed transformation to the arc-graph, a node, not originally detected as watershed point, is forcibly labelled as a watershed if its value is similar to a nearby watershed points by a factor. This eliminates the shortsightness problem described in [START_REF] Beucher | P algorithm, a dramatic enhancement of the waterfall transformation[END_REF], that was lacking from the initial idea of watershed arcs removal.

In section 2 a brief overview of Watershed Arcs Removal and Image Foresting Transform is provided as the foundation for the proposed extension. Section 3 contains the detailed methodology and essential algorithmic outlines. The correctness and the analysis of the computational complexity of the proposed method is discussed in Section 4. The experimental results by applying watershed arcs removal to the Wiezmann dataset of color images are presented in Section 5, along with a comparison of its performance to few existing methods in literature, followed by the concluding remarks and the scope of extension in section 6.

RELATED LITERATURES

In this section the theoritical basis of the proposed segmentation method is described briefly along with the necessary notations and terminologies.

For various aspects of image processing studies an image is generally defined on a digital grid with 4 or 8-connectivity [START_REF] Borgefors | Distance transformations in digital images[END_REF]. For an image-grid G = (V , N , I ), V is the set of pixels, N is the set of neighborhoods such that N v = {u : u is neighbor to v}, u, v ∈ V , and I : V → R + (for singleband images) or I : V → (R + ) d (for d-dimensional multiband images) are the pixel values. An image is often defined as G = (V , N , E ), where E : N → R + are the edgeweights. A path π ⟨u⇝v⟩ between two vertices/nodes u and v in G is a sequence of vertices ⟨p 0 , p 1 , ..., p n ⟩ where p 0 = u, p n = v and p i+1 ∈ N pi , for i = 0, 1, ..., n-

1. A connected component in G is a sub-graph G ′ ⊆ G such that ∀u, v ∈ G ′ , ∃π ⟨u⇝v⟩ within G ′ .

Hierarchical Partitioning using Watershed Arcs in Single-band Images [7]

In a single-band image a non-decreasing-non-increasing path (nDnI-path in short) between U, V ⊂ G , U ∩ V = ϕ, is a sequence of vertices ⟨u 0 , u 1 , ...u p , m, v 0 , v 1 , ..., v q ⟩ such that, u 0 ∈ vertex-set of U, v q ∈ vertex-set of V , and I up , I v0 < I m , and ∀i = 0, 1, ..., p -1, I ui ≤ I ui+1 , ∀i = 0, 1, ..., q -1,

I vi ≥ I vi+1 . A regional minima is a connected component U ⊂ G such that, ∀u ∈ vertex-set of U , ∀v ∈ vertex-set of G \ U , if v ∈ N u then I v > I u .
Let, M IN is the set of all regional minima and nDnI Π = {π U ⇝V } where U, V ∈ M IN and π U ⇝V is a nDnI-path. Let M = {p : p ∈ π U ⇝V such that ∀π U ⇝V ∈ nDnI Π, ∄q ∈ vertex-set of π U ⇝V such that I q > I p }, i.e, M is the set of pixels having maximum values on the distinct nDnI-paths between any two regional minima. The smallest subset W ⊆ M are the watershed points if ∀w 1 ∈ W, ∀w 2 ∈ M \ W, I w1 ≤ I w2 and ∀π ∈ nDnI Π, ∃w ∈ V π such that w ∈ W . This definition of watershed transformation ensures the set of watershed points to be a maximal vertex-cut. The algorithmic approach to determine such watershed points uses a priority queue where each local minima is extended towards higher altitude neighbor vertices labelling such vertices same as the minima; if a vertex found from a minima, that already is labelled by another minima, that vertex is marked as watershed point. Each point associated with a regional minima is marked with same label and is called the basin extension of the minima. The set of watershed points and the set of basin-extensions is denoted by W and B ext respectively. The set of watershed arcs, A , contains the set of watershed points having same pair of neighbor basins.

To obtain the hierarchical region merging process an arc-graph is created that constitutes of representatives of the arcs as the nodes and the neighborhood is created between the arcs having common basin neighbors. The value associated with the arc representatives are in R + thus the same watershed transformation could be applied on the arc-graph. Owing to extract hierarchical river basins from a digital elevation model, the value assigned to an arc was the minimum altitude of the pixels constituted that arc. This pixel is termed as the saddle point of that arc. The arcs corresponding to its representatives, that are not detected as the watershed points by the watershed transformation on the arc-graph are removed by a arc-removal operation, that preserves the maximal vertex-cut constraint.

The notations are added with level number as suffix to indicate that at a particular level. This process goes on until in some level no watershed points are detected in the arcgraph. The disjoint basin points along with the watershed points present at final level being merged with one of their neighbor basins, forms the resulting segmentation. This segmentation method takes linear computational time. In the proposed method in this study uses a similar arc removal strategy where difference lies in imposing a stricter criteria to select the arcs to be removed in a level.

Image Forest Transformation [12]

In an edge-weighted graph G = (V , N , E ) a path costfunction f is said to be monotonically increasing if, (i)

f (π. ⟨p, q⟩) ≥ f (π) and (ii) f (π 1 ) ≥ f (π 2 ) iff f (π 1 . ⟨p, q⟩) ≥ f (π 2 , ⟨p, q⟩) where p is considered as one end-point of π, π 1 , pi 2 and q ∈ N p . Example of such monotonically increasing cost-functions are additive cost f sum (π. ⟨p, q⟩) = f sum (π) + E ⟨p,q⟩ pass-value cost f max (π. ⟨p, q⟩) = max(f max (π), E ⟨p,q⟩ )
In the proposed method the cost-map of a pixel p, C p , i.e., the path-cost till p from a seed pixel is calculated by passvalue path cost.

In IFT, from a set of distinctly labelled seed-pixels S, all the non-seed pixels are traversed and a non-seed pixel v ∈ V \ S is labelled same as a seed-pixel s i ∈ S if ∄s j ∈ S such that f (π ⟨sj ⇝v⟩ ) > f (π ⟨si⇝v⟩ ), f being a monotonically increasing path cost function. IFT labelling takes linear computational time if implemented using a priority queue datastructure.

HIERARCHICAL PARTITIONING USING WATER-SHED ARCS IN MULTI-BAND IMAGES

This section includes the outline of the algorithms for various required steps of the proposed method.

The initial watershed arcs are determined in the edgeweighted image-grid following a region growing like IFT. The process of obtaining the set of seed points M uses F ⊖ operation on the grid graph, where the pixels most similar to its neighbor pixels are determined. From these spatially distributed seed points having local similarity, regions are expanded to detect the watershed points W (0) , in turn forming the watershed arcs of the initial level A (0) . Intuitively, the points on the watershed arcs have lesser local similarity than the seeds. For each arc in A (0) an arc value is calculated that captures the dissimilarity in neighboring basins to form I A (0) : A (0) → R + . Hence for the subsequent levels the process is similar to that described in 2.1. The local minima formed in the arc-graphs corresponds to the arcs having lesser dissimilarity hence rightly determined to be removed, where as, the arcs still existing must have higher dissimilarity than the removed ones. While removing an arc, as two neighbor basins are merged, all arc-values of the neighbor arcs of the merged basin needs to be updated. The stronger criteria to select the set of arcs to be removed is described in detail in 3.3.

Following is the overall layout of the proposed method that includes the different steps being processed. Require: V , I , E , N Ensure: Set of partitions P P.append(∪A (l-1) )

1: Determine seed-set S from F ⊖ (V , E , N ) 2: Determine W (0) , basinlabel (0) from V , E , N , S 3: Determine A (0) , I A (0) from W (0) , basinlabel (

7:

Create N A (l) from A (l-1) , basinlabel (l-1)

8:

Determine A from A (l-1) , I A (l-1) , N (l)

9:

if A == A (l-1) then A (l) , I A (l) , basinlabel (l) ← Remove Arcs(A)

13:
end if 14: end while

Extracting Seed Points

On a edge weighted graph [START_REF] Cousty | Watershed cuts: Minimum spanning forests and the drop of water principle[END_REF]. From this graph local minima can be detected in linear time by the method present in [START_REF] Moga | Implementation of a distributed watershed algorithm[END_REF]. A single pixel from each of the local minima constitutes the set of seed pixels S.

G = (V , N , E ) the F ⊖ oper- ation produces a node-weighted graph G = (V , N , I ′ ) such that ∀v ∈ V , I ′ v = min({E ⟨v,u⟩ , ∀u ∈ N v })

Generating Initial Watershed Arcs

From the set of seed pixels S using a priority queue Q all the non-seed points are traversed. A pixel reached from a seed s ∈ S if not reached from any other pixel already, is labelled same as s and if already was reached from some other seed, marked as watershed point. At the end of the following process the non-watershed points form initial basins. This process ensures each basin is disconnected from other basins and the set of watershed points W forms a maximal vertexcut, which is a prerequisite of the subsequent steps as described in 2.1.

Require: V , E , N , S Ensure: W ⊂ V : wshed v = T RU E, label v (∀v ∈ V \ W ) 1: C v ← ∞(∀v ∈ V \ S), C v ← 0(∀v ∈ S), label v ← v(∀v ∈ V ), f ixed v ← F ALSE(∀v ∈ V ), wshed v ← F ALSE(∀v ∈ V ) 2: Q.push(v, C v ) 3: while Q is not empty do 4: p, C p ← Q.pop() 5:
if not f ixed p then 6:

for n ∈ N p : notf ixed n do 7: if C n == ∞ then 8: C n ← C p .E ⟨p,n⟩ , label n ← label p 9: Q.push(n, C n ) 10:
else if label n ̸ = label p then 11:

wshed n ← T RU E, f ixed n ← T RU E 12: end if 13:
end for 14:

f ixed p ← T RU E 15:
end if 16: end while Note that, unlike IFT cost map of a pixel once set less than ∞ is not updated anymore, as by using the monotonically increasing path-cost only the boundary pixels in a segment could be re-labelled, where as those are the points detected as watershed points in the proposed algorithm. This ensures the computational linearity of this step.

Determininig Arcs to Remove

The method presented in [START_REF] Soor | Watershed arcs in hierarchical image segmentation[END_REF] after applying watershed transformation on the arc-graph selects corresponding arcs of all the points not marked as watershed point to remove. A vulnerability of this approach lies in the fact that, there may exist higher valued arc representatives marked as basin points than the arc-value of the watershed marked representatives neighbor to that basin. A similar issue is discussed in details in [START_REF] Beucher | P algorithm, a dramatic enhancement of the waterfall transformation[END_REF]. The presence of such higher valued arcs goes in contrast to the assumption that each arc being chosen to delete in a level should have less dissimilarity than its nearby arcs. Also for some complex image scenes, such as the image having invariant intensity or the objects in image having area leakage, eliminating every such non-watershed arcs is not a dependable approach.

In the proposed method, to deal with this vulnerability, a factor β is introduced. Each non-watershed point is assessed with respect to the saddle point of lowest relative altitude neighbor to its basin. The non-watershed point is remarked as a watershed point if its relative height is greater than the lowest neighbor saddle point by the factor β. If β is set to a very large value, this modified method behaves similar to the original, where as, a very low value set to β slows down the region merging process by creating higher number of levels. wshed a = T RU E 27: end if 28: end for With β lesser than 1 if any arc representative is remarked as watershed having lower relative altitude than its neighboring lowest saddle point, the arc value of the representative is upgraded to that of the saddle point. Otherwise, these arcs can serve as local minima in the subsequent iteration and be eliminated, rendering the modification useless.

Require: A

, I A , N A , β(> 0) Ensure: A ⊂ A : wshed a = T RU E(∀a ∈ A) 1: ∀a ∈ A do label a ← a,

Removing a Set of Arcs

Arc removal from a digital image preserving maximal vertex cut is thoroughly defined in [START_REF] Soor | Watershed arcs in hierarchical image segmentation[END_REF]. With the original approach of selecting the arcs to be removed, for an arc A to be removed if B 1 , B 2 are the two neighbor basins get merged and there exists arc A 1 between B, B 1 and arc A 2 between B, B 2 then by the construction of the arc-graph either both A 1 , A 2 are selected to be deleted or both are not. But with the modified approach presented in this study, either of A 1 , A 2 can be selected to be eliminated. In this modified approach, any of A 1 , A 2 is removed only if both are selected to be removed.

ANALYSIS OF THE PROPOSED ALGORITHM

In this section we examine of the correctness and the computational complexity of the proposed watershed arcs removal method.

Proof of Correctness

In the process of generating the initial watershed arcs given in 3.2 each non-seed pixels are reached from the seed pixels using a priority queue so that the pixel with least cost-map is fixed first and its neighboring pixels not already fixed are pushed in the queue and labelled . If a pixel that is already in the queue being reached form some seed is reached again by another seed, then only that pixel is marked as watershed. The watershed point from which it was initially visited and put into the priority queue was fixed already; and the point from which the watershed marked point is visited again before being marked, and the watershed point both are marked as fixed point and hence do not take further part in the traversing process, as a result the point from which the watershed point was visited remains as basin point. Hence each watershed has at least two neighbor points which are part of basin extensions. Between two different seeds in all the paths connecting them must have one watershed point. This structure ensures the set of watershed points being a maximal vertex-cut.

As β is set to be always > 0 in the arc-graph at least the corresponding arcs of different local minima are set to be removed in one level. If at some level no watershed points are found in the arc-graph of that level, which occurs if the arc-graph contains a single local minima, the hierarchical partitioning is terminated. The number of local minima decreases as more of the lesser valued arcs are removed from each level, until at some level there is only one local minima left. This ensures the convergence of the proposed method.

Time Complexity

In [START_REF] Soor | Watershed arcs in hierarchical image segmentation[END_REF] it was shown that the hierarchical partitioning on a node-weighted graph has a linear computational complexity. The process of generating the initial arcs, which is different than the original method, is of linear time complexity, as each node is processed only once. In the subsequent levels number of arcs remaining are decreased exponentially, and the number of level being logarithmic to the input size, the total procedure has a linear run-time. The plot given in Figure 1 validates the linear run-time of the proposed algorithm.

EXPERIMENTAL RESULTS

The effectiveness of the proposed method, watershed arcs removal, is assessed in this section using the Wiezmann segmentation evaluation database, which contains 100 each of 1-and 2-object color images that depict objects that differ from their surroundings in terms of intensity, texture, and other attributes, as well as the ground-truth segmentation. The code and results are available at [START_REF] Soor | Watershed arcs[END_REF].

The performence of the proposed method is compared with some of the procedurally related approaches such as Normalized-cuts [START_REF] Shi | Normalized cuts and image segmentation[END_REF], Region Adjacency Graph (RAG) merging [START_REF] Haris | Hybrid image segmentation using watersheds and fast region merging[END_REF] and Iterated Watershed [START_REF] Soor | Iterated watersheds, a connected variation of k-means for clustering gis data[END_REF] for better understanding. The normalized-cuts approach attempts to impose a global look during merging from an initial fine-segmented image by setting the cut criterion to assess both intersegment dissimilarity and intra-segment similarity. Region adjacency graph merging is conceptually very similar to the proposed method, where neighbor regions are merged based on some similarity criteria, rather watershed arc removal emphasizes on regions not to merge based on dissimilarity criteria. For normalized-cuts and RAG merging, the initial segments are determined by SLIC [START_REF] Achanta | Slic superpixels compared to state-of-the-art superpixel methods[END_REF], which with fine-tuning could have produced better results. Iterated Watershed (IW) which is essentially the connected variation of k-means clustering method [START_REF] Soor | Extending k-means to preserve spatial connectivity[END_REF] where from a set of randomly selected seed pixels initial regions are grown with a underlying connectivity constraint, and alike k-means in The evaluation metrics considered here are (i) Adjusted Mutual Information (AMI), that calculates the mutual information adjusted to account for the chance, (ii) Adjusted Rand Index (ARI) that computes rand index adjusted to chance, and (iii) Clustering Accuracy (CA), where accuracy is obtained from the segments having highest intersection with ground truth labels [START_REF] Fahad | A survey of clustering algorithms for big data: Taxonomy and empirical analysis[END_REF]. The comparison presented in Table-1, where for each method the median values for each evaluation metric is given, shows the potential of the proposed method for image segmentation. In Figure 2 endresults of segmentation for some images from the Wiezmann dataset are shown, and in figure 3 segmentation results obtained from two hyper-spectral images are shown.

CONCLUSION AND FUTURE WORKS

In this study, we extended the watershed arcs removal based hierarchical partitioning strategy to the edge-weighted graphs and applied in image segmentation that produced satisfactory results. The key emphasis of this extension is the reintroduction of several critical arcs in consideration along with the structural modification required to use the method for edge-weighted graphs. Another benefit of this proposed method is reduction of the run-time considerably.

The inclusion of the β factor to restrict some of the critical arcs from being eliminated was a huge advantage, although picking an adequate value for β is still an existing challenge. The results presented in this study were attained after experimenting with various set of values. An appropriate value if retrieved from the image characteristics could lend to better results. Additionally, a variant β that depending on spatial and local aspects could have been helpful rather than applying a constant β value to the entire image setup.

The node values in the arc-graphs were generated form the difference in mean color features between the corresponding neighboring basins, which can be improved fur-ther by other local and global complex features. Moreover, a directional watershed transformation with constrained growing is worthy for further research.

Fig. 1 .

 1 Fig. 1. Run-time on simulated images of different sizes. Experiments carried on a Intel E5-1650 v3 Processor with 32GB RAM. Python 3.7 is used for coding.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Sample segmentation results on Wiezmann dataset by Watershed arcs removal. (a)(c) Original images, (b)(d) Obtained Segmentation (β = 2)

  -minima labela ) ≥ β(l saddle labelaminima labela ) then

	22:	end if
	23: end while
	24: for a ∈ A : not wshed a do
	25:	if (I A a
	26:
		3:	p, p prio ← Q.pop()
		4:	if visited p then
		5:	continue
		6:	end if
		7:	if p prio > I A p then
		8:	Q.push(p, I A p ), shif ted p ← T RU E
		9:	minima p = I A p , l saddle p = ∞
		10:	else
		11:	visited p ← T RU E
		12:	for n ∈ N A
		13:	if not shif ted n then
		14:	Q.push(n, I A n ),
		20:	end if
		21:	end for

wshed a ← F ALSE, visited a ← F ALSE, shif ted a ← F ALSE, Q.push(a, I A a + 1) 2: while Q is not empty do p : label n ̸ = label p do 15: label n ← label p , shif ted n ← T RU E 16:

else 17: wshed n ← T RU E, visited n ← T RU E 18: l saddle labelp = min(I A n , l saddle labelp ) 19:

l saddle labeln = min(I A n , l saddle labeln )

TABLE 1

 1 Evaluation on color images in Weizmann Data-set. For each method the median metric-scores are given.

		Method	AMI ARI	CA
		Watershed arcs Removal 0.430 0.445 0.682
	1-Object	RAG Merging Normalized Cuts	0.330 0.265 0.494 0.363 0.344 0.603
		Iterated Watershed	0.247 0.313 0.628
		Watershed arcs Removal 0.559 0.607 0.843
	2-Object	RAG Merging Normalized Cuts	0.420 0.376 0.560 0.516 0.548 0.805
		Iterated Watershed	0.368 0.374 0.595

each of the subsequent iterations new seeds are calculated which are spatially or characteristically most centered to each of the grown regions in previous iteration. The dissimilarity between the mean feature-set between two neighbor regions of an arc is used as the arc values in higher levels, where as β value was set to 2.

ACKNOWLEDGMENTS

S.S. is grateful to Indian Statistical Institute for providing the junior and senior research fellowship.

The work of B. S. D. Sagar was supported by DST-ITPAR-IV under Grant number INT/Italy/ITPAR-IV/Telecommunication/2018.


Pascal Perrier

ABOUT SPEECH MOTOR CONTROL COMPLEXITY

A key issue in research about speech motor control is the one of the level of complexity that is required for the internal models: have these models to account accurately for all physical properties of the speech motor system, including the complex tongue-jaw biomechanics? Or would more simple internal representations be sufficient, which model only the static characteristics of the peripheral speech apparatus or which give a rough account of articulatory dynamics? On the basis of experimental and modeling studies of speech movements and human limb movements published in the literature, the adequacy of simplified internal representations for speech motor control is analyzed.

INTRODUCTION

In the past two decades, the analysis of speech motor control and its modeling were basically inspired by investigations and models of other skilled human movements, such as reaching, pointing or grasping. This research approach has been proved to be efficient and it allowed the emergence of a number of speech control models which served as theoretical backgrounds for numerous studies (see among others [START_REF] Saltzman | Task dynamic coordination of the speech articulators: a preliminary model[END_REF][START_REF] Saltzman | A dynamical approach to gesture patterning in speech production[END_REF][START_REF] Guenther | Speech sound acquisition, coarticulation and rate effects in a neural network model of speech production[END_REF][START_REF] Ostry | Coarticulation of jaw movements in speech production: is context-sensitivity in speech kinematics centrally planned[END_REF][START_REF] Perkell | Speech motor control: Acoustic goals, saturation effects, auditory feedback and internal models[END_REF][START_REF] Laboissière | Control of multi-muscle systems: human jaw and hyoid movements[END_REF][START_REF] Perrier | Control of tongue movements in speech: The Equilibrium Point hypothesis perspective[END_REF][START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF]. Consequently, the issues raised today in the domain of motor control research are very important for future studies of speech motor control, and addressing them is a prerequisite for the improvement of the current models. Among these issues, two questions are of particular interest: (1) How complex are the acquired internal representations of the motor system, built up during the acquisition of the motor task, and used to plan and achieve the movement? (2) What role does low-level, short-latency feedback play in achieving an accurate and stable movement control? In this paper, contributions to these issues are proposed, while taking into account some important peculiarities of the speech production task.

SPEECH PRODUCTION: A COMPLEX MOTOR TASK.

Compared to many other human motor tasks classically studied in motor control research, speech production has a number of peculiarities that make it particularly complex. Some findings suggesting such a complexity were already frequently discussed in the literature:

(1) Because of its semiotic nature, the goal of speech production is actually defined in an abstract domain. Hence, its physical characterization is not straightforward, and this has two consequences. First there is no unique physical correlate for a given elementary speech sound and a large variability of patterns can be observed in the neuro-20/07/2005 physiological, articulatory and acoustic domains (see [START_REF] Perkell | Invariance & Variability in speech processes[END_REF]. Second the issue of the physical space in which the motor task is planned becomes particularly complex, since the distal space can be defined either by articulatory positions, or by spectral properties of the speech signal, or by perceptual characteristics of this signal (see [START_REF] References Browman | Gestural specification using dynamically-defined articulatory structures[END_REF][START_REF] Stevens | On the quantal nature of speech[END_REF][START_REF] Savariaux | Compensation strategies for the perturbation of the rounded vowel [u] using a lip-tube: a study of the control space in speech production[END_REF][START_REF] Guenther | A theoretical investigation of reference frames for the planning of speech movements[END_REF][START_REF] Tremblay | Somatosensory basis of speech production[END_REF], or a multimodal space associating oro-sensory, auditory and even visual characterizations.

(2) Speech production has a large number of degrees of freedom that confer a many-to-one characteristic on the relationships between motor commands, articulatory positions and acoustic or auditory properties. This characteristic, together with the above mentioned intrinsic variability of the physical correlates of the production of a given sound, has the consequence that a large set of motor equivalence strategies can be used to implement a range of coarticulation strategies or to deal with artificial perturbations (such as a pipe or food in the mouth), or pathological perturbations (such as tongue or mandible surgery) or peripheral perturbations [START_REF] Perkell | Trading relations between tongue-body raising and lip rounding in production of the vowel /u/: A pilot "motor equivalence" study[END_REF][START_REF] Mcfarland | Speech compensation to structural modifications of the oral cavity[END_REF][START_REF] Guenther | Articulatory tradeoffs reduce acoustic variability during American English /r/ production[END_REF][START_REF] Savariaux | Compensation strategies for the perturbation of French [u] using a lip tube. II. Perceptual analysis[END_REF]. These multiple strategies obviously contribute to the complexity of speech motor control.

(3) Compared to other skilled human motor activities, speech movements in normal conditions can be very short, since vowels have a mean duration of approximately 80 ms and consonants have mean durations around 40ms [START_REF] O'shaughnessy | A study of French vowels and consonant durations[END_REF]. These characteristics seem to exclude any potential on-line contribution of long-latency oro-sensory feedback that would be processed by the cortex, and to limit the role of auditory feedback to a suprasegmental level and to an a posteriori monitoring used to correct segmental aspects of speech after it was produced [START_REF] Perkell | Speech motor control: Acoustic goals, saturation effects, auditory feedback and internal models[END_REF]. The absence of on-line use of auditory feedback to control speech at a segmental level is 20/07/2005 well supported by experimental work showing that speakers can produce intelligible speech even after hearing loss [START_REF] Lane | Speech deterioration in postlingually deafened adults[END_REF][START_REF] Manzella | Speech production before and aftrer deafening[END_REF].

Compatibly, work on stutterers and normal speakers shows that delaying auditory feedback in the range of 50 -200ms affects prosodic (speaking rate, fluency, rhythm, intonation and stress) rather than segmental features [START_REF] Hargrave | Effect of frequencyaltered feedback on stuttering frequency at normal and fast speech rate[END_REF][START_REF] Stager | Speech production changes under fluency-evoking conditions in non-stuttering speakers[END_REF]. At the same time, speech gestures have to be accurate enough in order to ensure that the associated acoustic signal can be correctly perceived by a listener.

How accuracy can be obtained without the use of long-latency feedback that would be processed by the cortex, is a key issue for speech production research, but not for other human motor tasks except for eye-saccades.

To deal with the complexity and the multimodality of speech task representations, with the numerous motor or auditory equivalence strategies, and with the accuracy requirements in the absence of long-latency feedback going through the cortex, the large majority of speech motor control models published in the literature assume the existence of internal representations of the speech apparatus, called internal models [START_REF] Jordan | Motor Learning and the Degrees of Freedom Problem[END_REF][START_REF] Kawato | Trajectory formation of arm movement by cascade neural network model based on minimum torque-change criterion[END_REF][START_REF] Laboissière | Modelling the Speaker-Listener Interaction in a Quantitative Model for Speech Motor Control: A Framework and Some Preliminary Results[END_REF][START_REF] Hirayama | Forward dynamics modeling of speech motor control using physiological data[END_REF][START_REF] Jordan | Forward Models: Supervised Learning with a Distal Teacher[END_REF][START_REF] Guenther | Speech sound acquisition, coarticulation and rate effects in a neural network model of speech production[END_REF][START_REF] Perkell | Speech motor control: Acoustic goals, saturation effects, auditory feedback and internal models[END_REF][START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF][START_REF] Perrier | Modéliser le physique pour comprendre le contrôle : le cas de l'anticipation en production de parole[END_REF].

INTERNAL MODELS AND SPEECH PRODUCTION CONTROL

A useful concept to deal with the control of complex motor tasks

The internal model concept was proposed in the research domain to deal with the nonbiunivocity between motor commands and position of the final effector, and with the delays associated with the processing of long-latency feedback. The basic hypothesis is that copies of the motor system, or of subsets of it, could be learned in the brain during the acquisition of the motor skill (in our case in the speech learning phase). Once they are learned, these models could be used to estimate predictions of the consequences of motor command changes on the trajectory of the final effector. According to different existing models of human motor control published in the literature, the role of internal models in the execution of the motor task could take different forms.

A first hypothesis suggests that internal models' predictions could be the basis of task planning strategies aiming at ensuring that the final effector moves along a specific desired trajectory in the task space. This is the so-called desired trajectory hypothesis [START_REF] Kawato | Internal models for motor control and trajectory planning[END_REF]. Since human motor systems have usually an excess of degrees of freedom, many different motor command sequences are likely to allow the achievement of the specified trajectory. The basic idea of the desired trajectory hypothesis is that the Central Nervous System would use internal models prior to the execution of the movement, in order to select an optimal one from all possible motor command sequences that would both generate the required trajectory and minimize a motor criterion, classically related to the concept of effort.

This kind of internal model is classical called inverse internal model, since it permits to go from the desired output to the motor commands.

For target oriented movements, an alternative hypothesis suggests that the Central Nervous System would use internal models as direct models during the planning to optimize the positioning accuracy of the final effector at the target in presence of neural noise [START_REF] Harris | Signal-dependent noise determines motor planning[END_REF]. From this perspective, internal models would also be used prior to the execution of movement, but the trajectory of the final effector toward the target would be a consequence of the planning strategy rather than the specification of the task itself. Another, more recent use of direct internal models was proposed by [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF] in the 20/07/2005 framework of their optimal control feedback model. According to this model the motor control strategy would not consist in specifying a desired trajectory in the task space and in selecting the appropriate commands for the motor system to follow this trajectory. It would instead make use of feedback information during the execution of movement to selectively modify motor commands in an optimal way, when deviations in the task space occur that would endanger the achievement of the task goal. The proposed use of feedback would require long-delay loops that are known to generate instabilities in closed-loops servomecanisms and that would consequently induce inadequate corrections from the controller.

To overcome these problems, Torodov & [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF] have proposed using the outputs of the internal models as afferent signals.

Thus, from the perspective of motor control models such as the desired trajectory hypothesis, or Harris & Wolpert's (1996) proposal, or the optimal control feedback model, internal models are very powerful tools for dealing both with the many-to-one nature of the relations between motor commands and vocal tract configurations or spectral characteristics of the acoustic signal, as well as with the long-latency of feedback processed by the cortex, and with the possible multimodality of the speech task representations. And there is considerable experimental evidence to suggest that internal models are set up and used in the control of speech production control.

Experimental findings support the existence of internal models for speech production control.

Most of the data supporting the internal model hypothesis for speech production were obtained in perturbation experiments where a noticeable change was artificially introduced either in the speech production apparatus or in speech perception feedback.

20/07/2005 [START_REF] Savariaux | Compensation strategies for the perturbation of the rounded vowel [u] using a lip-tube: a study of the control space in speech production[END_REF][START_REF] Savariaux | Compensation strategies for the perturbation of French [u] using a lip tube. II. Perceptual analysis[END_REF] perturbed the production of the French [u] by introducing a 2 cm diameter tube between the lips of their subjects. None of the subjects could compensate for the perturbation without a training phase where they could listen to their acoustic speech production. However, immediately after the insertion of the tube, the large majority of the subjects showed articulatory changes that tended to partly compensate for the perturbation by reducing the acoustical and perceptual gaps between their normal and their perturbed [u]'s. In addition, none of them provided changes in a direction that would have been incompatible with a compensation. These observations support the idea that prior to their first production with the lip tube, speakers would use an internal representation to indicate in which direction articulatory changes should be provided to compensate for the perturbation. [START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF] reported results about the spectral characteristics of speech produced by cochlear-implanted patients according to whether the cochlear implant was turned on or off.

It was shown that the degradation of speech without the auditory feedback was very slow, and that it was perceptually noticeable only in alveolar fricatives that require accurate tongue positioning. This supports the hypothesis of the existence of internal representations of the relations between motor commands and spectral characteristics of the signal which would allow the subjects to predict their speech signals in spite of the absence of auditory feedback.

Note that [START_REF] Perkell | A theory of speech motor control and supporting data from speakers with normal hearing and with profound hearing loss[END_REF] suggested that auditory feedback would be necessary to maintain these representations.

Houde & Jordan (1998) set up an original experimental protocol where subjects were asked to produce whispered steady state vowels while listening via headphones to acoustic feedback in that had been distorted by slightly shifting the formants.. Subjects were asked to compensate for the auditory perturbation so that they no longer perceived any perturbation.

All subjects could compensate after training, and an after-effect was also systematically observed once the auditory feedback perturbation was removed. These findings support the idea that due to the distortion of the auditory feedback, subjects learned new relations between their articulatory commands and the associated spectral characteristics of the speech signal, i.e. a new internal model, and that they relied after training on this new representation to control their speech production.

The three experimental paradigms described above could in fact be explained by static internal models describing the link between steady-state articulatory configurations of the vocal tract and spectral patterns. However, a recent experiment carried out by [START_REF] Tremblay | Somatosensory basis of speech production[END_REF] suggests that a more complex internal representation could be required for speech motor control. Subjects were asked to produced vowel-to-vowel transitions while a velocitydependent perturbing force was applied to their jaw. Thus, perturbations modified the dynamical state of the speech apparatus and not only the static relations between articulation and perception. After training, the subjects were able to compensate for this perturbation in such an accurate way that the jaw moved along the same articulatory path as under normal conditions. In addition, an after-effect was observed once the perturbation was removed. This suggests that the subjects were able to learn a new internal model taking into account the modified dynamical properties of their jaw, and to use it in their speech production.

Interestingly, the learning was not observed, when the task only consisted of moving the jaw from a closed to an open position without speaking.

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What kind of internal models for speech production control?

As shown in the preceding section, different sources of evidence show that the control of speech production could incorporate and make use of internal models. These models could be purely static, as suggested in the first three examples, or could require a more complex description of the physics as shown by [START_REF] Tremblay | Somatosensory basis of speech production[END_REF]. This raises the question of the level of complexity of the internal models of speech production. This is a key issue, and not only for speech production research, but for motor control research in general as illustrated by the controversy between Kawato and colleagues and Ostry and colleagues in their respective attempts to explain arm trajectories in pointing tasks [START_REF] Gomi | Equilibrium-point control hypothesis examined by measured arm stiffness during multijoint movement[END_REF][START_REF] Gribble | Are complex control signals required for human arm movement[END_REF]. Does the motor control system need internal models that describe the motor system in its whole complexity, including dynamical and biomechanical aspects (Kawato et al.'s hypothesis), or could a rough description be sufficient (Ostry et al.'s assumption)?

In the case of speech motor control this question becomes particularly acute, since the physical properties of speech apparatus are particularly complex:

(1) The tongue is a hydrostat, i.e. a soft body with highly complex, typically non linear, physical properties and deformation capabilities (see Gerard et al., in this issue, for more details). This articulator is carried by the mandible, a hard body with dynamical properties that differ strongly from the ones of the tongue. Hence, the mechanical coupling between these two articulators cannot be described by simple mechanical laws.

(2) The tongue moves in a narrow space delimited by the pharyngeal walls, the palate, the teeth, and the cheeks. Consequently, the tongue is often in contact with external structures. The amount of contacts as well as their location vary during speech 20/07/2005 production, and this induces over time changes in the physical constraints on tongue movements (damping, limitation of movement amplitude in a given direction…).

Thus, the mechanical and dynamical properties of the peripheral speech apparatus are not only, very complex. They also vary during the movements, since they depend on the positions of the articulators in relation to each other and on the strain of the soft tissues.

Consequently, building up accurate internal models of this apparatus means, first, accounting for a strong mechanical and dynamical complexity, and, second, making the models sufficiently flexible in such a way that they can adapt based on information about the internal structure of the soft tissues and about their relations to solid structures. Recent publications have shown evidence for the very good capacity of the Central Nervous System to learn adaptable internal models of complex and time variable dynamics to control accurately the arm or the limb [START_REF] Flanagan | The inertial anisotropy of the arm is accurately predicted during movement planning[END_REF][START_REF] Dingwell | Manipulating objects with internal degrees of freedom: evidence for model-based control[END_REF][START_REF] Wainscott | Internal models and contextual cues: encoding serial order and direction of movement[END_REF].

However, in the case of speech articulators, things seem to be even more complex, since, in this case, to be accurate, internal models would have to adapt very quickly and on the basis of short latency proprioceptive afferent signals. Hence, it seems reasonable to assume that internal models of speech production are inaccurate functional descriptions of the speech motor system. From this perspective, different levels of complexity, from a pure static model to a sophisticated dynamical one, can be considered. In this context, an important challenge consists in collecting evidence supporting one hypothesis or the other. In the following sections, results will be presented that support the hypothesis that simplified internal representations could be efficiently used for speech motor control.

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SIMPLIFIED INTERNAL MODELS COULD BE SUITED TO THE CONTROL OF SPEECH PRODUCTION

Complex trajectories do not necessarily require complex internal models

The hypothesis that planning of skilled movement sequences would be based on cost minimization principles using dynamic variables such as jerk [START_REF] Hogan | An organising principle for a class of voluntary movements[END_REF] or torque change [START_REF] Kawato | Trajectory formation of arm movement by cascade neural network model based on minimum torque-change criterion[END_REF]) is often proposed to explain specific kinematic characteristics of human movements such as the curved path observed for the finger in pointing tasks [START_REF] Gomi | Equilibrium-point control hypothesis examined by measured arm stiffness during multijoint movement[END_REF] or for the tongue in speech tasks [START_REF] Löfqvist | Control of oral closure in lingual stop consonant production[END_REF]. However, our recent work carried out with a 2D biomechanical model of the tongue does not support this point of view [START_REF] Perrier | Influences of tongue biomechanics on speech movements during the production of velar stop consonants: A modeling study[END_REF].

We simulated tongue movements in Vowel -Velar Stop -Vowel sequences with a 2D biomechanical model of the tongue, which includes the main muscles responsible for shaping and moving the tongue in the midsagittal plane (posterior and anterior parts of the genioglossus, styloglossus, hyoglossus, inferior and superior longitudinalis and verticalis).

Tissue elastic properties are accounted for in finite-element (FE) modeling. Muscles are modeled as general force generators which (1) act on anatomically specified sets of nodes of the FE structure, and (2) modify the stiffness of specific elements of the model to account for muscle insertions into tongue tissues. The model was controlled via a target-based model using Feldman's Equilibrium Point Hypothesis [START_REF] Feldman | Once more on the Equilibrium-Point Hypothesis (λ model) for motor control[END_REF]. In our simulations, each sound was specified within a given speech sequence by a set of motor commands that determine a static equilibrium position of the tongue. The transition from one sound to the next was produced by shifting the motor commands at a constant rate of shift. Hence, the temporal paths of the motor commands were strictly dependent on the time of occurrence of discrete sets of motor commands specifying the successive equilibrium positions associated with the phonemes of the sequence. These paths were not determined by any requirement on the articulatory trajectory itself. Under these conditions, the simulated tongue trajectories replicated nicely the well-known curved aspect observed in real speakers (see Perrier et al., 2003, for details). Consequently, there is no need to hypothesize any cost minimization based on dynamical variables, nor on dynamical internal models, to explain the curvature of the articulatory paths: it could be simply due to biomechanical properties of the tongue, i.e.

passive tongue elasticity, muscle arrangements within the tongue, and force generation mechanisms. These results do not disprove the hypothesis of complex internal representations, since jerk or torque minimization principles could act in combination with biomechanical factors. However, our simulations suggest that this hypothesis is not a necessity to explain curved articulatory trajectories in speech.

We do not want to discard the possibility that internal representations could include some knowledge about the dynamics of the speech apparatus. It is well-known that time control is a major issue in speech production, since phone durations can carry a significant part of the linguistic information. Since time is strongly dependent on the dynamics of the motor system, it is very unlikely that the dynamics could be excluded from the internal representations that the brain builds up to control the movements. Now, the question is whether coarsely specified dynamical internal models could be sufficient to reach the time and positioning accuracy required for speech production.

Inaccurate dynamical representations are compatible with movement accuracy

Inaccurate internal models of the speech production apparatus will induce errors in the estimation of motor commands from the specification of the desired gestures. Consequently, they will induce gesture and acoustic inaccuracy, unless motor commands can be corrected during the execution of movement. To deal with this problem, Kawato and colleagues (for example [START_REF] Kawato | Trajectory formation of arm movement by cascade neural network model based on minimum torque-change criterion[END_REF]1987;[START_REF] Gomi | Adaptive feedback control models of the vestibulocerebellum and spinocerebellum[END_REF][START_REF] Kawato | Internal models for motor control and trajectory planning[END_REF] proposed to use a combination of a feedforward controller, inferring motor commands from the desired output by using inverse dynamics, and of a feedback controller; taken into account the output errors of the motor system to correct on line the motor commands. This is definitely a good framework to further think about how to deal with internal model inaccuracy. Still, there are two important problems remaining :

(1) How to deal with feedback delays?

(2) Is such a feedback controller compatible with a sparse description of the desired output, in which only target points are specified?

As explained above, [START_REF] Todorov | Optimal feedback control as a theory of motor coordination[END_REF] proposed a solution to the first problem, by suggesting using the outputs of the internal model as feedback signals. However, it is not clear how their control model would behave, if the feedback signals taken into account are inaccurate due to internal model inaccuracy. An alternative, very interesting, contribution to this issue was proposed by [START_REF] Hanneton | Does the brain use sliding variables for the control of movements[END_REF], who analyzed hand movements in visuomanual tracking tasks. Instead of providing a gradual correction of the motor commands based on the continuous comparison between the actual position of the motor system and a desired trajectory , their feedback controller generates from time to time a very strong change to the motor commands in order to push the motor system back to right track.

Hanneton et al.'s subjects were seated in front of a computer screen on which a point moved continuously along a random trajectory. By moving a joystick commanding a simulated second order mechanical system, the subjects had to drive another point on the screen, so that it followed as accurately as possible the first point. [START_REF] Hanneton | Does the brain use sliding variables for the control of movements[END_REF] observed that hand correspond to the classical closed-loop response of the dynamical system to the motor commands.

This approach is very appealing, because it makes possible to reach gesture accuracy in spite of internal model inaccuracy, and because it seems to be compatible with a target-based model of control, since a sparse description of the output based on discrete target sequences would not allow a continuous feedback correction. Hanneton et al. (1987) have demonstrated that system stability is possible, even if feedback corrections to the inputs are not continuous and strong. However, it should be noted that [START_REF] Hanneton | Does the brain use sliding variables for the control of movements[END_REF] experiment involved long-latency visual feedback, since according to their model, the delay time of this feedback control would be 290 ms. This would be obviously far too long for speech production.

However, the corrective movements proposed by [START_REF] Hanneton | Does the brain use sliding variables for the control of movements[END_REF] do not seem to require a cortical processing of the feedback, since they are ballistic and fully determined by the initial impulse. Hence, it seems possible that this model could be applied to speech by taking into account low-level short latency feedback, such as oro-sensory afferences. Further work is obviously needed to test this hypothesis for speech production.

CONCLUSION

In this paper, we have provided various types of evidence in support of the hypothesis that internal models of the speech production apparatus, from the motor commands to the articulatory patterns or to the spectral characteristics of the acoustic signal, could exist and be used in the planning and/or the execution of speech production. We have also shown that complex articulatory patterns would not necessarily require complex internal models, since a part of the measured complexity could arise from the physical characteristics of the speech production system.

20/07/2005

Nevertheless, the necessity for speakers to control temporal events in speech production with accuracy raises the question of the level of complexity that is required for the internal models to take into account dynamics. The sliding variable hypothesis offers a robust framework to think about inaccurate dynamical descriptions of the speech production system, without the need to specify the complete desired articulatory paths. However, a role for short-latency feedback needs to be considered given that in speech production movement durations are shorter than long-latency feedback delays.

Very sensitive mechanoreceptors have been found in the human tongue (Trulsson & Essick, 1997). Neural linkages between the upper lip and the jaw have also been hypothesized (Gracco & Abbs, 1985), in order to explain compensatory movements of the upper lip in case of a sudden perturbation applied to the jaw (note that [START_REF] Gomi | Compensatory articulation during the bilabial fricative production by regulating muscle stiffness[END_REF], have suggested that, in addition to the neural linkage, a dynamical coupling between lips and jaw could contribute to the compensatory movement). Hence, short latency reflexes exist in the neural circuitry of the speech articulators, and they seem to contribute to the stability of the speech output.

Hence, it is worthwhile looking further for general neuro-physiological evidence that short latencies reflexes could be used in combination with a sparse description of the desired articulatory paths and simplified internal representations to produce accurate speech movements.

movements were made of a number of small successive sub-movements associated with high velocity peaks and separated from each other by short periods in which velocity was small. Spectral analysis of these trajectories showed that the dynamical system controlled by the subjects and consisting of the joystick and the simulated mechanical system had non linear dynamical properties.

The authors could reproduce the observed movement patterns with a control model based on the processing of an intermediate sliding variable [START_REF] Slotine | Applied non-linear control[END_REF], while assuming linear characteristics for the dynamical system controlled by the subjects.

"A sliding variable is a specific combination of the instantaneous error and its successive time derivatives. […] Sliding variables can be viewed either as a prediction about the error or as a criterion to be minimized by an algorithm adapted to the dynamics of the system being controlled. […] In the dynamic case, […] control mechanisms are built based on arrangements of feedforward and feedback components: the feedforward part provides an estimate of the inverse dynamics of the controlled system while the feedback part compensates for uncertainties about this estimate." (Hanneton et al., 1997, p. 382).

Thus, these authors proposed that the sub-movements associated with high velocity peaks could be produced in a ballistic-like way by the Central Nervous System when the absolute value of the sliding variable exceeds a constant threshold. Once triggered, these parts of the movements would be independent of any feedback (open-loop movements). The role of these rapid open-loop movements would be to bring rapidly the sliding variable value back below the threshold. The other parts of the movement, those having a small velocity, would