1. Introduction

Image segmentation is indispensable in applications that use imaging systems. It facilitates the extraction of useful information for subsequent high level image analysis. For instance, in pathological research, digital microscopy becomes increasingly popular since the introduction of high-throughput tissue microarrays (TMA) into bioimaging communities [1]. It is therefore crucial to segment each tissue image into a meaningful partition in an accurate, fast, automated and robust manner. In robotics, segmentation is often coupled with 3D scene reconstruction.

Texture may be defined as the variation of data at scales smaller than the scales of interest [2]. In segmentation, texture can be a useful cue for object recognition, but it can also be a nuisance since it creates many extra lines in the edge map. Most segmentation algorithms deal with texture by first transforming the intensity levels to high-dimensional feature vectors, followed by looking for regions which are homogeneous in the feature space. For example, texture features extracted by Gabor filters [3] or Laws’ filters [4] have been widely used [5, 6, 7, 8]. Numerous approaches for texture segmentation exist: clustering [5], region-growing [9], graph-partitioning [10] and optimization model-based [11, 12, 13, 14]. Recent surveys in this topic can be found in [15] and [16].

1.1. Motivation

In the literature, ample research has also been done on designing algorithms for segmenting non-texture images, i.e. images that can be well-approximated by piecewise smooth functions. Prominent examples include the Mumford-Shah model [17], watershed algorithm [18], snake/active contour models [19], and their thousands of descendants. Although many of these algorithms are originally designed for non-texture piecewise smooth images, almost all of them can be naturally generalized to handle textured images reasonably well. This is because of the optimism that when feature extraction is done properly, the feature vector over the image domain is more or less piecewise smooth. Of course, when dealing with high-dimensional feature spaces, some complications arise.

First of all, textured images and their transforms often have no sharp boundary. For example, features computed over a sliding window tend to be smeared even if the original intensity map has sharp edges. Second, the algorithm has to be robust to noise while still be able to capture small scale textures. Lastly, one has to face the feature selection problem described as followed.

1.2. Subspace Clustering

Different textures are better captured by different features. Hence, a high-dimensional feature space is usually considered for texture segmentation. However, for each given pixel the number of relevant features is usually small, making the data very sparse. It is well-known in high-dimensional data analysis that distances computed in high-dimension are less reliable since contributions from irrelevant dimensions become significant. Moreover, in practice users may want to include all kinds of features and let the algorithm decide which set of features are relevant. A successful approach to alleviate this problem is to use subspace clustering techniques so that objects are projected onto their own relevant subspace only.

Subspace clustering is a well-studied problem in data mining community [20, 21, 22, 23, 24]. For example, in clustering of text documents, each document is typically represented by a vector containing the (suitably normalized) frequency of a large universal set of keywords. Since documents in each cluster usually concentrate on a specific topic, they share a relatively small set of keywords. Hence different clusters are characterized by different sets of keywords. Another example is supplier relationship management where each supplier, and hence the cluster that it belongs, is associated with a set of items. In the context of subspace clustering, these are examples where different clusters live in different subspaces. In [25] Jing et al. proposed an entropy weighting k-means algorithm for subspace clustering of high-dimensional sparse data. They applied the algorithm to the problems of document clustering and supplier relationship management and yielded promising results showing the usefulness of the subspace selection procedure. The model they proposed is as follows. Let U = {u_ki}, c = {c_ij} and Λ = {λ_ij}. The objective to optimize is given by

Here, f _k = (f _k1,..., f_km) for k = 1,2,...,N are the N given m-dimensional data points; c _i = (c _i1,...,c_im) for i = 1,2,...,n are the n cluster centers; λ_ij is the weight of jth feature in the ith cluster; u_ki is the degree of membership of the kth point to the ith cluster; and γ > 0 is a parameter controlling the entropy of the feature weights. The unknowns U and Λ are required to satisfy the following constraints:

This model simultaneously minimizes the within-cluster dispersion and maximizes the entropy of the weights to stimulate more features to contribute to the identification of clusters. Setting γ = 0 results in a trivial solution where for each cluster one particular feature gets a full weight of 1. In contrast, setting γ = ∞ reduces to the original k-means where all features are equally weighted.

While subspace clustering algorithms are very useful for clustering high-dimensional data, most existing ones are designed for generic data that they do not consider any geometric properties of the segments in the spatial domain, e.g. regularity of the boundary. It is thus attractive to incorporate geometric features into the k-means subspace clustering model.

1.3. Vector-Valued Mumford-Shah Model

Among the various segmentation approaches, optimization model-based methods often give very promising results. An advantage of this approach is that one can incorporate domain knowledge into the model explicitly through rigorous mathematical modeling. This makes such kind of methods highly adaptive to deal with various kinds of segmentation problems [26, 27, 28].

The Mumford-Shah (MS) model [17] is one of the best optimization models for non-texture segmentation. A distinctive feature is that it is region-based which allows it to segment objects without sharp edges. Some promising results in segmenting objects without edges are obtained in [29, 30]. For the same reason, it can group a cluster of smaller objects into a larger object. Moreover, the geometry of the segments is regularized explicitly. Thus it is robust to noise that it does not require any pre-smoothing, which can dislocate edges. These features are very attractive for texture segmentation. It is thus tempted to further develop the Mumford-Shah model to handle high-dimensional sparse data.

In particular, we consider the vector-valued piecewise constant version of the Mumford-Shah model by Sandberg et al. [31] which seeks to minimize the following objective

The set of curves C partition the image domain Ω into n mutually exclusive segments Ω_i for i = 1,2,...,n. The idea is to partition the image so that each feature f_j in each segment Ω_i is well-approximated by a constant c_ij. The goodness-of-fit is measured by the fitting term ∑^m _j=1 ∫_Ωi|f_j(x,y)-c_ij|² dxdy. On the other hand, a minimum description length principle is employed which requires the curves C to be as short as possible. This increases the robustness to noise and avoids spurious segments. The parameter μ > 0 controls the trade-off between the goodness-of-fit and the length of the curves C. It can be easily shown that for each fixed C, the optimal constant c_ij is given by the average of f_j over Ω_i.

In [12], Sandberg et al. considered the application of this model to texture segmentation using Gabor features. The result showed that Gabor features are able to capture the characteristics of various kinds of textures. However, after some features have been selected (either manually or automatically by some criteria), all the selected features are used to describe all pixels. Hence, as mentioned in Section 1.2, computing similarity between pixels is problematic and erroneous. In summary, the vector-valued active contour without edges model is natural for the use of texture segmentation. Together with Gabor features as input channels, the model is able to obtain good segmentation results for certain kind of images if relevant features are selected properly. However, when different regions lay on different subspaces, this approach might fail to retrieve a good result. In this case, subspace clustering technique is indispensable.

1.4. Objective

In this paper, we propose the use of subspace clustering techniques for texture segmentation. Our first contribution is a novel model, called the Semi-Supervised Subspace Mumford-Shah model, which combines subspace clustering and Mumford-Shah segmentation techniques. However the natural unsupervised learning of a feature subspace can easily be trapped in a local solution. Our second contribution is a novel semi-supervised algorithm which is much more robust for optimizing the proposed model.

We propose to incorporate the entropy weighting k-means subspace clustering model [25] into the Mumford-Shah model to facilitate feature selection, which is important for the often high-dimensional problem of texture segmentation. The two methods combine naturally since they share the same data fidelity term. We use a soft subspace clustering approach so that relevant features are given a larger weighting and vice versa.

In most k-means type algorithms, the solution obtained is only a local minimum of the objective function and depends highly on the initial guess. This problem is even more severe in subspace clustering since the objective involves extra variables (the weights) in which the objective is not jointly convex. While such an unsupervised learning of the segmentation and the weights fails to work, an alternative is to first use a supervised approach to learn the weights based on some sample textures and then use an unsupervised method to construct the segmentation. This supervised approach can greatly reduce the “number of local minima” making the computed solution more robust to the initial guess. But it assumes that the sample segments are good representatives everywhere in the image. This can be unrealistic in many real images.

To alleviate these problems, we propose a semi-supervised approach that makes use of information derived from both the intermediate segmentation results and the regions-of-interest selected by the user to obtain the weights of the features. We show that the proposed model can separate textures which are otherwise difficult in the original full feature space.

2. Algorithm

A work flow of the Semi-Supervised Subspace Mumford-Shah method is shown in Fig. 1. The details are described in the following sections.

 

Fig. 1. A work flow of the Semi-Supervised Subspace Mumford-Shah method. Features such as Gabor and RGB are first extracted from the image. Then a vector-valued Subspace Mumford-Shah model is used to segment the image, where the feature weights {λ_ij} are learned with the help of the selected ROI’s.

Download Full Size | PDF

2.1. Feature Extraction

2.1.1. Gabor Features

Gabor filters have been widely used in extracting textured image features [3, 6, 12, 11, 13, 7] since they are designed to capture local patterns in different scales and orientations. A 2-D Gabor function G(x,y) is defined as [6]:

where W is the modulation frequency and i = √-1. Then a self-similar filter dictionary can be obtained by dilations and rotations:

where a > 1, p ∊ ℤ, q ∊ ℤ, θ = qπ/K and K is the total number of orientations. If σ_x, σ_y, W are fixed, then the parameters of a Gabor filter are the orientation q and the scale p. By varying p and q, we can obtain a set of Gabor functions {G_Pjqj}_j. Given an image u, its Gabor feature at location (x,y) is defined to be

which is the modulus of the Gabor transform of u. Here G¯ indicates the complex conjugate of G. Thus, we can obtain a set of input channels {f_j}_j for texture segmentation.

Gabor transforms contain many redundant information since the support of their filter responses in the frequency domain are highly overlapped. To reduce the redundancy, Manjunath and Ma [6] designed a scheme to select an appropriate subset of filters which ensures that the half-peak-magnitude support of the filter responses in the frequency spectrum touch each other. We apply this scheme in our experiments for parameter setting.

2.1.2. Features Derived from a Texture Database

A distance measure between two images in an image database has been proposed in [6]. The distance is simple to compute and has been shown to be quite effective in the context of image retrieval. We propose a novel set of features which utilizes such a distance measure. The feature vector at a pixel location is the set of distances between the local patch around the pixel and each texture patch in a texture database. The details will be discussed as follows.

We here present the distance proposed in [6]. Given a set of Gabor features {f_j} derived from an image u, we define for each Gabor feature the mean and standard deviation as follows:

We also denote the standard deviation of μ_j(∙) and σ_j(∙), calculated over the entire image database, by std(μ_j) and std(σ_j) respectively. Let u and v be two images (which may have different sizes). Let {f_j} and {g_j} be the Gabor features derived from u and v respectively. The distance between u and v is defined as

Now, given an image u and a texture database {v_k} where each v_k for k = 1,2,..., K is a texture patch, we define a feature value f_k(x,y) at pixel (x,y) by

for k = 1,2,..., K and for (x,y) ∊ Ω. Here u(∙,∙|x,y) is a local patch in u around (x,y). In some experiments below, we combine these K features with others such as Gabor features and Laws’ features to form a large feature space. We note that a subset of 64 in the Brodatz album [32] is used as the texture database, over which the quantities std(μ_j) and std(σ_j) are computed.

2.2. Semi-Supervised Subspace Mumford-Shah Segmentation Model

In this section, we present the proposed semi-supervised model. To better explain the ideas behind, we first introduce an unsupervised version.

Let C be a set of curves partitioning the image domain Ω into n mutually exclusive segments Ω_i for i = 1,2,..., n. Let c = {c_ij} and Λ = {λ_ij} for 1 ≤ i ≤ n and 1 ≤ j ≤ m be two sets of scalars. The Mumford-Shah image segmentation model (2) and the k-means subspace clustering model (1) can be naturally combined as follows:

We refer this model as the Unsupervised Subspace Mumford-Shah model (USSMS). This model inherits the essential features of models (2) and (1)—it enables cluster-wise feature selection and it regularizes the geometry of the segments in the spatial domain to avoid over-fitting. In this model, we seek for a segmentation of the image such that the jth feature value within the segment Ω_i is well-approximated by a constant c_ij, provided that the weight λ_ij is “significant”. If λ_ij is close to zero, then the determination of the ith segment is insensitive to the jth feature. Thus, a key feature of the model is that each segment can use a different subspace in computing the within-segment dispersion. Therefore, it opens up the possibility of using an optimal subset of features for each segment.

Unfortunately, this objective function is highly non-convex which is difficult to optimize. We found empirically that many spurious local minima can exist. We tried several standard optimization methods as well as with different initial conditions but their performance has been disappointing. The results often depend very much on the initial guess. Thus this model can be difficult to use in practice. To resolve this problem, we propose the following Semi-Supervised Subspace Mumford-Shah model (SSSMS) which amounts to the minimization of the following objective function:

Here ROI_k is the k-th user-specified region-of-interest (ROI) which serves as a sample of the k-th segment and

is the fitting error of the jth feature over ROI_i. The optimal fitting constant is given by

The parameter β ∊ (0,1) is to control the degree of involvement of the ROI’s. In the algorithm presented in the next section, the parameter β is gradually decreased from 1 towards 0. Thus the ROI’s eventually drop out from the objective F ^SSSMS which converges to its unsupervised version F ^USSMS. This can make the final weights depend on the image globally thus preventing any bias introduced by the ROI’s. One may think of this approach as using the ROI’s to specify an initial guess for the optimization of the unsupervised model F ^USSMS.

Except for the introduction of the constant |Ω|/∑_k|ROI_k| to normalize the difference between the size of the ROI’s and Ω, the proposed objective function is essentially a convex combination of F ^USSMS (C, c, Λ) and F ^USSMS (C̃, c̃, Λ) where c̃ = {c̃_ij} and C̃ is the set of curves defining the ROI’s and is therefore determined. Roughly speaking, the idea of the model is that the given ROI’s (i.e. C̃) allow us to estimate the weights Λ more accurately. Once the weights are in place, the problem essentially reduces to the original Mumford-Shah problem. Of course, the Mumford-Shah model itself can possess multiple local minima, but the problem is much more amenable. See [33] for global optimization of the Mumford-Shah model.

2.3. Optimization of the Objective

To optimize the objective F ^SSSMS, we use an alternating minimization approach (cf. the second and third boxes in Fig. 1). The main steps are as follows. The parameter β is reduced every other M steps.

The details of the implementation of Steps 4.1)–4.3) are given in the Appendix.

3. Methods

Five real-life images are used for evaluation. The endometrial tissue image is from [34]. The rat myocardium muscle image and the cuttlefish image are obtained from the World Wide Web. The zebra image is from [12]. The stem cell images are obtained within the first and second authors’ institute. Each selected image has different textures, some of them have noticeable pattern (e.g. zebra) and some have not (e.g. cuttlefish). This allows us to test the effectiveness of the proposed feature selection procedure on different types of textures.

We use a large set of features to see how the algorithm determines the optimal weights. As a result, some less informative and less discriminative features are also included. In this way, we can compare the results with and without subspace clustering. We expect that if the number of irrelevant features is increased, then the original MS model might perform worse since the distance is less reliable. However, our SSSMS model is much less sensitive to irrelevant features.

In all tests, we use the parameter setting in [6] (i.e. (σ_x,σ_y) = (1.4054,1.8544) and W = 0.4). The parameter β is set to 0.99 initially and is periodically decreased (e.g., every other 100 steps) at a rate of 0.9, i.e., β → 0.9β. The parameters μ and γ are chosen manually. The detailed setting of other parameters are reported in Section 4 individually for each example. In all examples, we normalized the feature values to between 0 and 1.

To validate our method, we compare it with the Mumford-Shah segmentation model (2) proposed in [12]. For the MS model, we set an initial segmentation u ⁰ to be 1 (resp. 0) within the area of the ROI which represents foreground (resp. background) and randomly set a value from {0,1} elsewhere. We also set the initial c_ij to be the mean feature value of the j-th feature taken over the corresponding ROI_i. For the SSSMS model, we use a random assignment as the initial segmentation u ⁰. Both methods use the same full feature spaces. In addition, we report the result of the MS model with nonlinear structure tensor features (MS_NLST) [35] as input channels and an initial segmentation as described above for the MS model. In the last example, we use the k-means clustering result as an initial segmentation for all three stem cell images. The ROI’s are selected from one of the three images and are used for all the three images for segmentation. The percentage symmetric difference between the manual segmentation and the computed segmentation is used as a measure of segmentation error. In the first four real life examples, we tried a wide range of μ for the MS and the MS_NLST models and reported the one with the least segmentation error. The errors of the models of SSSMS, MS and MS_NLST are shown in Table 2. All algorithms are implemented in MATLAB 7. The tests are done with a Pentium D 3GHz machine.

4. Results

4.1. Endometrial tissue image

In this test, we use an endometrial tissue image with size 225 × 225 containing some endometrial glands and stroma. In pathology research, the ratio of the area of these two kinds of tissue is crucial for aiding the diagnosis of some diseases such as hyperplasia and extraperitoneal endometriosis. To segment the image, we use the RGB channels and twelve Gabor features with 2 scales and 6 different orientations (θ ∊ {0, π/6, π/3, π/2,2π/3,5π/6}). In Fig. 2, the results show that both the MS and the MS_NLST models cannot find a clear boundary while the SSSMS model can. In the second row of the figure, we also show the top channel having the largest weighting for each region. The weights of the two most significant channels for each region are shown in Table 1.

 

Fig. 2. Segmentation of endometrial tissue: SSSMS and MS models with three RGB channels and twelve Gabor features as input features, and MS_NLST with nonlinear structure tensor features. μ = 0.04 (SSSMS) and 0.17 (MS) and 0.002 (MS_NLST), γ = 0.5.

Download Full Size | PDF

Table 1. The weights of the top two channels for each region.

View Table | View all tables in this article

4.2. Muscle image

In this test, we use a transmission electron microscopy image of rat myocardium with size 300 × 300. Our aim is to separate the nucleus in the upper part of the image from the striated muscle. To achieve this, we use three Gabor features with the same scale but different orientations (θ ∊ {0,π/3,2π/3}) and 4 Brodatz texture similarity measures (see Section 2.1 for details). The user specifies one ROI for each kind of cells. In Fig. 3, the results show that both the MS and the MS_NLST models cannot find a clear boundary while the SSSMS model can. In the figure, we also report for each region the top channel having the largest weighting. Note that in both cases, the top channel is a Brodatz texture similarity measure.

 

Fig. 3. Segmentation of a rat myocardium image: SSSMS and MS models with 3 Gabor features and 4 Brodatz texture similarity measures as input features, and MS_NLST with nonlinear structure tensor features. μ = 0.5 (SSSMS) and 1.5 (MS) and 0.0085 (MS_NLST), γ = 0.5.

Download Full Size | PDF

4.3. Cuttlefish camouflage

In this test, we use a cuttlefish image with size 260 × 260 containing two infant cuttlefishes where one is located in the center and (part of) the other is located in the upper left corner of the image. The cuttlefishes in the image can hardly be seen in the RGB space. It is because cuttlefish has the ability to regulate its disruptive skin patterns for camouflage on benthic substrate [36] such as sand in this case. To segment the image, we use 6 Gabor features with the same scale but different orientations (θ ∊ {0,π/6,π/3,π/2,2π/3,5π/6}), 14 Laws’ Mask features [4, 37] and 64 Brodatz texture similarity measures (see Section 2.1 for details). To optimize the performance of both the MS and the SSSMS models, we first select the top five channels which have the highest mean intensity differences between two selected ROI’s. In Fig. 4, the results show that the SSSMS model is more capable of distinguishing the two camouflaged cuttlefishes from the sand substrate than the MS and the MS_NLST models. In the figure, we also report for each region the top channel having the largest weighting. Note that in both cases, the top channel is a Brodatz texture similarity measure.

4.4. Zebra image

The purpose of this experiment is to test all the three models on an image which has been used by many researchers to evaluate texture segmentation methods and serves as a benchmark for direct comparison with other models. In this test, we use a 200 × 200 image of two zebras with noticeable contrast in textures between object and background. We use 6 Gabor filters with the same scale and different orientations (θ ∊ {0, π/6, π/3, π/2, 2π/3, 5π/6}). The results are shown in Fig. 5. As expected, all of the models are able to obtain a good segmentation result. But the proposed model has an additional advantage that it can reveal the dominated features of each segment. For example, in the zebra region, the Gabor features with θ ∊ {π/6, π/3, π/2} have the highest weight. In the grassland region, all features have a similar weight. These observations are consistent with the textures of the zebra and the grassland.

 

Fig. 4. Segmentation of two cuttlefishes: SSSMS and MS models with 84 input features before feature selection process, and MS_NLST with nonlinear structure tensor features. μ = 0.1 (both SSSMS and MS) and 0.006 (MS_NLST), γ = 0.5.

Download Full Size | PDF

The percentage errors of the segmentations in Fig. 2, 3,4 and 5 (with respect to some manual segmentations) are shown in Table 2. We remark that each result of the MS and the MS_NLST models in Fig. 2, 3, 4 and 5 has been optimized by choosing a parameter μ that minimizes the segmentation error.

Table 2. Percentage errors (percentage symmetric difference) w.r.t. manual segmentations.

View Table | View all tables in this article

4.5. Stem cell images

In practice, given a set of images with similar textures, one may want to select ROI’s once from one image and use it for all images for segmentation. The aim of this experiment is to show whether this strategy works in our case. In this test, we use three mesenchymal stem cell images with size 305×305 containing some differentiated cells and some undifferentiated cells under different conditions. The ratio of the area of these two kinds of cells is crucial for understanding cell differentiation mechanisms. To segment the images, we use the red channel and six Gabor features with same scale but different orientations (θ ∊ {0,π/6,π/3,π/2,2π/3,5π/6}) and specify one ROI for each kind of cells in one image. In Fig. 6, the results show that the SSSMS model can find a clear boundary for all three images. In the figure, we also report the percentage symmetric difference between the results and our manual segmentations. Since the ROI’s were selected from the leftmost image, it has the minimal error as expected. However, the results of the other two images are still promising.

 

Fig. 5. Segmentation of two zebras: SSSMS and MS models with 6 Gabor transforms as input features, and MS_NLST with nonlinear structure tensor features. μ = 0.02 (SSSMS) and 0.2 (MS) and 0.05 (MS_NLST), γ = 50.

Download Full Size | PDF

5. Conclusion

In this paper, we propose a model which incorporates the concept of subspace clustering into the Mumford-Shah segmentation model and propose an improved semi-supervised optimization algorithm to optimize the model. The algorithm uses the ROI’s with a decreasing degree of involvement to avoid the problem of over-specification of the ROI’s. In our experiments, we show that even when the images have a very low contrast in textures, the algorithm can give much better results than the original Mumford-Shah model since irrelevant features are filtered which in effect increases the contrast between different segments.

 

Fig. 6. Segmentation of three stem cell images: SSSMS model with red channel and 6 Gabor features as input features. μ = 0.3, γ = 5.

Download Full Size | PDF

A. Implementation Details of Alternating Minimization Algorithm

In Section 2.3, there are three subproblems that need to be solved alternatively. For the sub-problem in Step 4.1), the optimal c_ij is simply the mean feature value of the jth feature taken over the ith segment, i.e.,

where χ_i equals 1 in Ω_i and 0 otherwise.

For the subproblem in Step 4.2), the optimal weight λ_ij is given by the following closed-form formula:

where

This formula can be easily derived using some arguments similar to those used in the proof of Theorem 1 in [25]. Thus the optimal weights are easily computed for any given segmentation and ROI’s.

For the subproblem in Step 4.3), a closed-form solution does not exist. We present a fast global minimization algorithm to solve it. The objective function F ^SSSMS is non-convex and therefore may possess many local minima (even for fixed constants c and fixed weights Λ). To globally minimize the objective F ^SSSMS, one can use the Stochastic Level Set Method proposed in [33]. Here we devise a fast algorithm for minimizing F ^SSSMS w.r.t. C based on the results of Chan et al. [38], Chambolle [39], Aujol et al. [40], and Bresson et al. [41]. An advantage of the method is that a global minimum w.r.t. the curves C for fixed c is obtained. Moreover, using a dual formulation, a fast iterative method is devised. The method works extremely well in the case of two-phase segmentation.

The idea is to first reformulate the non-convex problem into a convex minimization problem [38]:

where

Then the problem is solved by solving its dual problem using a splitting method [40] and an iterative scheme [39]. It has been shown in [38] that by thresholding the optimal solution u of (6) at almost every t ∊ [0,1], we obtain a global optimal segmentation of F ^SSSMS (in the continuous setting). Here we fix t = 0.5. We refer the readers to [41] for details.

The algorithm for optimizing F ^SSSMS w.r.t. the segmentation with fixed c and Λ is outlined below. Here, θ is a small constant controlling the closeness between u and an auxiliary variable v, δ = 1/8 is a step size to ensure convergence, div is the discrete divergence operator, and ∇ is the discrete gradient operator.

In practice, we can often speed up the convergence of the outer alternating minimization in Section 2.3 by iterating only once in the inner loop in the Step 3 of the above algorithm. In this case, the alternating minimization is equivalent to the above algorithm but with the update of c, Λ and β inserted immediately after the update χ ⁿ⁺¹.