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There are domain-specific image classification problems such as facial emotion and house-number classifications, where the color information in the images may not be crucial for recognition. This motivates us to convert RGB images to gray-scale ones with a single Y channel to be fed into the pre-trained convolutional neural networks (CNN). Now, since the existing CNN models are pre-trained by three-channel color images, one can expect that some trained filters are more sensitive to colors than brightness. Therefore, adopting the single-channel gray-scale images as inputs, we can prune out some of the convolutional filters in the first layer of the pre-trained CNN. This first-layer pruning greatly facilitates the filter compression of the subsequent convolutional layers. Now, the pre-trained CNN with the compressed filters is fine-tuned with the single-channel images for a domain-specific dataset. Experimental results on the facial emotion and Street View House Numbers (SVHN) datasets show that we can achieve a significant compression of the pre-trained CNN filters by the proposed method. For example, compared with the fine-tuned VGG-16 model by color images, we can save 10.538 GFLOPs computations, while keeping the classification accuracy around 84% for the facial emotion RAF-DB dataset.

A pre-trained 2D CNN (Convolutional Neural Network) can be used for the spatial stream in the two-stream CNN structure for videos, treating the representative frame selected from the video as an input. However, the CNN for the temporal stream in the two-stream CNN needs training from scratch using the optical flow frames, which demands expensive computations. In this paper, we propose to adopt a pre-trained 2D CNN for the temporal stream to avoid the optical flow computations. Specifically, three RGB frames selected at three different times in the video sequence are converted into grayscale images and are assigned to three R(red), G(green), and B(blue) channels, respectively, to form a Stacked Grayscale 3-channel Image (SG3I). Then, the pre-trained 2D CNN is fine-tuned by SG3Is for the temporal stream CNN. Therefore, only pre-trained 2D CNNs are used for both spatial and temporal streams. To learn long-range temporal motions in videos, we can use multiple SG3Is by partitioning the video shot into sub-shots and a single SG3I is generated for each sub-shot. Experimental results show that our two-stream CNN with the proposed SG3Is is about 14.6 times faster than the first version of the two-stream CNN with the optical flow, and yet achieves a similar recognition accuracy for UCF-101 and a 5.7% better result for HMDB-51.

This paper deals with the problem of training convolutional neural networks (CNNs) with facial action units (AUs). In particular, we focus on the imbalance problem of the training datasets for facial emotion classification. Since training a CNN with an imbalanced dataset tends to yield a learning bias toward the major classes and eventually leads to deterioration in the classification accuracy, it is required to increase the number of training images for the minority classes to have evenly distributed training images over all classes. However, it is difficult to find the images with a similar facial emotion for the oversampling. In this paper, we propose to use the AU features to retrieve an image with a similar emotion. The query selection from the minority class and the AU-based retrieval processes repeat until the numbers of training data over all classes are balanced. Also, to improve the classification accuracy, the AU features are fused with the CNN features to train a support vector machine (SVM) for final classification. The experiments have been conducted on three imbalanced facial image datasets, RAF-DB, FER2013, and ExpW. The results demonstrate that the CNNs trained with the AU features improve the classification accuracy by 3%-4%.

Image segmentation is an important tool in image processing and can serve as an efficient front end to sophisticated algorithms and thereby simplify subsequent processing. We develop a multiclass image segmentation method using hidden Markov Gauss mixture models (HMGMMs) and provide examples of segmentation of aerial images and textures. HMGMMs incorporate supervised learning, fitting the observation probability distribution given each class by a Gauss mixture estimated using vector quantization with a minimum discrimination information (MDI) distortion. We formulate the image segmentation problem using a maximum a posteriori criteria and find the hidden states that maximize the posterior density given the observation. We estimate both the hidden Markov parameter and hidden states using a stochastic expectation-maximization algorithm. Our results demonstrate that HMGMM provides better classification in terms of Bayes risk and spatial homogeneity of the classified objects than do several popular methods, including classification and regression trees, learning vector quantization, causal hidden Markov models (HMMs), and multiresolution HMMs. The computational load of HMGMM is similar to that of the causal HMM.

Different similarity measures between the descriptors of the key-points certainly yield different performances in image matching. In this paper we introduce an effective similarity measurement, which considers the distances of each key-point in a query image and its matched key-point with the smallest distance in the test image. Therefore, the distances of all key-points in the query image to the corresponding matched key-points in the test image contribute to the final similarity measurement. On the other hand, the previous method considers only the distances less than a threshold value of all possible key-point pairs, which may ignore a significant part of the key-points in the query image. Our experiments show that the proposed measure yields better performance for image similarity matching and retrieval.

Image retrieval based on color histograms requires quantization of a color space. Uniform scalar quantization of each color channel is a popular method for the reduction of histogram dimensionality. With this method, however, no spatial information among pixels is considered in constructing the histograms. Vector quantization (VQ) provides a simple and effective means for exploiting spatial information by clustering groups of pixels. We propose the use of Gauss mixture vector quantization (GMVQ) as a quantization method for color histogram generation. GMVQ is known to be robust for quantizer mismatch, which motivates its use in making color histograms for both the query image and the images in the database. Results show that the histograms made by GMVQ with a penalized log-likelihood (LL) distortion yield better retrieval performance for color images than the conventional methods of uniform quantization and VQ with squared error distortion.

MPEG-7 Visual Standard specifies a set of descriptors that can be used to measure similarity in images or video. Among them, the Edge Histogram Descriptor describes edge distribution with a histogram based on local edge distribution in an image. Since the Edge Histogram Descriptor recommended for the MPEG-7 standard represents only local edge distribution in the image, the matching performance for image retrieval may not be satisfactory. This paper proposes the use of global and semi-local edge histograms generated directly from the local histogram bins to increase the matching performance. Then, the global, semi-global, and local histograms of images are combined to measure the image similarity and are compared with the MPEG-7 descriptor of the local-only histogram. Since we exploit the absolute location of the edge in the image as well as its global composition, the proposed matching method can retrieve semantically similar images. Experiments on MPEG-7 test images show that the proposed method yields better retrieval performance by an amount of 0.04 in ANMRR, which shows a significant difference in visual inspection.

Edge histogram descriptor (EHD) consists of histogram bins of local edges in an image. Instead of the conventional bin-to-bin matching for the similarity measure, this paper presents a binary string descriptor for the EHD to achieve a faster binary bit-to-bit matching.

A texture description is useful in many applications including similarity based image search and browsing. We present here three texture descriptors that are being considered for the final committee draft of the ISO/MPEG-7 standard. A comprehensive overview of the syntax and semantics of these texture descriptors is provided. The Homogeneous Texture Descriptor (HTD) and the Edge Histogram Descriptor (EHD) are useful in similarity search. The HTD characterizes homogeneous texture regions and is also useful in texture classification and recognition. The EHD is applicable when the underlying texture is not homogeneous and can also be used in sketch based retrieval. In addition, a compact descriptor that facilitates browsing applications is also defined. These descriptors are selected after a highly competitive test and evaluation phase within the MPEG group and we briefly summarize the evaluation criteria, the datasets used and the experimental results.

A novel block-based image segmentation algorithm using the maximum a posteriori (MAP) criterion is proposed. The conditional probability in the MAP criterion, which is formulated by the Bayesian framework, is in charge of classifying image blocks into edge, monotone, and textured blocks. On the other hand, the a priori probability is responsible for edge connectivity and homogeneous region continuity. After a few iterations to achieve a deterministic MAP optimization, we can obtain a block-based segmented image in terms of edge, monotone, or textured blocks. Then, using a connected block-labeling algorithm, we can assign a number to all connected homogeneous blocks to define an interior of a region. Finally, uncertainty blocks, which are not given any region number yet, are assigned to one of the neighboring homogeneous regions by a block-based region-growing method. During this process, we can also check the balance between the accuracy and the cost of the contour coding by adjusting the size of the uncertainty blocks. Experimental results show that the proposed algorithm yields larger homogeneous regions which are suitable for the object-based image compression.

This paper proposes a general unsupervised segmentation algorithm which estimates all model parameters, including the number of regions, as part of the segmentation. The general image model is a hierarchical one, consisting of Markov random fields as components of the model. The MAP criterion is adopted, in principle, for the simultaneous image segmentation and parameter estimation procedures. Due to the difficulty of implementing the MAP segmentation with a large number of unknown parameters, a novel modification of the MAP criterion is proposed. For the model parameters having closed-form ML estimates, these estimates are substituted back into the objective function to reduce the difficulty of the maximization. The remaining maximization is implemented by a recursive segmentation-parameter estimation algorithm, which yields a partial optimal solution (POS) to the maximization problem. In the special case where all model parameters have closed-form ML estimates, the proposed algorithm is equivalent to implementing the MAP criterion. The number of regions in the image is determined through a model fitting criterion tagged on to the segmentation algorithm. Special forms of the general unsupervised segmentation algorithm are developed for the segmentation of noisy and textured images. For noisy images, the image is assumed to consist of uniform graylevel regions modeled by a class of Gibbs random fields and corrupted by additive, white, region-dependent, Gaussian noise. For textured images, the image is assumed to consist of regions, modeled by a class of Gibbs random fields, which are filled with textures, modeled by Gaussian Markov random fields. The algorithms for both classes of images are applied to a wide range of images—generated according to the model, hand-drawn, natural and Brodatz textures, their combinations, and outdoor images—with notable success. Despite the large number of unknown parameters (as many as 14 for some noisy images and 36 for some textured images), the algorithms yield good segmentations, accurate estimates for the parameters, and the correct number of regions.

To automatically generate images with the same foreground but different backgrounds, a watermark bit (e.g., binary 1 for foreground and 0 for background) can be inserted for each pixel location. Then, the embedded watermark bit can be automatically extracted and the background can be separated from the object. Note that the object extraction can be done successfully only if the watermarked image is intact. However, if the watermarked image goes through some post-processing including JPEG compression and cropping, then the pixel-wise watermark decoding may fail. To overcome this problem, in this paper, a block-wise watermark insertion and a block-wise MAP (maximum a posteriori) watermark decoding are proposed. Experimental results show that the proposed method is more robust that the pixel-wise decoding for various post-processing attacks.

To reduce the semantic gap, image retrieval systems based on users' relevance feedback have been adopted. However, since this structure needs human intervention during the retrieval process, it cannot be applied to fully automated systems. To avoid this problem, we propose a feed-forward framework instead of the feed-back retrieval system, which adds a classifier to the traditional system for giving feed-forward information to maximize the average precision. That is, given a database, our proposed system improves the overall precision by selecting the best mode based on known statistics (average precision vs. recall for each category). Lloyd-clustered Gauss mixtures are used in the classifier to provide the feed-forward category information and in the quantization of color images for histogram generation.

According to the definition of the edge histogram descriptor (EHD) in MPEG-7, one can easily generate an extra histogram bin from the 5-bin local edge histogram of each 4 × 4 sub-image. This extra histogram bin defines the ratio of the non-edge area (i.e., monotonous region) in the sub-image. Forming a feature vector with 6 edge/non-edge types, we can generate 33 different feature vectors (or 33 × 6 = 198 feature elements) including 16 vectors from 4 × 4 sub-images, 1 vector from a global histogram, 13 vectors from semi-global histograms, 1 vector from entropy, and 2 vectors from centers of gravity. A statistical hypothesis testing is employed to see which feature vectors/elements are most informative to differentiate different image classes. Experimental results show that non-edge and entropy features are the most informative features among all 33/198 feature vectors/elements.

The multiscale Markov models introduced in Chapter 4 can be used to extract large scale image features by parameterizing inter scale and intra scale interactions in an image. Since there are multiple scales with different parameter values for different scales, the parameter estimation process in the multi-scale approach becomes a major computational burden. One way to alleviate this complexity is to adopt a block-wise image modeling paradigm. That is, dividing the image space into non-overlapping blocks, a random variable is assigned for each image block to represent the class label. Then, a representative feature for each image block is extracted and is treated as the observed image data. The collection of all block features constitute the realization of the random field Y. Also, the class label assigned for each image block is a realization of the unobservable random field X. Since the representative feature values for image blocks normally have spatial continuity, the random field for the block class labels can be modeled as an MRF.

The pixel-wise Markov image models introduced in the previous chapters have a limited ability in dealing with texture images. In order to describe large scale behavior in texture images, it is necessary to increase the size of the neighborhood. However, this dramatically increases the number of parameters and makes the computation for the parameter estimation quite difficult. Considering that the scales of the texture elements can be diverse, one possible way of dealing with various scales of texture elements is to express image data hierarchically using a multiscale transform. Laplacian pyramids [27] and wavelet multiresolution representations [124] are techniques that can decompose an image into multiple scales. Successive filtering and decimation operations in multiscale transforms provide a pyramid structure for image data. As a result, we have a sequence of image data for various scales and frequency bands. Specifically, coarse image components with smaller image resolution are located at the higher level and fine image components capturing detail can be found at the lower levels of the image pyramid. This multiscale image structure can provide useful information for the analysis and representation of the image data, especially for texture images. Processing successively from coarse to fine levels of the pyramid, image features at different scales and frequency bands are captured and class labels are assigned accordingly to large regions with low frequency and then refined to small image blocks and eventually to pixels with higher frequency information. This multiscale processing also allows us to save computations. That is, coarsening the image data for a multiscale image representation may smooth out local minima and result in faster convergence with a reduced sensitivity to an initial class label. Then, the optimal class labels at the current level of the pyramid to the next finer level can be obtained by searching in the vicinity of the class labels obtained at the previous coarser level. Since there exist strong inter scale and intra scale dependencies among vertically and horizontally connected neighboring data in the image pyramid, the multiscale images can be described more efficiently via Markov modeling. Adopting Markov models for the multiscale image data, the following issues arise: (i) how to describe various inter scale and intra scale interactions in terms of the Markov models, (ii) how to formulate the optimization problem for the class labels in the image pyramid, (iii) how to obtain the optimal class labels defined in (ii). These issues are treated in this chapter.

Automatic segmentation of digital cell images into four regions, namely nucleus, cytoplasm, red blood cell (rbc), and background, is an important step for pathological measurements. Using an adaptive thresholding of the histogram, the cell image can be roughly segmented into three regions: nucleus, a mixture of cytoplasm and rbc’s, and background. This segmentation is served as an initial segmentation for our iterative image segmentation algorithm. Specifically, MAP (maximum a posteriori) criterion formulated by the Bayesian framework with the original image data and local variance image field (LVIF) is used to update the class labels iteratively by a deterministic relaxation algorithm. Finally, we draw a line to separate the touching rbc from the cytoplasm.

Histogram-based image retrieval requires some form of quantization since the raw color images result in large dimensionality in the histogram representation. Simple uniform quantization disregards the spatial information among pixels in making histograms. Since traditional vector quantization (VQ) with squared-error distortion employs only the first moment, it neglects the relationship among vectors. We propose Gauss mixture vector quantization (GMVQ) as the quantization method for a histogram-based image retrieval to capture the spatial information in the image via the Gaussian covariance structure. Two common histogram distance measures are used to evaluate the similarity of histograms resulting from GMVQ. Our result shows that GMVQ with a quadratic discriminant analysis (QDA) distortion outperforms the two typical quantization methods in the histogram- based image retrieval.

The paper describes an automatic object segmentation algorithm for images with low depth of field (DOF). The low DOF images are segmented into two regions, namely, focused objects and defocused background. A local variance image field (LVIF) can represent the pixel-wise spatial distribution of the high-frequency components in the image. However, applying a thresholding method to the LVIF for segmentation often yields blob-like errors in both focused and defocused regions. To eliminate these errors, a block-wise MRF (Markov random field) image model is employed for maximum a posteriori (MAP) segmentation. After the block-wise MAP segmentation, the image blocks in the object boundary are divided into smaller blocks. Then, they are reassigned to one of the neighboring objects through the watershed algorithm, which eventually yields a pixel-level segmentation. Experimental results show that the proposed method yields more accurate segmentation than the multiresolution wavelet-based segmentation method.

We propose a texture classification method using multiple Gauss mixture vector quantizers (GMVQ). We designed a separate model codebook or Gauss mixture for each texture using the generalized Lloyd algorithm with a minimum discrimination information (MDI) distortion based on a training data set. The multi-codebook structure of the GMVQ classifier is an extension to images of the isolated utterance speech recognizer of J.E. Shore and D. Burton (see Proc. Int. Conf. Acoust., Speech, and Sig. Processing, IEEE82Ch.1746-7, p.907-10, 1982). We applied the algorithm to the Brodatz texture database and showed it to be competitive in performance in comparison to other texture classifiers. Its low complexity implementation and real-time operation make the approach suitable for content-based image retrieval.

Summary form only given. Taking the average of feature vectors from the center and neighboring blocks to a block being coded is proposed as a method of considering context information in block classification. The algorithm has the advantage of low complexity. Gauss mixture models (GMM) are adopted to extract features from image blocks, including an algorithm to handle singular covariance matrices. Two different distortion measures are used; namely log-likelihood quadratic discrimination analysis (QDA) and a dimension-compensated distortion measure defined by dividing the QDA distortion by the corresponding cell's dimension. Aerial images were used to train and test. Experimental results show that the proposed algorithm not only improves the classification performance, but also provides a solution to the singular covariance problem.

In this paper, a new variable size block segmentation for image compression is proposed. The decision whether the given image block is homogeneous or not is based on the model selection criterion. More specifically, calculating the log-likelihoods for all pre-determined region segmentations with the given image data, we apply a modified AIC criterion to select a best match. If the selected pattern turns out to be a texture or an edge, we further divide the given image block to yield a variable size block segmentation. Since the proposed algorithm takes into account the contextual information as well as the block variance for the classification, it can differentiate a texture from an edge. Moreover, due to the pre-determined block segmentations, we can further differentiate vertical, horizontal, or diagonal edges.

In this paper, a relevance feedback method is proposed to improve the retrieval performance of the MPEG-7 edge histogram descriptor. Specifically, by updating the query point and the weighting factors simultaneously, we can reduce the convergence speed of the relevance feedback retrieval. Experimental results show that the proposed method improves the performance significantly for natural and clip art images.

In this paper, we propose a composite histogram for image indexing and retrieval. The histogram represents the composition of edge and luminance features as well as colors in images. As a result of combining the different features of the image together, we can reduce the size of the histogram drastically without degrading the retrieval performance. Since the proposed histogram contains only 14 bins, it has great advantage in terms of memory and computational complexity in the image retrieval. Experimental results show that the proposed histogram with 14 bins yields better retrieval performance than the RGB color histogram with 256 bins and the HSV color histogram with 166 bins.

The purpose of this paper is to show how the edge histogram descriptor for MPEG-7 can be efficiently utilized for image matching. Since the edge histogram descriptor recommended for the MPEG-7 standard represents only local edge distribution in an image, the matching performance for image retrieval may not be satisfactory. In this paper, to increase the matching performance, we propose to use the global and semi-local edge histograms generated directly from the local histogram bins. Then, the global, semi-global, and local histograms of two images are compared to evaluate the similarity measure. Since we exploit the absolute locations of edge in the image as well as its global composition, the proposed matching method is considered to be a more image content-based retrieval. Experimental results support this claim. Experiments on test images for MPEG-7 core experiment show that the proposed method yields better retrieval performance especially for semantic similarity.

In this paper, we propose a new image feature extraction method for MPEG compressed video. To minimize the MPEG decoding process, we use only DC values for Y, Cr, and Cb components for each macroblock. Then, we can obtain a feature vector, using the decoded DC values of Y, Cr, and Cb components, for all macroblocks in I frame. The feature vector consists of histograms for various colors, luminance, and edge types. In obtaining histograms for colors and luminance features, we consider the ratio of contributing pure colors and luminance to the chroma DC values for each macroblock. Then, we update all contributing colors and/or luminance histograms accordingly. Otherwise, if the macroblock is classified as an edge block, we update the corresponding edge type histogram. To demonstrate the performance of the proposed feature extraction method, we apply it to a scene change detection problem.

In this paper, we propose a new way of obtaining an unsupervised image segmentation. Similar to the previous unsupervised image segmentation algorithms, the proposed algorithm is iterative. However, we adopt a different stopping rule such that if the region label at each pixel does not change for a sufficient number of iterations, then we confirm it as a final one and never visit the corresponding pixel for the update thereafter. Eventually, all pixels in the image will be confirmed to assure the convergence. Another feature of the proposed algorithm is that the Gibbs sampler is adopted to generate a region label for the update. This probabilistic region label update will resolve the problem of sticking at local maximum too early. Adding all these features together, the proposed algorithm updates region labels with Gibbs sampler using the current model parameter values and then, using the current region labels, the maximum likelihood estimates of the model parameters are obtained. This alternate procedure continues until all pixels in the image are declared as the confirmed final labels. Experimental results on the synthesized images with independent Gaussian noise show that the proposed algorithm yields significantly lower error rates than the previous deterministic unsupervised segmentation algorithms.

This paper presents a comparative study of three deterministic unsupervised image segmentation algorithms. All of the three algorithms basically make use of a Markov random field (MRF) and try to obtain an approximate solution to the maximum likelihood or the maximum a posteriori estimates. Although the three algorithms are based on the same stochastic image models, they adopt different ways to incorporate model parameter estimation into the iterative region label updating procedure. The differences among the three algorithms are identified and the convergence properties are compared both analytically and experimentally.

Images included in documents usually provide information that may not be readily expressible by words. For example, academic articles with similar pictures may be of interest for researchers. We deal with the problem of extracting images in digital document. Given a digital document, the optimal block size is first determined by finding the best fit of the horizontally projected gray-level pattern to a set of orthogonal basis vectors. Because the block with the optimal size is supposed to contain sufficient information to identify text regions, the proposed method is font-size independent regardless of the size of the words in the text lines. The blocks divided by the optimal block size are classified into one of image, text, and background blocks. This block classification result, in turn, is used for the initial configuration for blockwise document segmentation. The blockwise segmentation method is based on the maximum a posteriori (MAP) framework with a deterministic relaxation algorithm. After the blockwise segmentation, each boundary block in the image region is further divided into four subblocks and the class labels for these subblocks are updated. These subdivision and class updating processes are executed recursively until we have a pixel-level segmentation. Experimental results show that the proposed image extraction method yields 2.9% error rates for 232 documents in the Oulu database.

We propose a block-based unsupervised image segmentation algorithm in which the basic unit for the region labeling is an image block rather than a pixel. Adopting the image block, we can introduce a new paradigm for the unsupervised image segmentation, namely, an edge-and-region based segmentation. That is, in our block-based segmentation, we exploit both edge blocks in the region boundary and the homogeneous blocks inside the region. The interior texture and monotone blocks are used to identify regions and the edge blocks are used to find an accurate contour. To obtain a pixel level segmentation, we divide the edge and unlabeled blocks into quadrant and relabel those subblocks to one of the neighboring homogeneous regions. We repeat this process until we obtain a pixel-based segmentation. Experimental results show that the proposed segmentation yields accurate segmentation for natural images even if they contain some texture regions as well as monotone regions.

The authors propose a new image block classification method. The proposed algorithm incorporates image context into the classification via pixel-based segmentation. To obtain a segmented image they adopt the stochastic model-based unsupervised image segmentation algorithm. Since the block classifier considers the grey level distribution in the block, it can differentiate edges from textures. Also, since the segmentation is executed independently at each small block, a parallel processor can be applied to obtain a real-time block classification.

For texture classification, the observation probability distribution for each texture can be estimated using a Gauss mixture vector quantizer (GMVQ) designed with the generalized Lloyd algorithm with a minimum discrimination information (MDI) distortion. The designed multiple GMVQs are applied to classifying Brodatz textures. For low complexity implementation, Super-blocks are used to capture the macro features of the texture. In [1], the results were compared well to TSWT [2]. As an extension of [1], this paper shows that our multi-codebook GMVQ classifier, applied to the Brodatz texture database, outperforms state-ofthe-art texture classifier, Gabor wavelet classifier [3].

The previous block-based MAP image segmentation is improved in this paper. The first improvement is achieved by adopting a criterion to differentiate various texture blocks using multi-resolution 2/spl times/2 DCT coefficients. To do that, we obtain an octave-band multi-resolution representation of each image block by faking a 2/spl times/2 DCT-rearrangement process for the DC values recursively. Then, comparing the energy distributions of the coefficients for each frequency band, we can differentiate the neighboring texture image blocks. Secondly, we propose a scheme to avoid the under-segmentation problem of the previous block-based MAP segmentation. So, as the block size reduces, we can also identify small monotone regions without affecting previously segmented large texture regions.

We propose a novel image classification method using a non-causal hidden Markov Gauss mixture model (HMGMM) We apply supervised learning assuming that the observation probability distribution given each class can be estimated using Gauss mixture vector quantization (GMVQ) designed using the generalized Lloyd algorithm with a minimum discrimination information (MDI) distortion. The maximum a posteriori (MAP) hidden states in an Ising model are estimated by a stochastic EM algorithm. We demonstrate that HMGMM obtains better classification than several popular methods, including CART, LVQ, causal HMM, and multiresolution HMM, in terms of Bayes risk and the spatial homogeneity of the classified objects. A heuristic solution for the number of clusters achieves a robust image classification.