We propose a deep convolutional neural network architecture codenamed Inception that achieves the new state of the art for classification and detection in the ImageNet Large-Scale Visual Recognition Challenge 2014 (ILSVRC14). The main hallmark of this architecture is the improved utilization of the computing resources inside the network. By a carefully crafted design, we increased the depth and width of the network while keeping the computational budget constant. To optimize quality, the architectural decisions were based on the Hebbian principle and the intuition of multi-scale processing. One particular incarnation used in our submission for ILSVRC14 is called GoogLeNet, a 22 layers deep network, the quality of which is assessed in the context of classification and detection.

/pdf/going-deeper-with-convolutions-1yobw2o2ds.pdf

Going deeper with convolutions

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

计量经济分析 = Econometric analysis

Digital radiography is a popular diagnostic imaging method. Denoising and enhancement have an important potential in obtaining as much easily interpretable diagnostic information as possible with reasonable absorbed doses of ionising radiation. Due to the increasing usage of high resolution and high precision images with a limited number of human experts, the computational efficiency of the denoising and enhancement becomes important. In this paper, a local adaptive image enhancement and simultaneous denoising algorithm for fulfilling the requirements of digital X-ray image enhancement is introduced. The algorithm is based on modification of the wavelet transform coefficients by a pointwise nonlinear transformation and reconstructing the enhanced image from the modified wavelet transform coefficients. The implementation of algorithm in software is simple, quick, and universal.

/pdf/an-approach-to-adaptive-enhancement-of-diagnostic-x-ray-52v1vbrb9c.pdf

An approach to adaptive enhancement of diagnostic X-ray images

An algorithm for image noise-removal based on local adaptive window size filtering is developed. Two features for use in local spatial/transform-domain filtering are suggested. First, filtering is performed on images corrupted not only by additive white noise, but also by image-dependent (e.g. film-grain noise) or multiplicative noise. Second, the used transforms are equipped with a varying adaptive window size obtained by the intersection of confidence intervals (ICI) rule. Finally, we combine all estimates available for each pixel from neighboring overlapping windows by weighted averaging these estimates. Comparison of the algorithm with the known techniques for noise removal from images shows the advantage of the new algorithm, both quantitatively and visually.

Adaptive window size image denoising based on ICI rule

A discrete diffraction transform (DDT) is a novel discrete wavefield propagation model that is aliasing free for a pixelwise invariant object distribution. For this class of distribution, the model is precise and has no typical discretization effects because it corresponds to accurate calculation of the diffraction integral. A spatial light modulator (SLM) is a good example of a system where a pixelwise invariant distribution appears. Frequency domain regularized inverse algorithms are developed for reconstruction of the object wavefield distribution from the distribution given in the sensor plane. The efficiency of developed frequency domain algorithms is demonstrated by simulation.

Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions.

A novel anisotropic estimator for image restoration is presented. The proposed approach originates from the geometric idea of a starshaped estimation neighborhood topology. In this perspective, an optimal adaptation is achieved by selecting in a pointwise fashion the ideal starshaped neighborhood for the estimation point. In practice, this neighborhood is approximated by a sectorial structure composed by conical sectors of adaptive size. Special varying-scale kernels, supported on these sectors, are exploited in order to bring the original geometrical problem to a practical multiscale optimization. It is proposed to use this adaptive estimator iteratively. This recursion results in the anisotropic enlargement of the estimation neighborhood, an effect that can be interpreted as a special diffusion process. The resulting estimators are truly anisotropic, providing clean and accurate edge adaptation and excellent restoration performance. Their implementation is fast as it is based on simple convolutions and scalar optimizations. Although we focus on image processing, the approach is general and can be extended to higher-dimensional data. 1. Motivation and idea We consider the denoising problem of restoration of the image intensity y from the noisy observations z(x)= y(x)+ση(x), η ∼N (0,1). Our main intention is to develop algorithms efÞcient for highly anisotropic images. When estimating y, a trade-off between noise suppression (variance) and smoothing (bias) has to be considered. Usual images are nonstationary, often characterized by localized features. Therefore, images should be treated adaptively: for example, one would achieve a higher noise suppression where the original image is smooth than in the vicinity of sharp transitions such as edges, where oversmoothing should be avoided. So, the desired balance between variance and bias depends on the images local features. How to control this balance is a key problem in adaptive signal processing. A novel strategy to achieve such adaptation is presented in this paper. 1.1 Estimates with support optimization Let X ⊂R2 (or ⊂ Z2) be the image domain. Consider a conventional kernel estimator (Þlter) in the form ŷ(x)= ! 1Ux (x−v)z(v)dv = ! 1Ũx (v)z(v)dv = ! Ũx z(v)dv/μ(Ux ) , (1.1) where Ux is a neighborhood of the origin, and the uniform smoothing kernel 1Ux has support Ux and constant value 1/μ(Ux) on Ux (μ(Ux) stands for the Lebesgue measure of Ux ). We use the decoration ∼ to denote the translated and mirrored neighborhood about the reference point x , Ũx (·)=Ux (x−·), distinguishing it from Ux that is always about the origin. The term neighborhood (of a point) is used in a generic sense, meaning a simply connected set (containing the point). Relations between sets are always considered up to a null-set. Bias and variance of the estimate (1.1) are, respectively, mŷ(x) = y(x)− " 1Ũx (v)y(v)dv and σ 2 ŷ (x) = σ /μ(Ux). The ideal support U∗ x , yielding the best mean squared error, can be found by minimization of the quadratic risk lŷ(x): U∗ x = argmin Ux lŷ(x), l ŷ(x)= m 2 ŷ(x)+σ ŷ(x). (1.2) Thus ŷ (x) = " 1U∗ x (x − v)z(v)dv is the best local mean estimate of y (x). The optimization (1.2) can be quite difÞcult to achieve. In order to make it practical further speciÞcations of the problem are required. 1.1.1 Starshaped unbiased estimates and the Ux topology We discuss here a simpliÞed model, which will serve as a ground for the development of a more general approach. Let y be a binary black-and-white image, i.e. y (x) ∈ {0,1} ∀x , and let us restrict our attention to starshaped unbiased estimates. It means that we consider only sets Ux which are starshaped with respect to the origin and such that mŷ(x)= 0. The best estimate is obtained by minimization of the variance only or, equivalently, by maximization (with respect to the set inclusion ⊂) of the set Ux . Unbiasedness holds if and only if y (v)= y (x) for almost every v ∈ Ũx . Under mild regularity assumptions on y (e.g. piecewise regular boundary of level sets), such equality has to hold for every v ∈ Ũx . Thus, the best unbiased estimate corresponds to the largest starshaped Ux such that y(Ũx (v))= y (x) ∀v . This procedure can be formalized nicely in a topological manner. Let Ux be the topology constituted by all sets Ux such that: (i) Ux \ {0} is an open set in the Euclidean topology, (ii) Ux is starshaped with respect to 0 and (iii) y(x− v) = y (x) ∀v ∈ Ux . The maximum (w.r.t. ⊂) element in Ux corresponds to the ideal starshaped unbiased estimate of y (x), U∗ x =maxUx . This suggests a risk minimization strategy based on a progressive set enlargement within this topology. It may be achieved also by decomposing Ux as follows. Let {Si }K i=1 be a collection of K starshaped neighborhoods of the origin such that ∪K i=1Si = R2 (e.g. a collection of conical sectors). Then, Ui x = {Ui x = Ux ∩ Si : Ux ∈ Ux } are also topologies, and U∗ x = maxUx = ∪K i=1maxUS x . It means that the optimization can be performed independently on each subcomponent Ui x . Examples of the ideal Ũ∗ x are given in Figure 1 for two images: the characteristic function of an open disc and the Cheese image. Although different points x 2, x 22 may Figure 1: Examples of the ideal starshaped neighborhoods Ũ∗ x resulting from U∗ x =maxUx . Figure 2: Piecewise constant approximation of r∗(θ) and its representation by varying size sectors (a-b-c); adaptive fusing of uniform sectorial kernels gh∗(x,θ i ),θ i produces the equivalent uniform anisotropic kernel g∗ x (d-e). have Ũ∗ x 2 = Ũ∗ x 22 , the corresponding U∗ x 2 and U∗ x 22 are not equal, and in both examples each point x has its own different ideal neighborhood U∗ x . Adapting perfectly to the edges, they are typically non-convex and their shape can be rather complex. Despite the apparent simplicity of these speculations, the practical realization of this approach can be still hard to achieve, since the function y is usually unknown and only its noisy observation z is available. In particular, unless y is known to belong to some very speciÞc class, ensuring unbiasedness is not possible, and biased estimates have to be considered. 1.2 Estimates with kernel scale optimization Another way to adapt to the signals varying local features, following the majority of multiscale techniques, is to use kernels equipped with a scale parameter h (e.g. gh(·)= g(·/h)/h2). This estimate can be presented in the form ŷ(x) = " gh(x)(x − v)z(v)dv . The scale optimization can be formulated, similarly to (1.2), as h∗(x) = argminh lŷ . The bias and variance are, respectively, mŷ(x)= y(x)− " gh(x)(x− v)y(v)dv , and σ ŷ (x)= σ 2 " g2 h(x)(v)dv . This kind of optimization is known to be practical and can give good results through algorithms of reasonable complexity (e.g. Mallat, 1999). When the support of the kernel gh is bounded, the scale parameter h(x) controls the size of the neighborhood for estimation at the point x . The support of the optimal scale kernel gh∗(x) can be thought as an approximation of the optimalU∗ x considered in (1.2). However, traditional kernels have supports of simple convex geometry (square, rectangle, circle, oval, etc.) whereas the optimal neighborhoods can be quite complex, especially near edges or corners. Thus, this approximation of U∗ x can be quite poor. 2. Anisotropic estimator based on adaptive directional scale A reasonable compromise between the geometrical approach discussed in section 1.1 and the above kernel-based method, is obtained with a directional adaptive scale estimator (Katkovnik et al., 2004). The considerations from section 1.1.1 shed some insight on how this sort of compromise is produced and clarify the geometrical properties of the estimator. The starshapedness of U∗ x allows to describe this set using polar coordinates: there exists a function r∗(θ), θ ∈ [0,2π) (see Figure 2a), such thatU∗ x ={v ∈ X,v = (v1,v2)= (rv cosθv ,rv sinθv) : rv < r∗(θv)}. Instinctively, one may assume some sort of continuity of r∗(θ) with respect to its argument. This regularity, however, fails in the vicinity of edges where, as in the examples shown in Figure 1, r∗(θ) presents sharp transitions. This irregular behaviour is a direct manifestation of the anisotropy of y or, roughly speaking, that the functions properties are different in different directions. The most natural model, allowing good approximation of such rapid transitions and also discontinuities is to assume r∗(θ) as a piecewise constant function of its angular argument, i.e. assuming that the optimal neighborhood U∗ x has a sectorial structure, as shown in Figure 2(b-c). In our approach we exploit this sectorial decomposition. A collection of directional LPA (local polynomial approximation, see Fan & Gijbels, 1996) kernels {gh,θ i }h∈H,i=1,...,K supported on such sectors is designed. Each kernel is characterized by a direction θk and a scale parameter h. The corresponding estimate is the convolution ŷh,θ i (x)= (gh,θ i ~ z)(x). The statistical ICI rule (Goldenshluger & Nemirovski, 1997; Katkovnik, 1999) is used to select a pointwise optimal scale h∗(x,θ i )≈ r∗(θ i ) for each direction. Let ŷh∗(x,θ i ),θ i (x) be the directional optimal scale estimate and σ 2 i (x) its variance. All these estimates can be fused in the Þnal one as follows: ŷ(x)= # iλ(x,θ i )ŷh∗(x,θ i ),θ i(x), λ(x,θ i )=σ−2 i (x)/ # jσ −2 j (x). (2.1) The weights λ(x,θ i ) in the above convex combination are data-driven adaptive, as σ−2 i (x) depend on the adaptive h∗(x,θ i ). The estimate (2.1) is equivalent to the adaptive anisotropic kernel estimate ŷ (x) = " g∗ x (x−v) z (v)dv , where g∗ x = $ i λ(x,θ i )gh∗(x,θ i ),θ i . When uniform kernels gh,θk are used, the adaptive weights λ(x,θ i ) make so that also the anisotropic kernel g∗ x is uniform on its support, i.e. g∗ x = 1∪i suppgh∗(x,θ i ),θ i , as shown in Figure 2(d-e). Figure 3 shows the estimation neighborhoods resulting from the proposed anisotropic LPA-ICI approach for noisy images (

A novel anisotropic local polynomial estimator based on directional multiscale optimizations

In this paper we consider the problem of suppressing additive noise in video data. We propose a transformbased video denoising method in sliding, local 3D variable-sized windows. For every spatial position in each frame we use a block-matching algorithm to collect highly correlated blocks from neighboring frames and form 3D arrays for all predefined window sizes by stacking the matched blocks. An optimal window size is then selected according to the ICI rule and a 3D unitary transform is applied to the selected 3D array. Hard-thresholding on its coefficients attenuates the noise and an inverse 3D transform reconstructs a local estimate of the noise-free signal in the array. The final estimate is a weighted average of the overlapping local ones. Our experiments show that the proposed algorithm outperforms all, known to the authors, video denoising methods, both in terms of objective criteria (L 2 distance) and visual quality.

Karen Egiazarian

Papers

An approach to adaptive enhancement of diagnostic X-ray images

Adaptive window size image denoising based on ICI rule

Discrete diffraction transform for propagation, reconstruction, and design of wavefield distributions.

A novel anisotropic local polynomial estimator based on directional multiscale optimizations

Moving-window varying size 3d transform-based video denoising