Normalized cut (Ncut) is one of the popular image segmentation models in which the parametric similarity matrix should be given in advance. In this paper, we propose a normalized cut segmentation algorithm with an adaptive similarity measure and spatial regularization. The statistical Parzen-Rosenblatt window, expectation maximum (EM) method and the regularization technique in partial differential equation (PDE) are unified in a proposed variational framework. The introduced EM strategy makes our method can adaptively update the similarity matrix, and it can produce a better classification criterion than the existing Ncut based methods. While the regularization can guarantee the proposed algorithm has a robust performance under noise and it can also rectify the similarity measure with a priori spatial location. In addition, the existence theorem of the proposed model is given in the paper. Compared with some existing spectral clustering methods such as the traditional Ncut algorithm and the variational based Chan-Vese model, the numerical experiments show that our methods can provide promising segmentation results.
In this section, we give some related works. First of all, we review the Chan-Vese model @cite_3 briefly, and then we introduce classical Ncut model @cite_55 . As mentioned earlier, the similarity matrix in traditional Ncut is given in advance, and the choice of parameters is always troublesome. To solve this problem, [rgb] 0.00,0.00,0.00 we adopt Parzen-Rosenblatt window method (or kernel density estimation) @cite_54 @cite_24 to approximation the image intensity. Inspired by the Gaussian mixture model @cite_50 and EM algorithm @cite_46 , we construct a functional which is associated with the similarity of pixels. We will review Gaussian mixture model and EM algorithm in subsections and .
@cite_55 @cite_54 @cite_3 @cite_24 @cite_50 @cite_46
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How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its "width" --- namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers --- is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [, 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. A subsequent paper [, 2019] gave heuristic Monte Carlo methods to estimate the NTK and its extension, Convolutional Neural Tangent Kernel (CNTK) and used this to try to understand the limiting behavior on datasets like CIFAR-10. The current paper gives the first efficient exact algorithm (based upon dynamic programming) for computing CNTK as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for performance of a pure kernel-based method on CIFAR-10, being 10 higher than the methods reported in [, 2019], and only 5 lower than the performance of the corresponding finite deep net architecture (once batch normalization etc. are turned off). We give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK. Our experiments also demonstrate that earlier Monte Carlo approximation can degrade the performance significantly, thus highlighting the power of our exact kernel computation, which we have applied even to the full CIFAR-10 dataset and 20-layer nets. In previous work on "transformed mixtures of Gaussians" and "transformed hidden Markov models", we showed how the EM algorithm in a discrete latent variable model can be used to jointly normalize data (e.g., center images, pitch-normalize spectrograms) and learn a mixture model of the normalized data. The only input to the algorithm is the data, a list of possible transformations, and the number of clusters to find. The main criticism of this work was that the exhaustive computation of the posterior probabilities over transformations would make scaling up to large feature vectors and large sets of transformations intractable. Here, we describe how a tremendous speed-up is acheived through the use of a variational technique for decoupling transformations, and a fast Fourier transform method for computing posterior probabilities. For N × N images, learning C clusters under N rotations, N scales, N x-translations and N y-translations takes only (C + 2 log N)N2 scalar operations per iteration. In contrast, the original algorithm takes CN6 operations to account for these transformations. We give results on learning a 4-component mixture model from a video sequence with frames of size 320×240. The model accounts for 360 rotations and 76,800 translations. Each iteration of EM takes only 10 seconds per frame in MATLAB, which is over 5 million times faster than the original algorithm. We address the problem of segmenting multiple similar objects by optimizing a Chan-Vese-like [1] functional with respect to a mixture of level set functions. We solve the variational formulation under this model allowing for similarity transforms. This allows shape priors to be enforced even in the presence of mutual occlusion, lifting the limitation in [2]. We show numerical results on example images to demonstrate the promise of our approach. In this work we introduce a structured prediction model that endows the Deep Gaussian Conditional Random Field (G-CRF) with a densely connected graph structure. We keep memory and computational complexity under control by expressing the pairwise interactions as inner products of low-dimensional, learnable embeddings. The G-CRF system matrix is therefore low-rank, allowing us to solve the resulting system in a few milliseconds on the GPU by using conjugate gradient. As in G-CRF, inference is exact, the unary and pairwise terms are jointly trained end-to-end by using analytic expressions for the gradients, while we also develop even faster, Potts-type variants of our embeddings. We show that the learned embeddings capture pixel-to-pixel affinities in a task-specific manner, while our approach achieves state of the art results on three challenging benchmarks, namely semantic segmentation, human part segmentation, and saliency estimation. Our implementation is fully GPU based, built on top of the Caffe library, and is available at https: github.com siddharthachandra gcrf-v2.0.