Some preliminary tasks, based on the understanding of the physical phenomenon in the advance problem, are used to evaluate the solution of such problems. Limiting the size of the neighborhood allowed our algorithm to learn quickly from fewer samples.
Introduction
Reverse Problems: An Introduction 2 adapted for other image enhancement tasks such as denoising and deblurring. 1In multi-image SR, several images of the same scene are available, such as frames from a video.
Contributions of the thesis
We exploited previously underutilized intra-scale dependencies of the wavelet coefficients in conjunction with widespread inter-scale dependencies to estimate the desired detail coefficients. Inverse Problems: An Introduction 6 faster running (test) time performance compared to state-of-the-art algorithms.
Organization of the thesis
Inverse Problems: An Introduction 7 Simple rules are used to obtain the pixel values of the HR patch. Self-similarity is the appearance of the same local structures at different scales due to the variability of object distances from the image plane.
Conclusion
This led to a significant reduction in the number of training samples and time required without compromising accuracy. Polynomial neural networks for complex and nonlinear SISR 22 function using a learning algorithm that is efficient in terms of time and number of training samples [52].
Theory and Proposed Algorithm
Polynomial neural networks for SISR 23 are not enough input because many combinations of HR correction pixel values can be downscaled to get the same parent LR correction. Polynomial neural networks for SISR 25 trained using pairs of original HR corrections and their corresponding interpolated LR corrections in a machine learning framework as done in SRCNN [39,40].
Experiments
TABLE3.3: SR performance as a function of neural network type for 2×2 SR: Average PSNR (and SSIM) on 100 test images. Reducing the input-output patch sizes reduced the training time of the CNN in Table 3.3 to 38 hours instead of the 96 hours that the CNN used for results shown in other tables. A clear ranking of the techniques emerged when we compared average PSNR and SSIM for the two zoom factors.
Conclusions and Discussion
When the SR factor is an integer, we need to train 2PNNs, one for each of them. We exploited previously used intrascale dependencies of wavelet coefficients in conjunction with widely used interscale dependencies to estimate the desired detailed coefficients with higher accuracy. Then, the exponential decay and parent-child relationship of the wavelet coefficients along the scale (propertyP3) were used to estimate the pdf parameters at the desired finer resolution.
Proposed Algorithm
During testing, the given LR image is treated as Level 1 approximation coefficients of the desired HR image whose detail coefficients are missing. Patch Extraction: Compute the DWT of the given LR image to obtain the corresponding approximation and detail sub-bands. Rearrange the obtained wavelet coefficients in the corresponding locations for each of the desired wavelet sub-bands (horizontal, vertical and diagonal).
Experiments and Results
SISR in the Wavelet Domain 56 To overcome this limitation of absolute metrics, we also compared the relative performance metrics of each technique against the reference technique. Therefore, for the remaining experiments, the size of the input patch for MSDP was set to 5 × 5 (w = 2 in Algorithm 3). Therefore, we evaluated the qualitative performance of the proposed MSDP with the other best performers—K-SVD, KK, and SRCNN—on real images (HR ground reality was not available) for 4×4SR, as shown in Figure 4.8.
Discussion and Conclusion
For example, instead of using the parent pixel at the next coarser scale to estimate a desired detail coefficient, we used the approximation coefficients at the same scale and the collocated coefficients in other sub-bands of detail at the next coarser scale. thick as an introduction to the proposed learning framework. Moreover, the most important input comes from the approximation sub-band at the same scale, compared to the parent detail or other detail sub-bands at the next coarser level. This is because the approximation sub-band at a given level can independently provide full information to calculate the three detail sub-bands at the next coarser level, but not vice versa.
Proposed Algorithm
Stack the eigenvectors of CYˆ into columns of matrixU and let D be the diagonal matrix of the corresponding eigenvalues. Similarly, we calculate the vectorized ZCA whitened patches TX(xk) = ZXˆxk from vectorized HR patches xk ∈ R3s.
Experiments and Results
For testing, we randomly obtained 240 sub-images of size 140×140 from the LR test images and their corresponding HR test sub-images as ground truth. Each predicted HR sub-image was compared to the ground truth to evaluate QSSIM and PSNR metrics. The HR predictions on the test data were compared with the ground truth to obtain the QSSIM and PSNR metrics.
Discussion and Conclusions
Uniqueness of Nonnegative Matrix Factorization: An Introduction 77 According to this model, the goal of origin separation is to find matrices A and S that together satisfy the mixing model (6.1) and satisfy the constraints (6.3). In the case of the exact decomposition of the data matrix, this formulation is similar to factoring a non-negative matrix into the product of two non-negative matrices. Uniqueness of Non-Negative Matrix Factorization: An Introduction 78 sufficient conditions guaranteeing the uniqueness of the solution are formulated, and a discussion of what types of information can help increase the ambiguity.
Problem Statement and Preliminaries
This chapter is organized as follows: Section 6.2 describes the indeterminacy problem in the general case, reformulates scale and order indeterminacy in the framework of non-negative source separation, and provides some necessary definitions and properties. Before we proceed further, we must first reformulate the scaling and ordering indeterminacy in the case of non-negative source separation. Such a permutation matrix is reversible, and AT−1 is obtained by applying the same permutation to the columns of A.
Uniqueness Conditions
The last condition (Full Factorial Sampling) imposes that there are at least (p−1) distinct vectors (rows) of X in the pfacets (dimensions (p−1)) of the cone PS. Each face of the cone has dimension (p−1), each forming vector of the cone belongs to (p −1) different faces. According to this geometric interpretation, we can derive different situations where the uniqueness of the non-negative source separation solution is guaranteed.
Admissible Solutions
Uniqueness of Non-Negative Matrix Factorization: An Introduction 87 Similarly, the non-negativity of the transformed mixture coefficients corresponds to. Uniqueness of non-negative matrix factorization: An introduction 88 Finally, the set of all admissible solutions corresponds to. Reducing the solution space of Non-Negative Matrix Factorization 91 data matrix to be sparse.
Prior work on reducing NMF solutions
Our primary insight is to realize that a unique solution can be obtained by translating the objective to find a transformation such that the transformed data fills its positive orthant. At a first viewing of expressions (6.27) and (6.29), one can notice that the limits onα and β will take small values if the source signals and the mixing coefficients present components with low amplitudes. One can thus expect that the set of admissible solutions can be significantly reduced if, instead of factoring the original data matrixX =AS, the NMF algorithm is applied to a matrixX˜ obtained from a sparse representation of X in a dictionary F such that X = FX˜ .
Proposed approach
Thus, one can expect that the set of admissible solutions could be significantly reduced, if instead of factoring the original data matrix X = AS, the NMF algorithm is applied to a matrix so that X = FX˜. The NMF factorization is thus given as X˜ = AS˜ where S˜ contains the coefficients of the sparse representation of the original sources inF according to S =FS˜. Reducing the solution space of non-negative matrix factorization 93. It is easy to see that the atoms of an adaptive dictionary can provide a better fit of the hyperspectral signals using the sparse recovery algorithm5. Reducing the solution space of non-negative matrix factorization 95 Algorithm 6 Reducing the set of admissible solutions for a given spectral unmixing problem by NMF.
Experimental Results
The feasible resource solutions evaluated for a given two- and three-source NMF problem are compared qualitatively in Figure 7.3(a) and Figure 7.3(b). It is clear that the set of admissible solutions is significantly reduced when NMF is performed on sparse mixture data compared to NMF performed on the original data. This observation confirms our hypothesis that we can reduce the set of admissible solutions of an NMF problem by using X˜ instead of X .
Conclusion
34; Single Image Super Resolution Based on Fast Learning", Neeraj Kumar, Amit Sethi, in IEEE Transactions on Multimedia, May 2016, Single Image Super Resolution Using Sparse Regression and Natural Image Pre. Pattern Analysis and Machine Intelligence, IEEE Transactions June 2010. Fast single-image super-resolution via case-only learning and sparse representation.IEEE Transactions on Multimedia, 16(8), December 2014.
SRCCN was trained on a computer with 16 GB RAM, hexa-core Intel Xeon®CPU, with NVIDIA Tesla®C2075 GPU (hereafter the “GPU machine”), because deep learning-based techniques take long training time without GPUs. Further from Table 4.4 and Figure 4.9a, it is clear that MSDP was the most accurate and second fastest method to train. SRCNN was the second best performer in terms of SISR reconstruction, but the slowest to train, even though it was the only technique to use the GPU, more RAM, and more CPU cores.
Performance comparison of learning-based SISR methods
Uniqueness of Non-Negative Matrix Factorization: An Introduction 85 a source is exactly zero, which is unrealistic for example in the case of kinetic reactions. In this work, we have presented algorithms for solving two inverse problems (1) single image super resolution and (2) reducing the solution space of non-negative matrix factorization. 34;On reducing the number of admissible solutions of nonnegative matrix factorization", Neeraj Kumar, Amit Sethi, Said Moussaoui, David Brie, Jerome Idier (Journal Paper).
Non-negative source allocation: Range of acceptable solutions and conditions for uniqueness of solution. Sparse non-negative matrix factorizations via alternating constrained non-negative least squares for microarray data analysis.
Comparison of SRNN with (M1) and without (M2) ZCA whitening 74
Geometric interpretation of NMF, its solution space, and a trans-
Algorithm 5 Sparse non-negative joint representation Input: data matrixX ∈RL×N; dictionaryF ∈RL×Q Output: Sparse decomposition matrixX˜ ∈RQ×N. For this chapter, the dictionary was provided by the non-negative sparse coding algorithm [104]; taking all spectral signatures from the USGS (U.S. Geological Survey) library (500 spectra in 224 spectral bands)1[105] as data matrixX. Therefore, we conclude that adaptive dictionaries are better for a sparse joint representation of hyperspectral signals.
Decomposition dictionaries and reconstruction error
The dictionary on which the sparse representation is performed can be set using two strategies: (i) Analytical dictionary: Each atom in the dictionary is chosen as a Gaussian pattern with varying means and variances. ii) Adaptive Dictionary: The dictionary is obtained by using specialized dictionary learning algorithms taking into account the positivity constraint. Similar observations are also made from the results of mixing coefficient estimation (results not shown due to space limitations).
NMF admissible solutions for source signals of (a) two-source
34;On the spatial neighborhood of patch-based super-resolution", Neeraj Kumarand Amit Sethi inIEEE International Conference on Image Processing (ICIP)-2015, Montreal Canada. 34;On image-guided choice of wavelet basis for image super-resolution", Neeraj Kumarand Amit Sethi, 9th IEEE International Conference on Signal Processing and Communications (SPCOM), JULY 22-25, 2012, IISc Bangalore, India. 34;Learning-based super-resolution using interscale and intrascale dependence of wavelet coefficients", Neeraj Kumar and Amit Sethi discussed in IEEE Transactions on Cybernetics.
