Frobenius Norm Filtering (for Images)


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Frobenius Norm Filtering (for Images)

  1. 1. By : Sutanshu S. Raj, Divij Babbar (USIT) & Palak Jain (IGIT) GGS-IP Univ.
  2. 2. 1. A Novel Spatially Selective Noise Filtration Technique based on Relational Context Spatial Domain Analysis is introduced, for the removal of Additive, Multiplicative and Uncorrelated Noise – Frobenius Norm Filter (FNF). 2. In extension, the Filter is applied to various polluted Images – Medical, SAR, etc – and the results are found to be comparable with those of existing Filtering methods. 3. Also, the Filter is employed upon a Class of Noises having varied Probability Density Functions (PDF’s) to show the versatility of the FN Filter. 4. Mathematically Prove the existence of a Minimizer, and its Convergence, for the Frobenius Norm Filter.
  3. 3. 1. Measure Theory: a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, volume, etc. 2. Euclidean axioms leave no freedom, they determine uniquely all geometric properties of the Space. Every Euclidean space is also a Topological Space, which are of analytic nature. 3. Topological Spaces, by definition, have Open Sets leading to notions of continuous functions, paths, maps, convergent sequences, limits, interior, boundary. 4. Open Set provides a fundamental way to speak of nearness of points in a Topological Space, without explicitly having a concept of distance defined – done on a neighborhood basis.
  4. 4. 1. Frobenius Norm: A Vector Norm treats an m×n Matrix as a Vector of size mn, and using mapping function p = 2 gives - 2. The Frobenius Norm is similar to the Euclidean Norm on Kn and comes from an inner product on the space of all Matrices. The Frobenius Norm is sub-multiplicative and is easier to compute than Induced Norms. 3. The set of all n-by-n Matrices, together with such a sub- multiplicative Norm, is a Banach Space – which, unlike the Euclidean Space, does not support Orthonormal Basis.
  5. 5. 1. L2 space: Lp spaces are Function Spaces defined using generalizations of Norms for finite-dimensional Vector Spaces. On a space X, the set of square-integrable functions is an L2 space. Taken together with the inner product w.r.t a measure μ, the L2 space forms a Hilbert Space: 2. L2 space consists of equivalence classes of functions and we can think of an L2 function as a density function, so only its integral on sets with positive measure matter. 3. L2 function in Euclidean Space can be represented by a continuous function f and we can think of L2 (Rn) as the completion of the continuous functions with respect to the L2 norm. 4. CTM - Every isomorphism between two Euclidean Spaces is also an isomorphism between the corresponding Topological Spaces.
  6. 6. 1. Eigenvector & Eigenvalue: A Matrix acts on certain Vectors, called Eigenvectors, by changing only their magnitude, and leaving their direction unchanged. It does so by multiplying the magnitude of the Eigenvector with a factor, which is either positive or negative, called Eigenvalue. Mathematically, Ax = λx. 2. Frobenius Norm works on Eigenvalues and Eigenvectors, which are unique to a given system and are sensitive to perturbations. 3. The Mean of a sequence depends upon the number of elements on either side of the cardinal element, whereas the Median depends upon the magnitude of the elements on either side. 4. A Noise Model’s PDF can be measured / is compatible with the Frobenius L2 (Rn) Norm. 5. In short, the FNF is an adaptive order statistic filter functioning on the L2 space which can modulate itself according to the Noise Level.
  7. 7. 1. Noise: an unwanted sound / perturbation to a Signal / Image. It is the random variation of brightness and color information in the Image. Mathematically, Image Degradation can be modeled as: g(x,y) = H[f(x,y)] + η(x,y) ; where H is a linear, spatially invariant process; f is the input image; g is the output image and η is the Noise. 2. The FNF is applied under a given Window Set / Kernel and uses Pixel Connectivity for removal of additive / multiplicative Noise. 3. The minimization of outlier effects is accomplished by replacing the above linear form, for (s,t)∊ Sxy , with = Frobenius Norm{g(s,t)}, such that the PSNR value is maximum; subject to optical evaluation. Histogram Matching is also accounted for.
  8. 8. 1. Noise Models: describe the distribution (PDF) of random numbers added to the gray level of each pixel of the Image. They are decided based on understanding the physics of the source of the Noise, for e.g., Gaussian Noise is due to poor illumination of high temperature, Speckle, Poisson and Rayleigh Noise is due to Range Imaging ( in SAR, Medical Images), Gamma / Erlang Noise is due to Laser Imaging, Impulse Noise is due to ‘quick’ transients b/w processes. 2. Since Frobenius Norm works on the L2 space, as do Wavelets; we combine both to propose a De-noising algorithm. Wavelet Transform is applied to smoothen the edges of the FN Filtered Image. 3. PDF’s: Gaussian Noise: where, μ = mean. Rayleigh Noise: Erlang Noise:
  9. 9. Proposed Algorithm. If denote the dynamic range of , the gray level of a M×N Image, then we have y as a Noisy Image and as a window of size w×w centered at (i, j). The algorithm identifies the noisy pixels and then adaptively replaces them with the Frobenius Norm of the pixels in . 1. For each pixel location (i, j), initialize w = 3. 2. Compute , which are the minimum, Frobenius Norm, maximum of the pixel values in , respectively. 3. Compute . 4. If , then proceed to step [5], else set i = i+1. 5. If , then is not a noisy candidate, else we replace the pixel with . The neighborhood may include coefficients from other subbands, corresponding to basis functions at nearby scales and orientations.
  10. 10. Image corrupted with varying Noise Density (d) and thereafter filtered with FNF. Image Corrupted with d = 0.1 Frobenius Norm Filtered Image. Median Filtered Image. Image Corrupted with d = 0.5 Frobenius Norm Filtered Image. Median Filtered Image.
  11. 11. Comparative Performance Study of various Denoising Algorithms (d=0.05)
  12. 12. The Experimental Evaluation reveals that the proposed Frobenius Norm Filter shows better results than the conventional Median Filter & Adaptive Median Filter when the Images are highly corrupted and having Noise Density d ∊[0.4, 1). Graphical Plots (PSNR vs. Noise Density) of ADP and FNF at different values of d for bior6.8 and db2, both at n=2.
  13. 13. Frobenius Norm Filter.Median Filter.Poisson Noise. Rayleigh Noise. Median Filter Frobenius Norm Filter. 13
  14. 14. Parameter Settings (Mean and Variance) for the Simulations were Constant. Compatibility, in terms of De-noising, with Noise Models having PDF’s measurable in the L2 (Rn) space is evident from the results. The Noise is considered to be stationary. FNF depicts edge & feature sensitive selectivity in passing High Frequency Data.
  15. 15. Bottom – Down: MRI Images corrupted with Rician, AWGN and Gaussian Noise, resp.
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  17. 17. 1. Complexity: Sorting N pixel values in a neighbourhood in to Numerical Order requires a Temporal Complexity of O(N log(N)). Median Filters, and its variants, are a 2-phase process which are difficult to treat analytically, are expensive and complex to compute. The Frobenius Norm calculations do not exceed O(N) and it is hence, computationally (read time) more efficient. 2. Connectivity: It is observed that Median Filter’s Connectivity tends to break- down when the Image is highly corrupted. Whereas, the Frobenius Norm works on Eigenvalues and Eigenvectors, which are unique to a given system and when applied to a neighborhood of pixels, the Connectivity is preserved. 3. Compatibility: Most set of Noises are compatible with the FNF as they are measured either in the L1 (Ω) or L2 (Ω) space. 4. Correspondence: a one-to-one correspondence exists between the Banach Space, when the Frobenius Norm is measured, and the Euclidean Hilbert Space, where the Image is represented w.r.t the Image Intensity Function Ω. 5. Coupling: we use a Neighborhood of Coefficients drawn from two sub-bands at adjacent scales, thus taking advantage of the strong statistical coupling observed in Multi-resolution Analysis innate to Wavelet Analysis.
  18. 18. 1. Frobenius Norm Filters adapts itself to the Local Properties, Information surrounding the central pixel of a Mask in order to calculate a new pixel value. 2. FNF is far superior in preservation of the Salient Image features such as Sharpness, Details, Edges (- sharp contrast variation) and Contours / Gradients. 3. There is NO equivalent to Gibbs’Ringing at the edges after filtration and the loss of Spatial Resolution is almost unnoticeable – no over-smoothing of edges. 4. We have used connectivity in the Spatial Domain and exploited the Group Behaviour of pixel neighbourhood, leading to Adaptive Optimization of the code 5. FNF’s good Localization Characteristic and Protection of Sharp Edges will allow the Wavelet Filters to be very competitive in Edge Detection, Pattern Recognition, and Computer Vision. 6. The FNF method will be applied on the Compression Framework – to work towards a ‘Simultaneous Decompression-Filtering of Images’ Algorithm. 7. The Filters effect on Multi-dimensional Images will also be studied.
  19. 19. 1. Paper titled ‘Analysis of Wavelet Family with Frobenius Norm for the Removal of Impulse Noise’ has been accepted for publication at the IEEE – IET International Conference on Audio, Language and Image Processing ’10, Shanghai, China. 2. Paper titled ‘Image De-noising for a Class of Noises using the Frobenius Norm Filter’ has been accepted in the International Journal of Computer Applications (IJCA). 3. Paper titled ‘Enhancement of Medical & SAR Images Using a Novel Frobenius Norm Filtering Method’ has been submitted to IEEE International Conference on Communication and Signal Processing ’11, Calicut, India.
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