Frobenius Norm Filtering.

8/8/2019 Frobenius Norm Filtering.

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By : Sutanshu S. Raj,

Divij Babbar (USIT)&

Palak Jain (IGIT)

GGS-IP Univ.


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1. A Novel Spatially Selective Noise Filtration Techniquebased onRelational 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 pollutedImages 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 variedProbability Density Functions (PDFs) to

show the versatility of the FN Filter.

4. Mathematically Prove the existence of aMinimizer, and

its Convergence, for the Frobenius Norm Filter.


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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 ageneralization of the

concepts of length, area, volume, etc.

2. Euclidean axioms leave no freedom, they determine uniquely allgeometric properties of the Space. Every Euclidean space is also a

Topological Space, which are ofanalytic nature.

3. Topological Spaces, by definition, have Open Sets leading to notions

ofcontinuous functions, paths, maps, convergent sequences, limits,

interior, boundary.4. Open Set provides a fundamental way to speak ofnearness of points

in a Topological Space, without explicitly having a concept of

distance defined done on a neighborhood basis.


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1. Frobenius Norm: A Vector Norm treats an mn Matrix as a

Vector of size mn, and using mapping functionp = 2 gives -

2. The Frobenius Norm is similar to the Euclidean Norm on

Kn and comes from an inner producton the space of all

Matrices. The Frobenius Norm issub-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 aBanach Space which, unlike the

Euclidean Space, does not support Orthonormal Basis.


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1. L2

space: Lp

spaces areFunction Spaces defined usinggeneralizations of Norms forfinite-dimensionalVector Spaces. On a

spaceX, the set ofsquare-integrable functions is an L2 space. Taken

together with the inner productw.r.t a measure, the L2 space formsa Hilbert Space:

2. L2 space consists ofequivalence classes of functions and we canthink of an L2 function as a density function, so only its integral on

sets withpositive measure matter.

3. L2 function in Euclidean Space can be represented by a continuous

functionfand we can think ofL2 (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

isomorphismbetween the corresponding Topological Spaces.


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1. Eigenvector & Eigenvalue: A Matrix acts on certain Vectors, calledEigenvectors, by changing only their magnitude, and leaving their

direction unchanged. It does so by multiplying the magnitude of the

Eigenvector with afactor, which is either positive or negative,

called Eigenvalue. Mathematically,Ax = x.

2. Frobenius Norm works on Eigenvalues and Eigenvectors, which areunique to a given system and aresensitive to perturbations.

3. The Mean of a sequence depends upon the number of elements on

either side of the cardinalelement, whereas the Median depends

upon the magnitude of the elements on either side.

4. A Noise Models PDF can be measured/ is compatible with theFrobenius L2 (Rn) Norm.

5. In short, the FNF is an adaptive order statistic filterfunctioning on

the L2 space which can modulate itself according to the Noise Level.


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1. Noise: an unwantedsound/perturbation to a Signal / Image. It isthe random variation ofbrightness and color information in the

Image. Mathematically, Image Degradation can be modeled as:

g(x,y) = H[f(x,y)] + (x,y) ; where H is a linear, spatially invariant

process;fis the input image;gis the output image and is the Noise.

2. The FNF is applied under a given Window Set/Kerneland uses

Pixel Connectivity for removal ofadditive / multiplicativeNoise.

3. The minimization ofoutlier effects is accomplished by replacingthe above linear form, for(s,t)Sxy, with =Frobenius

Norm{g(s,t)}, such that thePSNR value is maximum; subject to

optical evaluation. Histogram Matching is also accounted for.


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1. Noise Models: describe the distribution (PDF) of random numbersadded to thegray levelof each pixel of the Image. They are decided

based on understanding thephysics of the source of the Noise, for e.g.,

GaussianNoise is due topoor illumination ofhigh temperature,

Speckle, Poisson and RayleighNoise is due toRange Imaging( in

SAR, MedicalImages), Gamma / ErlangNoise is due to LaserImaging, ImpulseNoise is due to quick transientsb/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 tosmoothen the edges of the FN Filtered Image.

3. PDFs: Gaussian Noise:

mean.

Rayleigh Noise: Erlang Noise:


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Proposed Algorithm.Proposed Algorithm.

If denote the dynamic range of , the gray level of aMNImage, then we havey as a Noisy Image and as a window of size ww

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 nota noisy candidate, else wereplace the pixel with .

The neighborhood may include coefficients from othersubbands,

corresponding to basis functions at nearbyscales and orientations.


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Image corrupted with varyingNoise 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.


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Comparative Performance Study of various Denoising Algorithms (d=0.05)


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The Experimental Evaluation reveals that the proposed Frobenius Norm Filter shows

better results than the conventional Median Filter & Adaptive Median Filter whenthe Images are highly corrupted and having Noise Density d

Graphical Plots (PSNR vs. Noise Density) of ADP and FNF at different values ofdfor bior6.8 and db2, both at n=2.


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Frobenius Norm Filter.Median Filter.Poisson Noise.

Rayleigh Noise. Median Filter Frobenius Norm Filter.

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Parameter Settings (Mean and Variance) for the Simulations were Constant.

Compatibility, in terms of De-noising, with Noise Models having PDFs measurable

in the L2 (Rn) space is evident from the results. The Noise is considered to bestationary.FNF depicts edge & feature sensitive selectivity in passing High Frequency Data.


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Bottom Down: MRI Images corrupted with Rician, AWGN and Gaussian Noise, resp.


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1. Complexity: Sorting N pixel values in a neighbourhood in to Numerical Order

requires a Temporal Complexity ofO(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 Filters Connectivity tends to break-

down when the Image is highly corrupted. Whereas, the Frobenius Normworks on Eigenvalues and Eigenvectors, which are unique to a given system

and when applied to a neighborhood of pixels, the Connectivity ispreserved.

3. Compatibility: Most set of Noises are compatible with the FNF as they are

measured either in the L1 () orL2 () space.

4. Correspondence: a one-to-one correspondence exists between the BanachSpace, when the Frobenius Norm is measured, and the Euclidean Hilbert

Space, where the Image is represented w.r.t theImage Intensity Function .

5. Coupling: we use a Neighborhood of Coefficients drawn from two sub-bands

at adjacent scales, thus taking advantage of thestrong statistical coupling

observed inMulti-resolution Analysis innate to Wavelet Analysis.


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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 Ringingat the edges after filtration and the

loss ofSpatial Resolution is almost unnoticeable no over-smoothing of edges.

4. We have used connectivity in the Spatial Domain and exploited the Group

Behaviourof pixel neighbourhood, leading toAdaptive Optimization of the code

5. FNFs good Localization Characteristic andProtection of Sharp Edges will

allow the Wavelet Filters to be very competitive inEdge 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 onMulti-dimensional Images will also be studied.


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1. Paper titled Analysis of Wavelet Family with Frobenius Norm for the

Removal of Impulse Noisehas 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 Filterhas been accepted in the InternationalJournal of Computer Applications (IJCA).

3. Paper titled Enhancement of Medical & SAR Images Using a Novel

Frobenius Norm Filtering Methodhas been submitted to IEEEInternational Conference on Communication and Signal

Processing 11, Calicut, India.


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Frobenius Norm Filtering.

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