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WO1999046731A1 - Procedes de mise en oeuvre d'un filtrage et d'un remplissage de donnees de fonctions daf - Google Patents

Procedes de mise en oeuvre d'un filtrage et d'un remplissage de donnees de fonctions daf Download PDF

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WO1999046731A1
WO1999046731A1 PCT/US1999/005426 US9905426W WO9946731A1 WO 1999046731 A1 WO1999046731 A1 WO 1999046731A1 US 9905426 W US9905426 W US 9905426W WO 9946731 A1 WO9946731 A1 WO 9946731A1
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data
values
noise
daf
unknown
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PCT/US1999/005426
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David K. Hoffman
Donald J. Kouri
Gemunu H. Gunaratne
Mark E. Arnold
Desheng Zhang
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The University Of Houston System
Iowa State University
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T5/00—Image enhancement or restoration
    • G06T5/10—Image enhancement or restoration using non-spatial domain filtering
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T5/00—Image enhancement or restoration
    • G06T5/70—Denoising; Smoothing
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/20—Special algorithmic details
    • G06T2207/20048—Transform domain processing
    • G06T2207/20056—Discrete and fast Fourier transform, [DFT, FFT]
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/20—Special algorithmic details
    • G06T2207/20048—Transform domain processing
    • G06T2207/20064—Wavelet transform [DWT]

Definitions

  • TITLE METHODS FOR PERFORMING DAF DATA FILTERING
  • the present invention relates to methods using distributed approximating functionals (DAF), DAF-wavelets and DAF-neural networks for filtering, denoising, processing, restoring, enhancing, padding, or other signal processing procedures directed to images, signals, ID, 2D, 3D . . . nD spectra, X-ray spectra, CAT scans; MRI scans, NMR, and other applications that require data processing at or near the theoretical limit of resolutions.
  • DAF distributed approximating functionals
  • DAF-wavelets and DAF-neural networks for filtering, denoising, processing, restoring, enhancing, padding, or other signal processing procedures directed to images, signals, ID, 2D, 3D . . . nD spectra, X-ray spectra, CAT scans; MRI scans, NMR, and other applications that require data processing at or near the theoretical limit of resolutions.
  • the present invention relates to the use of infinitely smooth DAFs in combination with other signal processing techniques to provide methods and apparatuses utilizing such methods that can enhance image, spectral, or other signal data and decrease the time need to acquire images, spectra or other signals.
  • the present invention provides a method implemented on a digital processing device or stored in a memory readable by a digital processing unit which uses distributed approximating functionals (DAFs) to enhance and improve signal, image and multi- dimensional data constructs processing and to decrease acquisition time of real world spectrometric techniques that operate on square wave type frequency domains which require a large acquisition time to capture the signal from noise. Short the acquisition time, the more noise and less resolution and definition the spectra will have.
  • the limit for simultaneously acquiring frequency information in time is given by a variant of Heisenberg's uncertainty principle, i.e., ⁇ t ⁇ 1.
  • the methods of the present provides methods for image and signal processing where the accuracy and precision of the final signal and image closely approaches the uncertainty principles maximum accuracy and precision. The methods can be made to approach uncertainty principle accuracy via increased computational cost, but the real power of the methods of this invention is to yield improved measurements at a given ⁇ and ⁇ t product.
  • the present invention also provides methods for improving X-ray and magnetic imaging techniques, especially mammogram images using the DAF and DAF processing techniques set forth herein.
  • the present invention also provides a mammogram imaging system of conventional design, the X-ray data derived thereform is then enhanced via DAF processing in an associated digital processing unit.
  • Figure 1 depicts the Hermite DAF in (a) coordinate space and (b) frequency space respectively.
  • the solid line is close to the interpolation region and the dashed line is in the low pass filtering region.
  • the frequency in (b) has been multiplied by a factor of the grid spacing.
  • Figure 2 depicts extrapolation results for the function in Eq. (15).
  • the solid line is the exact function.
  • Figure 4 depicts the L ⁇ error and (b) the signature of the periodic padding of the noisy sine function as a function of DAF parameter ⁇ / ⁇ .
  • the M is fixed to be-6.
  • Figure 5(a) depicts the E ⁇ error and Figure 5(b) the signature of the DAF smoothing to the periodically extended noisy sine function as a function of ⁇ / ⁇ .
  • the M is fixed to be
  • Figure 6 depicts periodic extension of the nonperiodic function (with noised added) in Eq. (15).
  • (a) The 220 known values of the function in the range [-7J0] with 20% random noise (solid line) and the 36 exact values of the function(dashed line). Note that the function is not periodic at all in the range of 256 grid points,
  • Figure 7 depicts the periodic extension signature of the noisy function in Figure 5as a function of ⁇ / ⁇ .
  • the M is fixed to be 6.
  • Figure 8(a) depicts the L ⁇ error and Figure 8(b) the signature of the DAF SMOOTHING to the periodically extended noisy function in Figure 6(b) as a function of ⁇ / ⁇ .
  • the M is fixed to be 12for the DAF-smoothing.
  • Figure 2 depicts image restoration from lower noise Lena image: (a) corrupted image
  • Figure 1 depicts ⁇ . band-limited interpolating wavelets (a) Sine function - _ (b) Sinclet wavelet;
  • Figure 13 depicts Nonlinear Masking Functionals (a) Donoho Hard Logic Nonlinearity (b) Softer Logic Nonlinearity
  • Figure 14 depicts 2D Lagrange wavelets for image processing (a) Scaling, (b) Vertical, (c) Horizontal and (d) Diagonal wavelets
  • Figure 15 depicts NGN image processing for Lena (a) noisy Lena (b) Median filtering result (c) our method
  • Figure 16 depicts NGN processing for Barbar (a) noisysy Barbara (b) Median filtering result (c) Our method IMAGE ENHANCEMENT NORMALIZATION
  • Figure 2 depicts image restoration from 40% impulse noise corrupted Lena image.
  • FIG. 1 Image restoration from 60% impulse noise, (a) Corrupted image, (b) Filtering 8 by Sun and Nevou's median switch scheme, (c) Our filtering, (d) Our modified filtering.
  • the noiseless time domain signal was sampled between -45 ⁇ t ⁇ 45. The two are visually indistinguishable.
  • Fig. 3 Dotted line: The calculated spectrum f( ⁇ ) obtained from the time signal corrupted by random noise of 20%. Solid line: The calculated spectrum obtained from the noise-free time signal. Both clean and corrupted signals were sampled between -45 ⁇ t ⁇ 45.
  • Fig. 5 Cross-hatched line: the calculated spectrum DPI ( ⁇ ) obtained using the DAF -padding values for 5 ⁇
  • the inventors have found that a signals, images and multidimensional imaging data can be processed at or near the uncertainty principle limits with DAFs and various adaptation thereof which are described in the various section of this disclosure.
  • the basic idea is to introduce a pseudo-signal by adding gaps at the ends of the known data, and assuming the augmented signal to be periodic.
  • the unknown gap data are determined by solving linear algebraic equations that extremize a cost function. This procedure thus imposes a periodic boundary condition on the extended signal. Once periodic boundary conditions are enforced, the pseudo-signal is known everywhere and can be used fdr a variety of numerical applications. The detailed values in the gap are usually not of particular interest.
  • the advantage of the algorithm is that the extended signal adds virtually no aliasing to the true signal, which is an important problem in signal processing. Two of the main sources of aliasing are too small a sampling frequency and truncation of the signal duration. Another source of error is contamination of the true signal by numerical or experimental noise. Here we are concerned only with how to avoid the truncation induced and noisy aliasing of the true signal.
  • DAFs Distributed approximating functions
  • DAFs also provide a well-tempered approximation to various linear transformations of the function. DAF representations of 14 derivatives to various orders will yield approximately similar orders of accuracy as long as the resulting derivatives remain in the DAF class. The DAF approximation to a function and a finite number of derivatives can be made to be of machine accuracy with a proper choice of the DAF parameters.
  • DAFs are easy to apply because they yield integral operators for derivatives. These important features of DAFs have made them successful computational tools for solving various linear and nonlinear partial differential equations (PDEs) [14-17], for pattern recognition and analysis [6], and for potential energy surface fitting [8].
  • PDEs linear and nonlinear partial differential equations
  • the well-tempered DAFs also are low-pass filters.
  • the usefulness of DAFs as low pass filters is also studied when they are applied to a periodically extended noisy signal. For the present purpose, we assumerthat the weak noise is mostly in the high frequency region and the true signal is bandwidth limited in frequency space, and is larger than the noise in this same frequency region. To determine when the noise is eliminated, we introduce a signature to identify the optimum DAF parameters.
  • This concept is based on computing the root-mean-square of the smoothed data for given DAF parameters. By examining its behavior as a function of the DAF parameters, it is possible to obtain the overall frequency distribution of the original noisy signal. This signature helps us to periodically extend and filter noise in our test examples.
  • the first example is a simple, noisy periodic signal, for which the DAF periodic extension is a special case of extrapolation.
  • the second is a nonperiodic noisy signal. After performing the periodic extension and filtering, it is seen that the resulting signal is closely recreates the true signal.
  • DAFs can be viewed as "approximate identity kernels" used to approximate a continuous function in terms of discrete sampling on a grid [10-13].
  • One class of DAFs that has been particularly useful is known as the well-tempered DAFs, which provide an approximation to a function having the same order of accuracy both on and off the grid points.
  • a particularly useful member of this class of DAFs is constructed using Hermite polynomials, and prior to discretization, is given by
  • ⁇ are the DAF parameters
  • H 2n is the usual (even) Hermite polynomial.
  • the Hermite DAF is dominated by its Gaussian envelope, exp(-(x-e') 2 /-2 ⁇ 2 ), which effectively determines the extent of the function.
  • the continuous, analytic approximation to a function/(x) generated by the Hermite DAF is (2) ⁇ M (x - x' ⁇ )f ⁇ x')dx'. -oo
  • f DAF ( ⁇ ) A ⁇ ⁇ M (x - x J W)f(xj)- ( 3 ) j
  • ⁇ is the uniform grid spacing (non-uniform and even random sampling can also be used by an appropriate extension of the theory). The summation is over all grid points (but only those close to x effectively contribute).
  • f(x,y) For a two-dimensional function f(x,y), one can write
  • Eq.(3) or (4) can then be used to eliminate the high frequency noise of that data set. As long as the frequency distribution of the noise lies outside the DAF plateau ( Figure 1(b)), the Hermite DAF will eliminate the noise regardless of its magnitude.
  • the approximate linear transformations of a continuous function can also be generated using the Hermite DAF.
  • One particular example is derivatives of a function to various orders, given by
  • ⁇ j (x - x 1 1 ⁇ ) is the ⁇ h derivative of ⁇ M (x - x'
  • a pseudo-signal is introduced outside the domain of the nonperiodic experimental signal in order to force the signal to satisfy periodic boundary conditions and/or-to have the appropriate number of samples.
  • the required algorithm is similar to that for filling a gap, as discussed above.
  • the values ⁇ f ⁇ + ],f ⁇ + 2 ,--,f ⁇ + j- ⁇ ⁇ are also, by fiat, known to be equal to ⁇ f ⁇ , f 2 ,--, fj ⁇ ⁇ -
  • the pseudo-signal is only used to extend the data periodically retaining essentially the same frequency distributions.
  • the utility of the present periodic extension algorithm is that it provides an artificial boundary condition for the signal without significant aliasing.
  • the resulting signal can be used with any filter requiring information concerning the future and past behavior of the signal.
  • the signature essentially measures the smoothness of the DAF filtered result.
  • the first extremely rapid decrease is due to the fact the DAF is interpolating and not well tempered. It is the region beyond the initial rapid decrease that is important (i.e., of A ⁇ 1.5). To understand the behavior in this region, we ⁇ write S ⁇ dA)
  • the weight W was taken as discussed above.
  • the continuation of the solid curve from points x 220 to x 256 shows the function without noise. We shall predict the remaining 36 points (excluding x 256 because the function there must equal the function at x 0 ) by the periodic extension algorithm presented in this paper.
  • Figure 6(a) shows the function, with 20 ⁇ % random noise in the range (-7J0) (solid line). These noisy values are assumed known at only 220 grid points in this range. Also plotted in Figure 6(a) are the true values of the function (dajshed line) on the 36 points to be predicted. In our calculations, these are treated, of course, as unknown and are shown here only for reference. It is clearly seen that the original function is not periodic on the range of 256 grid points. We force the noisy function to be periodic by padding the values of the function on these last 36 points, using only the known, noisy 220 values to periodically surround the gap.
  • the periodic-extension is simply a scheme to provide an artificial boundary condition in a way. that- does not significantly corrupt the frequency distribution of the underlying true signal in the sampled region.
  • the D AF-window is better able to simulate an ideal band-pass fi.lter (while still being infinitely smooth and with exponential decay in physical and Fourier space.)
  • This paper presents a DAF-padding procedure for periodically extending a discrete segment of a signal (which is nonperiodic).
  • the resulting periodic signal can be used in many other numerical applications which require periodic boundary conditions and/or a given number of signal samples in one period.
  • the power of the present algorithm is that it essentially avoids the introduction of aliasing into the true signal. It is the well-tempered property of the DAFs that makes them robust computational tools for such applications.
  • Application of an appropriate well-tempered DAF to the periodically extended signal shows that they are also excellent low pass filters. Two examples are presented to demonstrate the use of our algorithm. The first one is a truncated noisy periodic function. In this case, the extension is equivalent to an extrapolation.
  • a new class of generalized symmetric interpolating wavelets are described, which are generated from a generalized, window-modulated -interpolating shell.
  • various interpolating shells such as Lagrange polynomials and the Sine function, etc.
  • bell-shaped, smooth window modulation leads to wavelets with arbitrary smoothness in both time and frequency.
  • Our method leads to a powerful and easily implemented series of interpolating wavelet.
  • this novel designing technique can be extended to generate other non-interpolating multiresolution analyses as well (such as the Hermite shell).
  • the approximation problem which is a rth-order-accurate reconstruction of the discretization.
  • the approximating functional is a piecewise polynomial. If we use the same reconstruction technique at all the points and 27 at all levels of the dyadic sequence of uniform grids, the prediction will have a Toplitz- like structure.
  • DAFs distributed approximating functionals
  • DAFs both interpolating and non-interpolating
  • DAF-wavelets can therefore serve as an alternative basis for improved performance in signal and image processing.
  • the DAF wavelet approach can be applied directly to treat bounded domains.
  • the wavelet transform is adaptively adjusted around the boundaries of finite-length signals by conveniently shifting the modulated window.
  • the biorthogonal wavelets in the interval are obtained by using a one-sided stencil near the boundaries.
  • Lagrange interpolation polynomials and band-limited Sine functionals in Pal ey- Wiener space are two commonly used interpolating shells for signal approximation and smoothing, etc. Because of their importance in numerical analysis, we use these two kinds of interpolating shells to introduce our discussion.
  • Other modulated windows such as the square, triangle, B-spline and Gaussian are under study with regard to the time- frequency characteristics of generalized interpolating wavelets.
  • ⁇ can be represented- _as a linear combination of dilates and translates of itself, with a weight given by the value of ⁇ at k/2.
  • ⁇ x ) ⁇ ⁇ ( k ,- > ) ⁇ (2x - k ) ( 2 )
  • the interpolating wavelet transform possesses the following characteristics: 1.
  • the wavelet transform coefficients are generated by the linear combination of signal samplings,
  • W k ⁇ R ⁇ k (x)f2- ] k)dx (5)
  • ⁇ y k (x) 2 jl2 ⁇ (2 J x-k)
  • threshold masking and quantization are nearly optimal approximations for a wide variety of regularity algorithms.
  • ⁇ P(k) ⁇ is a sequence that satisfies the infinite summation condition
  • Deslauriers-Dubuc functional Let D be an odd integer, D>0.
  • F D such that if F D has already been defined at all binary rationals with denominator 2 J , it can be extended by polynomial interpolation, to all binary rationals with denominator 2 J+l , i.e. all points halfryay between previously defined points [9, 13].
  • F D the function at (k+M2)l2 J when it is already defined at all ⁇ kX 1 ⁇
  • This subdivision scheme defines a function which is uniformly continuous at the rationals and has a unique continuous extension;
  • F D is a compactly supportecSchval polynomial and is regular; It is the auto-correlation function of the Daubechies wavelet of order D+l . It is at least as smooth as the corresponding Daubechies wavelets (roughly- 31 twice as smooth).
  • Daubechies wavelets lead to, respectively, the interpolating Schauder, interpolating spline, C°° interpolating, and Deslauriers-Dubuc wavelets.
  • the transfer function of the symmetric FIR filter h(n)-h(-n) has the form
  • Halfband Lagrange wavelets can be regarded as extensions of the Dubuc interpolating functionals [9, 13], the auto-correlation shell wavelet analysis [40], and halfband filters [1].
  • B-spline Lagrange Wavelets are generated by a B-splirie-windowed Lagrange functional which increases the smoothness and localization properties of the simple Lagrange scaling function and its related wavelets.
  • Lagrange Distributed Approximating Functionals (LDAF)-Gaussian modulated Lagrange polynomials have been successfully applied for numerically solving various linear and nonlinear partial differential equations. Typical examples include DAF-simulations of 3-dimensional reactive quantum scattering and the solution of a 2-dimensional Navier-Stokes equation with non-periodic boundary conditions.
  • DAFs can be regarded as particular scaling functions (wavelet-DAFs) and the associated DAF- wavelets can be generated in a number of ways [20, 21, 22, 65, 66, 67].
  • a special case of halfband filters can be obtained by choosing the filter coefficients according to the Lagrange interpolation formula.
  • the filter coefficients are given by
  • interpolating wavelet 33 decomposition can be regarded as a class of the auto-correlated shell of orthogonal wavelets, such as the Daubechies wavelets [7].
  • the interpolating wavelet transform can also be extended to higher order cases using different Lagrange polynomials, as [40]
  • the predictive interpolation can be expressed as
  • T is a projection and S, is the_ th layer low-pass coefficients.
  • S is the_ th layer low-pass coefficients.
  • Z)—H- ⁇ the interpolating functional tends to a band-limited Sine function and its domain of definition is on the real line.
  • the biorthogonal subband filters can be expressed as 34
  • the Donoho interpolating wavelets have some drawbacks. Because the low- pass coefficients are generated by a sampling operation only, as the decomposition layer increases, the correlation between low-pass coefficients become weaker. The interpolating (prediction) error (high-pass coefficients) strongly increases, which is deleterious to the efficient representation of the signal. Further, it can not be used to generate a Riesz basis for E 2 (R) space.
  • R 0 is the interpolating prediction process, and the R, filter is called the updating filter, used to smooth the down-sampling low-pass coefficients. If we choose R 0 to be the same as R, then the new interpolating subband filters can be depicted as
  • the newly developed filters h x , g ⁇ , h , and S also generate the biorthogonal dual pair for a perfect reconstruction. Examples of biorthogonal lifting wavelets with regularity
  • N is the B-spline order and, ⁇ is the scaling factor to control the window width.
  • fvy e- ⁇ 2,2 ⁇ '
  • PJA is the Lagrange inte ⁇ olation kernel.
  • the DAF scaling function has been successfully introduced as an efficient and powerful grid method for quantum dynamical propagations [40]. Using Swelden's lifting scheme [32], a wavelet basis is generated.
  • the Gaussian window in our DAF-wavelets efficiently smoothes out the Gibbs oscillations, which plague most conventional wavelet bases.
  • the following equation shows the connection between the B-spline window function and the Gaussian window [34] :
  • the scaling Sine function is the well-known ideal low-pass filter, which possesses the ideal square filter response as
  • GSDF Gaussian-Sinc DAF
  • WJA is a window function which is selected as a Gaussian
  • the Gaussian Sine wavelets generated by the lifting scheme will be similar to the B-spline Sine wavelets.
  • the Gaussian Sine DAF displays a slightly better smoothness and rapid decay than the B-spline Lagrange wavelets. If we select more sophisticated window shapes, the Sine wavelets can be generalized further. We call these extensions Bell- windowed Sine wavelets.
  • the available choices can be any of the popularly used DFT (discrete Fourier transform) windows, such as Bartlett, Harming, Hamming, Blackman, Chebychev, and Besel windows.
  • Dubuc utilized an iterative inte ⁇ olating function, F D on the finite interval to generate an inte ⁇ olation on the set of dyadic rationals D
  • the inte ⁇ olation in the neighborhood of the boundaries is treated using a boundary-adjusted functional, which leads to regularity of the same order as in the interval. This avoids the discontinuity that results from periodization or extending by zero. It is well known that this results in weaker edge effects, and that no extra wavelet coefficients (to deal with the boundary) have to be introduced, provided the filters used are symmetric.
  • K j represent the number of coefficients at resolution layer/, where K-2 1 .
  • 2 J >2D+2 define the non-interacting decomposition. If we let j 0 hold the noninteraction case 2 /0 >2D+2, then there exist functions " erva ? ⁇ "" TM , such that for
  • the , ⁇ are called the interval inte ⁇ olating scahngs and wavelets
  • the interval scaling is defined as
  • the inte ⁇ olating wavelet transform can be extended to high order cases by two kinds of Lagrange polynomials, where the inner-polynomials are defined as [14]
  • FIG. 17 One example for a Sinc-DAF wavelet is shown in Figure 17.
  • Figurel 8 is the boundary filter response comparison between the halfband Lagrange wavelet and our DAF wavelet. It is easy to establish that our boundary response decreases the overshoot of the low-pass band filter, and so is more stable for boundary frequency analysis and approximation.
  • ⁇ k and E k are the Mi eigenfunction and eigenvalue respectively.
  • the eigenvalues are given exactly by
  • Image Filtering is a difficult problem for signal processing. Due to the complicated structure of image and background noise, an optimal filtering technique does not currently exist. Generally, the possible noise sources include photoelectric exchange, photo spots, image communication error, etc. Such noise causes the visual perception to generate speckles, blips, ripples, bumps, ringing and aliasing. The noise distortion not only affects the visual quality of images, but also degrades the efficiency of data compression and coding. De-noising and smoothing are extremely important for image processing.
  • Biorthogonal inte ⁇ olating wavelets and corresponding filters are constructed based on Gauss-Lagrange distributed approximating functionals (DAFs).
  • DAFs Gauss-Lagrange distributed approximating functionals
  • The-utility of these DAF wavelets and filters is tested for digital image de-noising in combination with a novel blind restoration technique. This takes account of the response of human - 45 vision system so as to remove the perceptual redundancy and obtain better visual performance in image processing.
  • the test results for a color photo are shown in Figure20. It is evident that our Color Visual Group Normalization technique yields excellent contrast and edge-preservation and provides a natural color result for the restored image [48].
  • Mammograms are complex in appearance and signs of early disease are often small and/or subtle.
  • Digital mammogram image enhancement is particularly important for solving storage and logistics problems, and for the possible development of an automated-detection expert system.
  • the DAF-wavelet based mammogram . enhancement is implemented in the following manner. First we generate a perceptual lossless quantization matrix Q ) m to adjust the original transform coefficients C, Struktur,(&). This treatment provides a simple human- vision-based threshold technique for the restoration of the most important perceptual information in an image. For grayscale image contrast stretching, we appropriately normalize the decomposition coefficients according to the length scale, I, of the display device [16] so that they fall in the interval of [OJ] of the device frame
  • NQ.m r m X J (NC m ) (62) where the constant ⁇ and function X j m are appropriately chosen so that the desired portion of the grayscale gradient is stretched or compressed.
  • FIG. 21(a) is an original 1024x1024 side-view breast image which has been digitized to a 200 micron pixel edge with 8 bits of gray scale.
  • the 46 enhanced image result is shown in Figure 22(b). In this case we again obtain a significant improvement in image quality as described herein.
  • Local averaging is one of the simplest filtering techniques. It preserves the mean gray level while suppressing the variability in flat regions.
  • simple linear local averaging is undesirable for image smoothing because it is incapable of preserving image edges.
  • Linear local averaging filters are essentially low pass filters, which tend to blur the edges and fine structures in the original image.
  • nonlinear filters A number of nonlinear selective averaging methods [1-8] have been devised for this pu ⁇ ose. The basic idea for these methods is t select only a portion of the gray level values in the local window to use in a (weighted) average.
  • references [3-5] an alpha-trimmed mean filter ( -TMF), which uses a "median basket” to select a predetermined number of pixels above and below the median pixel to the sorted pixels of the moving window, was proposed.
  • the values in the basket are averaged to give the ⁇ -TMF filtering output.
  • An asymmetric way to select the averaging pixels whose values are close to that of the median pixel was presented in references [4-6] and was named the modified trimmed mean filter (MTMF) in reference[4].
  • GTMF modified trimmed mean filter
  • the selected pixels and the center pixel in the window are weighted and averaged to give the filtering output. It has been shown--[8] that the GTMF performs better than other well known filters for the removal of either impulse noise or additive noise.
  • a new nonlinear filtering technique is disclosed, called the "dual window selective averaging filter" (DWSAF), to remove additive noise (e.g., Gaussian noise).
  • DWSAF dual window selective averaging filter
  • two normal concentric moving windows and a pixel container are employed to determine the values to be used in replacing the gray level value of the center pixel I c . Three steps are employed in this filtering algorithm. First, the GTMF is implemented within the smaller. window
  • the GTMF also includes the center pixel I c in the averaging operation and its weight is usually larger than those of other pixels in the median basket, which is important for the removal of additive noise.
  • a threshold T based on the GTMF output G c is used to select the pixels for the container.
  • a pixel is selected for inclusion in the container if its value is in the range of [G c -T, G c +T ⁇ .
  • An averaging operation is then utilized to generate an adjusted replacement A c for _/.
  • a 3-entry median basket ⁇ -TMF is implemented according to
  • the ⁇ -TMF outperforms the median filter.
  • the ⁇ -TMF does not take the I c as a special pixel.
  • I c has the largest probability of being the closest to the true value among all the pixels in the window. Neglecting the influence of the center pixel is a 54 mistake if one desires to filter additive noise.
  • the GTMF uses a median basket to collect a group of pixels from the sorted pixel values of the window associated with I c , in the same way as the ⁇ -TMF.
  • the values of the selected pixels and I c are then weighted and averaged to give the GTMF output.
  • DWSAF dual window selective averaging filter
  • MTMF modified trimmed mean filter
  • D c is the output of the DWSAF to replace I c and A c is the average of the gray level values in the container.
  • D c is the output of the DWSAF to replace I c
  • a c is the average of the gray level values in the container.
  • the reason for employing two windows is as follows. As in the case of median filter, the implementation of the GTMF using a larger window blurs the image more than using a smaller window. However, the output of the GTMF in a smaller window can be used as a criteria to improve the filtering result by use of the standard selective averaging algorithm in the larger window. This reduces the image blurring because we only average those gray level values that are close to the GTMF output.
  • the DWSAF algorithm can be employed iteratively, which generally improves its performance. If the weight w c of the center pixel is too high, the output values at some pixels may remain close to their original input values after a number of iterations, and thus some isolated impulse-like noise may be present. Numerical 56 experimentation shows that changing the threshold Jto zero after a number of iterations greatly alleviates this problem.
  • the sizes of the two moving windows we used are 7 and 19 for W s and W L respectively.
  • the DWSAF algorithm is implemented recursively.
  • the sizes of the two windows for both corrupted images are 3 for W s and 5 for
  • the PSNR is increased about 8.52 dB for the lower noise image and 1 Q J4 dB for the higher noise image, both of which are better than the best median, ⁇ -TMF and MTMF filtering results.
  • the distributed approximating functional (DAF) approach to the time evolution of wave packets is a powerful, new method aimed at taking advantage of the local nature of the potential in the coordinate representation for ordinary chemical collisions, and the localized nature of the kinetic energy in this same representation.
  • DAFs distributed approximating functional
  • the propagator is banded, its application to the propagation on the grids (1) scales like the number of grid points, N, in any dimension (which is the ultimate for any grid methods; the scaling constant depends on the band width); (2) requires reduces communication time when implemented on massively parallel computer; and (3) minimizes the storage requirement for the propagator.
  • DAF can be regarded as a radial inte ⁇ olating basis, as well as the scaling function.
  • the corresponding DAF wavelets can be implemented using generalized DAFs.
  • RBF radial basis functional
  • DAF neural networks are constructed and applied in signal processing.
  • Symmetric quincunx-DAF Dirichlet classes are carefully designed for hyper spherical surface representation.
  • the function H 2 deris the Hermite polynomial of even order, 2n.
  • the quantity ⁇ is the width of the Gaussian window of the Hermite polynomial.
  • the qualitative behavior of one particular Hermite DAF is shown in Figure 1.
  • the Hermite polynomial H is generated by the usual recursion
  • V x is the x component of the gradient operator.
  • this kind of functional can be regarded as the smoothing operator or the scaling function in wavelet theory. It can be used to generate the corresponding wavelets (differential functionals) for signal analysis.
  • the discrete wavelet transform is implemented using filterbanks. Hoffman and Kouri gave a tensor-product extension of DAF in multi-dimensional Hilbert space. It can be explained in detail as following.
  • a new algorithm for impulse noise removal was presented.
  • a group of significant values in the neighboring window of one pixel are bundled and weighted to obtain a modified luminance estimation (MLE).
  • MLE modified luminance estimation
  • a threshold selective-pass technique is employed to determine whether any given pixel should be replaced by its MLE. Iterative processing improves the performance of our algorithm for -highly corrupted images. Numerical experiments show that our technique is extremely robust and efficient to implement, and leads to significant improvement over other well-known methods.
  • inte ⁇ olating wavelets based on a subdivision scheme has attracted much attention recently. It possesses the attractive characteristic that the wavelet coefficients are obtained from linear combinations of discrete samples rather than from traditional inner product integrals.
  • various inte ⁇ olating wavelets can be formulated in a biorthogonal setting. Harten has described a kind of piecewise biorthogonal wavelet construction method [12]. Swelden independently has developed essentially this method into the well known "lifting scheme" theory [32], which can be regarded as a special case of the Neville filters [19].
  • the lifting scheme enables one to construct a custom-designed biorthogonal wavelet transforms assuming only a single low-pass filter without iterations.
  • the lifting-inte ⁇ olating wavelet theory is closely related to: the finite element technique for the numerical solution of partial differential equations, the subdivision scheme for inte ⁇ olation and approximation, multi-grid generation and surface fitting techniques.
  • the most attractive feature of the approach is that discrete samplings are made which all identical to wavelet multiresolution analysis. Without any of the pre-conditioning or post-conditioning processes required for accurate wavelet analysis, the inte ⁇ olating wavelet coefficients can be implemented using a parallel computational scheme.
  • Lagrange inte ⁇ olation polynomials are commonly used for signal approximation and smoothing, etc.
  • inte ⁇ olating Lagrange functionals By carefully designing the inte ⁇ olating Lagrange functionals, one can obtain smooth inte ⁇ olating scaling functions with arbitrary order of regularity.
  • we will present three different kinds of biorthogonal inte ⁇ olating Lagrange wavelets (Halfband Lagrange wavelets, B-spline Lagrange wavelets and Gaussian-Lagrange DAF wavelets) as specific examples of generalized inte ⁇ olating Lagrange wavelets.
  • Halfband Lagrange wavelets can be regarded as an extension of Dubuc inte ⁇ olating functionals [8, 11], auto-correlation shell wavelet analysis [26], and halfband filters [1].
  • B-spline Lagrange Wavelets are generated by a B-spline- 66 windowed Lagrange functional which increases the smoothness and localization properties of the simple Lagrange scaling function and
  • Lagrange Distributed Approximating Functionals have been successfully applied for numerically solving various linear and nonlinear partial differential equations [40].
  • Typical examples include DAF-simulations of 3-dimensional reactive quantum scattering and 2-dimensional Navier-Stokes fluid flow with non-periodic boundary conditions.
  • DAFs can be regarded as particular scaling functions (wavelet-DAFs); the associated DAF-wavelets can be generated in a number of ways [41]. Both DAFs and DAF-wavelets are smooth and decay rapidly in both the time and frequency representations.
  • PN perceptual normalization
  • HVS Human Vision System
  • Perceptual signal processing has the potential of overcoming the limits of the traditional Shannon Rate-distortion (R-D) theory for perception-dependent 67 information, such as images and acoustic signals.
  • R-D Shannon Rate-distortion
  • VGN Visual Group Normalization
  • VGN Visual Group Normalization
  • VR-D Visual Rate-Distortion
  • Softer Logic Masking is an adjusted de-noising technique [29], designed to improve the filtering performance of Donoho's Soft Threshold (ST) method [9].
  • the SLM technique efficiently preserves important information, particularly at an edge transition, in a manner particularly suited to human visual perception.
  • ⁇ can be represented as a line ' ar combination of dilates and translates of itself, while the weight is the value of ⁇ at a subdivision integer of order 2.
  • Inte ⁇ olating wavelets are particularly efficient for signal representation since their multiresolution analysis can be simply realized by discrete sampling. This makes it easy to generate a subband decomposition of the signal without requiring tedious iterations. Moreover, adaptive boundary treatments and non-uniform samplings can be easily implemented using inte ⁇ olating methods.
  • the inte ⁇ olating wavelet transform possesses the following characteristics:
  • a parallel-computing algorithm can be easily constructed.
  • the calculation and compression of coefficients are not coupled.
  • the calculation of the wavelet coefficients, W j k does not exceed N+2 multiply/adds for each.
  • the wavelet coefficients decay rapidly.
  • threshold masking and quantization are nearly optimal for a wide variety of regularization algorithms.
  • inte ⁇ olating wavelets are closely related to the following wavelet types: Band-limit Shannon wavelets
  • This subdivision scheme defines a function that is uniformly continuous at the rationals and has a unique continuous extension.
  • the function F D is a compactly supported interval polynomial and is regular; it is the auto-correlation function of the Daubechies wavelet of order D+L This function is at least as smooth as the corresponding Daubechies wavelets.
  • Lamarie-Battle, Meyer, and Daubechies wavelets lead, respectively, to the inte ⁇ olating Schauder, inte ⁇ olating spline, C°° inte ⁇ olating, and Deslauriers-Dubuc wavelets.
  • Lagrange half-band filters Ansari, Guillemot, and Kaiser [1] have used Lagrange symmetric halfband
  • a special case of symmetric halfband filters can be obtained by choosing the filter coefficients- according to the Lagrange inte ⁇ olation formula. The filter coefficients are then given by
  • These filters have the property of maximal flatness. They possess a balance between the degree of flatness at zero frequency and flatness at the Nyquist frequency (half sampling). These half-band filters can be utilized to generate the inte ⁇ olating wavelet decomposition, which can be regarded as a class of auto-correlated shell orthogonal wavelets such as the Daubechies wavelets [6].
  • the inte ⁇ olating wavelet transform can also be generated by different Lagrange polynomials, such as [26] according to
  • the subband filters generated by Lagrange inte ⁇ olating functionals satisfy (1) Inte ⁇ olation: h( ⁇ )+h( ⁇ A)-l
  • the biorthogonal subband filters can be expressed as
  • the Donoho inte ⁇ olating wavelets have some drawbacks, because the low- pass coefficients are generated by a sampling operation only, as the decomposition layer increases, the correlation between low-pass coefficients becomes weaker.
  • the inte ⁇ olating (prediction) error (high-pass coefficients) strongly increases, which destroys the compact representation of the signal. Additionally, it does not lead to a Riesz basis for E 2 (R) space.
  • the newly developed filters h , g , fo , and g also construct the biorthogonal dual pair for perfect reconstruction. Examples of generated biorthogonal lifting wavelets with different regularity are shown in Figure 6 and Figure 7.
  • Lagrange polynomials are natural inte ⁇ olating expressions. Utilizing a different expression for the Lagrange polynomials, we can construct other types of inte ⁇ olating wavelets. We define a class of symmetric Lagrange inte ⁇ olating functional shells as
  • the scaling function (mother wavelet) can be defined as an inte ⁇ olating B-Spline Lagrange functional (BSLF)
  • N is the B-spline order
  • ⁇ is the scaling factor to control the window width
  • Gaussian-Lagrange DAF Wavelets We can also select a distributed approximating functional-Gaussian Lagrange
  • W_(x) is a window function which is selected to be a Gaussian
  • ⁇ is a window width parameter
  • PJA is the Lagrange inte ⁇ olation kernel.
  • the DAF scaling function has been successfully introduced as the basis for an efficient and powerful grid method for quantum dynamical propagations [40]. Using the lifting scheme [32], a wavelet basis is generated.
  • the Gaussian window in our DAF- wavelets efficiently smooths out the Gibbs oscillations, which plague most conventional wavelet bases.
  • the following equation shows the connection between the B-spline function and the Gaussian window [34]:
  • the Gaussian Lagrange wavelets generated by the lifting scheme will be much like the B-spline Lagrange wavelets.
  • the Gaussian Lagrange DAF based wavelets are smoother and decay more rapidly than B-spline Lagrange wavelets. If we select more sophisticated window shapes, the Lagrange wavelets can be generalized further. We shall call these extensions Bell-windowed Lagrange wavelets.
  • wavelet coefficients can be regarded as the results of the signal passing through equivalent decomposition filters (EDF).
  • EDF equivalent decomposition filters
  • the responses of the EDF LC i m ( ⁇ ) axe the combination of several recurrent subband filters at different stages.
  • the EDF amplitudes of different sub- blocks are different.
  • the magnitude of the decomposition coefficients in each of the sub-blocks will not exactly reproduce the true strength of the signal components.
  • various EDFs are incompatible with each other in the wavelet transform.
  • the decomposition coefficients are re-scaled with respect to a common magnitude standard.
  • EDF coefficients, C j m (k), on layer y and block m should be multiplied by a magnitude scaling factor, ⁇ , to obtain an adjusted magnitude representation [28].
  • This factor can be chosen as the reciprocal of the maximum magnitude of the frequency response of the equivalent filter on node (j,m)
  • An image can be regarded as the result of a real object processed by a human visual system.
  • the latter has essentially many subband filters.
  • the responses of these human filters to various frequency distributions are not at all uniform. Therefore, an appropriate normalization of the wavelet coefficients is necessary.
  • the human visual system is adaptive and has variable lens and focuses for different visual environments. Using a just-noticeable distortion profile, we can efficiently remove the visual redundancy from decomposition coefficients and normalize them with respect to a standard of perception importance.
  • a practical, simple model for perception efficiency has been presented by Watson, et al. [5] for data compression. This model is adapted here to construct the "perceptual lossless" response magnitude Y for normalizing according to the visual response function [39],
  • This treatment provides a simple, human- vision-based threshold technique [39] for the restoration of the most important perceptual information in an image.
  • the luminance (magnitude) of the image pixels is the principal concern.
  • VGN Visual Group Normalization
  • Masking is essential to many signal-processing algorithms. Appropriate masking will result in reduction of noise and undesired components. Certainly, it is very easy to set up masking if the spectral distribution of a signal and its noise is known. However, in most cases, such prior knowledge is not available. Statistical 77 properties of the signal and its noise are assumed so that the noise is taken to be relatively more random than the signal in each subband. Hard logic masking and soft logic masking techniques are discussed in the following two subsections. Hard Logic Masking The Visual Group Normalization method provides an efficient approach for re-normalizing the wavelet decomposition coefficients so that various subband filters have appropriate perceptual impulse responses. However, this algorithm alone does not yield the best SNR in real signal processing.
  • noise and/or interference components can be embedded in different nodes of the subband decomposition tree.
  • a filtering process is needed to reduce the noise and preserve the main signal information.
  • Noise due to random processes has a comparatively wide-band distribution over the decomposition tree, whereas mechanical noise can have a narrow-band distribution over a few specific subband components. Therefore, time- varying masking techniques are utilized to reduce noise. We discuss a few useful masking methods in the rest of this subsection.
  • a single zone threshold masking is the simplest masking method. With a given decomposition tree, a constant threshold r is selected for our magnitude normalized wavelet decomposition coefficients NC j m (k). That is, if the absolute value ofNC j JA) is greater than the threshold r, the original decomposition coefficient will be kept; otherwise it will be set to zero. That is
  • the threshold r can be adaptively adjusted to the strength 78 of the noisy environment.
  • the threshold r should be set higher to suppress a noisier signal, and in general, r should vary as a function of the statistical properties of the wavelet decomposition coefficients, the simplest and most important of which are the mean and second variance. These are inco ⁇ orated in the present work.
  • N 2 ⁇ J N and N is the total length of a filter.
  • the corresponding reconstruction coefficients are selected by the rules
  • an alarm threshold, r based on the mean value of multiple measurements of the background signal.
  • This approach is similar to a background-contrasted signal processing in which only the differences of the signal's optimal tree decomposition coefficients, NC, chief,(&), and yv Cy" ) > the background decomposition coefficients of the same tree structure, are used for background- " contrasted signal reconstruction.
  • ⁇ y sgn(y i ⁇ y ⁇ - ⁇ + .- ⁇ ⁇ ⁇ l (42)
  • SLM Soft Logic Masking
  • the softer logic mapping, S:[0,l]-»[0,1], is a non-linear, monotonlcally increasing sigmoid functional.
  • a comparison of the hard and softer logic masking functionals is depicted in Figure 13.
  • ⁇ r is an upper frame boundary of the wavelet transform, i.e. the upper boundary singular value of the wavelet decomposition matrix.
  • the possible noise sources include photoelectric exchange, photo spots, the error of image communication, etc.
  • the noise causes the visual perception to generate speckles, blips, ripples, bumps, ringings and aliasing.
  • the noise distortion not only affects the visual quality of the images, but also degrades the efficiency of data compression and coding.
  • De-noising and smoothing are extremely important for image processing.
  • the traditional image processing techniques can be classified as two kinds: linear or non-linear.
  • the principle methods of linear processing are local averaging, low-pass filtering, band-limit filtering or multi-frame averaging. Local averaging and low-pass filtering only preserve the low band frequency components of the image signal.
  • the original pixel strength is substituted by an average of its neighboring pixels (within a square window).
  • the mean error may be improved but the averaging process will blur the silhouette and finer details of the image.
  • Band-limited filters are utilized to remove the regularly appearing dot matrix, texture and skew lines. They are useless for noise 82 whose correlation is weaker. Multi-frame averaging requires that the images be still, and the noise distribution stationary. These conditions are violated for motion picture images or for a space (time)-varying noisy background.
  • MSE mean square error
  • the MSE can be defined to be
  • More efficient human- vision-system-based image processing techniques possess the advantages of 1) large range de-correlation for convenience of compression and filtering; 2) high perceptual sensitivity and robustness; 3) filtering according to human visual response. It therefore can enhance the most important visual information, such as edges, while suppressing the large scale of flat regions and background. In addition 4) it can be carried out with real-time processing.
  • the space (time)-scale logarithmic response characteristic of the wavelet transform is similar to the HVS response. Visual perception is sensitive to narrow band low-pass components, and is insensitive to wide band high frequency components. Moreover, from research in neurophysiology and psychophysical studies, the direction- selective cortex filtering is very much like a 2D-wavelet decomposition..
  • the high-pass coefficients of the wavelet transform can be regarded as the visible difference predictor (VDP).
  • VGN-WT modified wavelet analysis- Visual Group-Normalized Wavelet Transform
  • Halfband Lagrange wavelets can be regarded as an extension of " Dubuc inte ⁇ olating functionals, auto-correlation shell wavelet analysis and halfband filters.
  • B- spline Lagrange Wavelets are generated by B-spline windowing of a Lagrange functional, and lead to increased smoothness and localization compared to the basic Lagrange wavelets.
  • Lagrange Distributed Approximating Functionals can be regarded as scaling functions (wavelet-DAFs) and associated DAF-wavelets can be generated in a number of ways [41]. Both DAFs and DAF-wavelets are smoothly decay in both time and frequency representations.
  • the present work extends the DAF approach to signal and image processing by constructing new biorthogonal DAF-wavelets and associated DAF- filters using a lifting scheme [32].
  • VGN Visual Group Normalization
  • Medical image such computed tomography (CT), magnetic resonance image (MRJ), X-ray mammogram, ultrasound and angiography, is one of major methods for field diagnosis.
  • CT computed tomography
  • MRJ magnetic resonance image
  • X-ray mammogram is widely recognized as being the only effective method for the early detection of breast cancer.
  • Major advances in screen/film mammograms have been made over the past decade, which result in significant improvements in image resolution and film contrast without much increase in X-ray dose.
  • mammogram films have the highest resolution in comparison to various other screen/film techniques.
  • many breast cancers cannot be detected just based on mammogram images because of poor visualization quality of the image.
  • EDGE ENHANCEMENT NORMALIZATION Mallat and Zhong realized that Wavelet multiresolution analysis provides a natural characterization for multiscale image edges, and those manipulations can be easily achieved by various differentiations [15]. Their idea was extended by Laine et al [7] to develop directional edge parameters based on subspace energy measurement. An enhancement scheme based on complex Daubechies wavelets was proposed by Gagnon et al. [9]. These authors made use of the difference between real part and imaginary part of the wavelet coefficients. One way or another, distorted wavelet transforms are designed to achieve desired edge enhancement.
  • the human visual system is adaptive and has variable lens and focuses for different visual environments. Using a just-noticeable distortion profile, one can efficiently remove the visual redundancy from the decomposition coefficients [17] and normalize them with respect to the importance of perception.
  • a practical simple model for perception efficiency has been presented to construct the "perceptual lossless" response magnitude Y j m for normalizing according to visual response,
  • A is a constant
  • R is the Display Visual Resolution (DNR)
  • f 0 is the spatial frequency
  • d m is the directional response factor.
  • a perceptual lossless quantization matrix Q j m is [10] QyiYj, ⁇ 0) where is a magnitude normalized factor.
  • NC m (k) Q m C m (k)IL (4)
  • NCj, m ⁇ h y ⁇ m (NC lm ) (5)
  • constant ⁇ and function ⁇ m is appropriately chosen so that desired portion of the grayscale gradient is stretched or compressed. For example, function
  • FIG. 1(b) is an original 1024 ⁇ 1024 side- view breast image which has been digitized to 200 micron pixel edge with 8 bits of gray scale. Enhanced image result is depicted in Figure 2(b). In this case we obtain a similar result in the previous one.
  • Edge enhancement normalization (EEN) and device adapted visual group normalization (D AVGN) are proposed for image enhancement without prior knowledge of the spatial distribution of the image.
  • Our algorithm is a natural extension of earlier normalization techniques for image processing. Biorthogonal inte ⁇ olating distributed approximating functional wavelets are used for our data representation. Excellent experimental performance is found for digital mammogram image enhancement.
  • GTMF generalized trimmed mean filter
  • ⁇ -TMF alpha-trimmed mean filter
  • the GTMF outperforms many well known methods in removing highly impulse noise corrupted images, it can be further modified to improve the filtering performance.
  • the averaging weights are predetermined and fixed throughout the filtering procedure.
  • a varying weight function is designed and applied to GTMF.
  • VWTMF varying weight trimmed mean filter
  • the argument of the weight function is the absolute difference between the luminance values in the median basket and the median value. Because we only 95 concentrate on filtering impulse-noise-corrupted images in this paper, the weight of the center pixel in the moving window is always assumed to be zero.
  • a switching scheme [9] as a impulse detector and an iterative procedure [8] is employed to improve the filtering performance.
  • the pixels ⁇ /,, I 2 , . . . , I m ⁇ , I m , I m+l , . . . , I. ⁇ in the moving window associated with a pixel I c have been sorted in an ascending (or descending) order in the same way as in the conventional median filtering technique, with I m being the median value.
  • the key generalization to the median filter which is introduced in the alpha-trimmed mean filter ( ⁇ -TMF) [5,6] is to design a median basket in which to collect a given number of pixels above and below the median pixel. The values of these pixels are then averaged to give the filtering output, A c , as an adjusted replacementvalue to I c , according to
  • V r . m+L 0
  • ⁇ J is a value in the range of [OJ] defined by
  • the weight w(x) in Eq. (3) is a decreasing function in the range [0, 1 ] and is taken to be
  • the median value has the least probability to be impulse noise corrupted because the impulses are typically presented near the two ends of the sorted pixels.
  • the median value may not be optimal because it may differ significantly from the noise- free value.
  • the ⁇ -TMF will not perform better than the median filter when treating impulse noise corrupted images in the absence of any other technique (such as the switching scheme) because the corrupted pixels may be also selected to the median basket for the averaging operation.
  • the VWTMF which uses a weighted averaging operation can alleviate the shortcomings of both filters.
  • the weight of the median value is the large stand the weights of other luminance values in the median basket vary according to their differences from the median value. If a impulse corrupted pixel I y happens to be selected for inclusion in the median basket, its contribution to the average will be small because ⁇ J is large.
  • the weight function can assist the VWTMF in eliminating the impulse noise while providing a well adjusted replacement value for the center pixel I c .
  • the standard 8 bit, gray-scale "Lena” image (size 512x512) is used as an example to test the usefulness of our filtering algorithm. We degraded it with various percentages of fixed-value (0 or 255) impulse noise.
  • the proposed algorithm is compared with the median filtering and ⁇ -TMF algorithms, and their peak signal-to-noise ratio
  • the ⁇ -TMF performs even worse than the median filter. However, it performs better than the median 99 filter when the switching scheme is used. This reflects the fact that although the ⁇ -TMF may not perform well in impulse noise removal, it is a good impulse detector.
  • the VWTMF performs better than either the median filter or the ⁇ -TMF, ⁇ vhether the switching scheme is used or not. It is simple, robust and efficient.
  • the VWTMF performs well in removing impulse noise, and is simultaneously a good impulse detector. It is especially efficient for filtering highly impulse noise corrupted images.
  • Figure 2 shows the original noise-free image, the impulse noise corrupted image (40% impulse noise), and the filtered results for several algorithms.
  • This paper presents a new filtering algorithm for removing impulse noise from corrupted images. It is based on varying the weights of the generalized trimmed mean filter (GTMF)continuously according to the absolute difference between the luminance values of selected pixels in the median basket and the median value. Numerical results show that the VWTMF is robust and efficient for noise removal and as an impulse detector. Although the VWTMF is only used for removing impulse noise in this paper, we expect that it also will be useful for removing additive noise.
  • GTMF generalized trimmed mean filter
  • Linear local averaging filters are essentially low pass filters. Because the impulse responses of the low pass filters are spatially invariant, rapidly changing signals such as image edges and details, cannot be well preserved. Consequently, impulse noise cannot be effectively removed by linear methods; nonlinear techniques have been found to provide more satisfactory results.
  • the median filter [1] and its various generalizations [2-8], which are well known to have the required properties for edge preservation and impulse noise removal.
  • the median filter is not optimal since it is typically implemented uniformly across the image. It suppresses the true signal as well as noise in many applications. In the presence of impulse noise, the median filter tends to modify pixels that are not degraded. Furthermore, it is prone to produce edge jitter when the percentage of impulse noise is large.
  • two median based filters namely the ⁇ -trimmed mean ( ⁇ -TM) [5-6] filter and the modified trimmed mean (MTM) [6] filter, which select from the window only the luminance values close to the median value, have been proposed.
  • the selected pixels are then averaged to provide the filtering output.
  • these algorithms still have problems.
  • the MTM outperforms the median filter in removing additive noise at the cost of increased computational complexity, but not as well as the median filter in removing impulse noise.
  • the ⁇ -TM filter is in general superior to the median filter as an impulse detector, but its performance in removing impulse noise is not so well. As is shown in the test example of this paper, the ⁇ -TM filter performs even worse than the median filter when no impulse detection techniques are employed. Nevertheless, when the level of impulse noise is high, the ⁇ -TM filter is not optimal for detection, since the selected pixels may have a large probability of being corrupted by impulse noise. Removal of additive noise is a problem for the ⁇ -TM and MTM filters because of not taking account of the central pixel in the window. 102
  • a new filtering technique is disclosed, using what we call a generalized trimmed mean (GTM) filter, which in general outperforms ⁇ -TM and MTM filters for images that are highly corrupted by impulse or additive noise.
  • the GTM filter- is based on a generalization of the ⁇ -TM filter.
  • a symmetric median basket is employed to collect a predetermined number of pixels on both sides of the median value of the sorted pixels in the window. The luminance values of the collected pixels and the central pixel in the window are weighted and averaged to obtain an adjusted value G(m,n) for the central pixel (m, ⁇ ).
  • the central pixel participate in the averaging operation because the probability of the luminance value of the central pixel to be the closest to the noise- free value is larger than for the other pixels.
  • the proposed filter is combined with a switching scheme [9] to produce an impulse detector to preserve the noise-free pixels exactly, while providing an optimal approximation for the noise-corrupted pixels.
  • Many impulse detection algorithms have been proposed [9-12].
  • a median filter-based switching impulse detector is employed. The basic idea is to calculate the absolute difference between the median filtered value and the input value for each pixel. If the -difference is larger than a given threshold, the output is the median filtered value; otherwise, the output equals the input value. We do the same except that we replace the median filter with the GTM filter.
  • a new iteration scheme which generally improves the performance of the filter.
  • the GTM filter is applied to the output of last iteration to give an intermediate output. If the absolute difference between the initial input and the intermediate output is larger than a predetermined threshold T, the current output is the intermediate output, otherwise, the current output is the initial input. In contrast to the traditional iteration technique, our iteration scheme does not use the output of the last 103 iteration to do the switching operation (the second step).
  • the luminance value of a pixel is replaced by the median value in a neighboring spatial window
  • M(m,n) Median ( ⁇ I(iJ) ⁇ ⁇ I(iJ) e ⁇ (m,n) (2) where I(i,j) is the luminance value at pixel (i,j).
  • I(i,j) is the luminance value at pixel (i,j).
  • ( ⁇ -TM) filter [5-6] is to design a symmetric median basket according to luminance value in order to combine a given number of pixels on both sides of the median value of the • sorted pixels in the window. The collected pixels are then averaged to give the filtering output, as follows
  • L ⁇ _aNj with O ⁇ dr ⁇ 0.5.
  • the ar-TM filter outperforms the median filter in detecting the impulse.
  • its capability of removing the impulse noise is even worse than for the median filter when no additional procedure is employed (such as the switching scheme).
  • the ⁇ -TM filter does not perform well, since the pixels being selected for the median basket now have a large probability of being impulse corrupted. It is therefore unreasonable for the ⁇ -TM filter to have the pixels in the basket equally weighted.
  • MTM modified trimmed mean
  • T(m,n) Mean ( ⁇ I(iJ) ⁇ I(i ) e C(m,n) (5)
  • the MTM filter is useful for removing additive noise but does not perform as well as the median filter when impulse noise removal is required, since impulse noise corrupted pixels are independent of the noise-free pixels.
  • ⁇ -TM and MTM filters where additive noise removal is concerned; they do not take special account of the luminance value of the central pixel in the window.
  • the value of the central pixel in general, has a larger probability of being the closest to its true value than those of all other pixels in the window.
  • GTM generalized trimmed mean
  • w c 0 and all w's are the same, it becomes the ⁇ -TM filter.
  • the GTM filter is usually more efficient when highly impulse noise corrupted images are to be filtered.
  • the weight w 0 for the median value is usually chosen to be higher than those of other pixels in the median basket because the median pixel usually has the least probability to be corrupted by impulse noise.
  • Smaller weights for the pixels other than the median pixel can serve as an 105 adjustment for the filtering output, which is important in the iteration calculation.
  • the problems of applying the ⁇ -TM filter to highly impulse noise corrupted images is that the weights for the pixels other than the median pixel are too large, which may give filtering results that are still highly corrupted.
  • the GTM filter includes the central pixel in the averaging operation, it is reasonable to expect that its performance in removing additive noise will be improved compared to the ⁇ -TM and MTM filters.
  • a switching scheme [9] based on our GTM filter is employed to detect the impulse noise corrupted pixels.
  • the filtering output I(m,n) for a pixel (m,n) is generated by the following algorithm:
  • I(m,n) ⁇ G(m,n), ⁇ I 1 (m, ) - G(m, ) ⁇ T
  • I(m,n ⁇ t) is the system output at time t
  • G(m,n ⁇ t) is the intermediate output obtained by applying [6] to I(i ⁇ t- ⁇ ) with (i )e ⁇ (m,n).
  • I(ij ⁇ 0)-1(i ) is the intermediate output obtained by applying [6] to I(i ⁇ t- ⁇ ) with (i )e ⁇ (m,n).
  • I(m, ) ⁇ t) I(m,n)S(m,n) ⁇ t) + G(m,n) ⁇ t)[l - S(m, «)
  • the benchmark 8bpp gray-scale image, "Lena”, size 512x512 is corrupted with different percentages of fixed value impulse noise (40% and 60%), and the Gaussian noise with peak signal-to-noise ratio (PSNR) 18.82 dB respectively.
  • the symmetric moving window size is 3x3, with a 3-entry median basket used for both cases.
  • the PSNR, mean square error (MSE) and mean absolute error (MAE) comparisons of several different filtering algorithms are shown in TABLE I for impulse noise and TABLE II for Gaussian noise.
  • the filtering parameters are also shown in the TABLES. All algorithms are implemented recursively for optimal PSNR and MSE performance. The new iteration scheme proposed in this paper also has been implemented with ⁇ -TM filter for filtering of impulse noise. It is evident from TABLE I that our GTM filter-based switching scheme yields improved results compared to both the median and ⁇ -TM filter-based switching schemes. In general, the ⁇ -TM filter-based switching scheme performs better than the median filter- based switching scheme. However, without the switching scheme, the ⁇ -TM filter performs even worse than the median filter for such highly impulse-noise-corrupted images, showing that although the ⁇ -TM filter is a good impulse detector, it is not good at removing impulse noise by itself.
  • Figure 1(a) the original Lena image is degraded by adding 60% impulse noise.
  • the 3x3 window is not suitable for applying the median filtering algorithms for such a high-noise image.
  • These speckles can be removed by increasing the window size from 3x3 to 5x5 but at the expense of even more blurring after many iterations.
  • Figurel(c) shows our filtering result with the same switching threshold. It is seen that our algorithm yields improved results.
  • signal filtering and processing may be regarded as a kind of approximation problem with noise suppression.
  • DAF theory [1] a signal approximation in DAF space can be expressed as
  • ⁇ (x) ⁇ gt ⁇ a (x-x,)
  • ⁇ jy is a generalized symmetric Delta functional sequence.
  • DAF distributed approximating functional
  • the function H 2n is the Hermite polynomial of even order, 2n.
  • the qualitative behavior of one particular Hermite DAF is shown in Figurel .
  • this kind of functional can be regarded as the smoothing operator or the scaling function in wavelet theory. It can be used to generate the corresponding wavelets (differential functionals) for signal analysis.
  • the discrete wavelet transform is implemented using filterbanks.
  • DAF wavelet neural nets possess a modified form the commonly used DAF approximation, given as

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Abstract

L'invention se rapporte à un procédé de remplissage, filtrage, suppression de bruit, amélioration d'image et acquisition temps-fréquence accrue pour données numérisées d'un ensemble de données, selon lequel on estime des données inconnues au moyen de données réelles en ajoutant des points de données inconnues d'une manière telle que le programme de remplissage peut fonder l'estimation de l'ensemble de données intérieures comprenant les données connues et inconnues, à une précision donnée, sur les points de données connues. Le procédé consiste également à filtrer au moyen de filtres passe-bas- non interpolateurs et à fonctions de lissage réparties bien-ajustées (NIDAF). Ce procédé consiste également en une extension symétrique et/ou antisymétrique de l'ensemble de données de façon à ce que ledit ensemble de données puisse être mieux affiné et puisse être filtré par filtrage de Fourier ou au moyen d'autres types de filtres de basses fréquences ou d'harmoniques.
PCT/US1999/005426 1998-03-13 1999-03-12 Procedes de mise en oeuvre d'un filtrage et d'un remplissage de donnees de fonctions daf WO1999046731A1 (fr)

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US7613504B2 (en) 2001-06-05 2009-11-03 Lumidigm, Inc. Spectroscopic cross-channel method and apparatus for improved optical measurements of tissue
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RU2618390C2 (ru) * 2015-10-06 2017-05-03 Федеральное государственное казенное военное образовательное учреждение высшего профессионального образования "Военный учебно-научный центр Военно-воздушных сил "Военно-воздушная академия имени профессора Н.Е. Жуковского и Ю.А. Гагарина" (г. Воронеж) Министерства обороны Российской Федерации Способ устранения импульсных помех на цветных изображениях
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US6865408B1 (en) 2001-04-11 2005-03-08 Inlight Solutions, Inc. System for non-invasive measurement of glucose in humans
US6983176B2 (en) 2001-04-11 2006-01-03 Rio Grande Medical Technologies, Inc. Optically similar reference samples and related methods for multivariate calibration models used in optical spectroscopy
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