Noncausal Vector Linear Prediction Filters Kh. Manglem Singh and Prabin K. Bora,

Dept. of Electronics and Communication Engineering, IIT, Guwahati – India,

Abstract: A new impulse noise detection technique from gray-scale and multichannel images is presented. It is based on block correlation of the noncausal linear prediction. The linear prediction coefficients are calculated using a block of the image at one instant. These coefficients are valid within that block only and are used to predict a pixel in a sliding window within the block. The sample is predicted from its neighboring pixels, considering past and future values. The current sample is corrupted if its difference from the predicted value is bigger than a pre-defined threshold. Key - words: Scalar, Vector, Linear prediction, Threshold, Center–weighted median filter, Vector median filter, noncausal. 1. Introduction Single channel linear prediction has been extensively used for modeling, estimating and coding of one-dimensional random signals, notably in speech coding and understanding, geophysics and biomedical signal processing applications [1-6]. In speech processing, the linear prediction coding (LPC) model is widely used for short term modeling of voiced speech. The model is used for compression, synthesis and understanding of the speech signal and also to detect impulse noise in speech sample [7-14]. Recently, linear prediction model (LPM) has been extended to two-dimensional cases [15-19]. It has been used in low bit rate image compression, image segmentation and classification and spectrum estimation [20-22]. There is strong correlation among image pixels over two-dimensional neighborhood that makes the linear prediction model very natural for estimating pixel values at any location of the image. Scalar linear prediction is restricted to use in one-dimensional signals and single channel multidimensional signals only. Wiggin and Robinson have used modified the Levinson recursion algorithm for solving

multichannel linear predictive system equations [23]. The system with vector prediction was found to give superior subjective picture quality over systems with scalar prediction when applied to image compression [24-27]. Majority of applications use causal linear prediction that estimates the current value as a weighted linear combination of past values. Causal linear prediction is sometime used on a block-by-block basis, in which case all the data samples in an entire block are available for analysis. Small block sometime results in unstable one. Hsue and Yagle have opined that a better estimate of the sample would be expected if linear prediction is based on both the past and future samples in the neighborhood of the current sample that motivates the use of nocausal linear prediction [28-32]. Causal and scalar linear prediction based on block-by-block has been already used [33] in detection of impulse noise from single channel images in filtering applications. New contribution here is to develop the noncausal linear prediction filter. The block diagram of the proposed filter is shown in the Fig.1. In this, the output of

the median filter is the input of the noncausal linear predictor that predicts the value of the current vector sample. The performance of the linear predictor is enhanced with the median prefilter as a pre-filter. The impulse noise detector compares the difference of the corrupted image and the predicted one and the current sample is declared corrupted if their difference is bigger than a pre-defined threshold.

2. Linear Prediction

Various studies on one-dimensional and two-dimensional multichannel linear predictions have shown that image pixels formed by grouping pixels together exhibit block correlation over some limited region of support [34-36]. There are several choices of the prediction strategy with multichannel images. There are both intrachannel and interchannel correlations. To take into account the interchannel correlation, interchannel predictor should be used. We first review the noncausal linear scalar prediction for a single channel in the following subsection, before we describe the noncausal linear vector prediction for the multichannel case in the next subsection. 2.1 Noncausal Linear Scalar Prediction The image pixel y[m, n] of an image, assuming in a noncausal region of support, as depicted in Fig. 2 is predicted for a first-order prediction as follows:

a],[]),1[],1[(

])1,[]1,[(],[

0,1

1,0^

nmYnmynmya

nmynmyanmyT=++−

+++−= R ]1,1[2

])0,2[]0,0[(]0,1[

0,1

1,0

xx

xxxxxx

Ra

RRa

+

+=

(5) (1) where Y[m, n] = [{y[m, n - 1] + y[m, n + 1]}, {y[m - 1, n] + y[m +1, n]}]T, considering samples at horizontal row and

vertical column only in the window, and a = [a0,1, a1,0] T. The error between the actual image pixel and its estimate can be written as follows:

]),1[

],1[(])1,[

]1,[(],[],[],[

],[],[],[

0,1

1,0

^

nmy

nmyanmy

nmyanmy nmYnmy

nmynmynmeT

++

−−++

−−=−=

−=

a

(2) The expected value of the product of e[m, n] and data is zero. For examples, E(e[m, n] × y[m - 1, n]) = 0 and E(e[m, n] × y[m, n - 1]) = 0. These can be expressed mathematically as follows:

]),1[]),1[],1[(

])1,[]1,[(],[((]),1[]),[],[((0

0,1

1,0

^

nmy) nmynmya

nmynmyanmyE nmynmynmyE

−×++−−

++−−=−×−=

(3) and

) nmy) nmynmya

nmynmyanmyE nmynmynmyE

]1,[]),1[],1[(

])1,[]1,[(],[((])1,[]),[],[((0

0,1

1,0

^

−×++−−

++−−=−×−=

(4) Equations (3) and (4) can be expressed in the following forms:

])2,0[]0,0[(

]1,1[2]1,0[

0,1

1,0

xxxx

xxxx

RRa

RaR

++

=

(6) or in the matrix form as follows:

rRa = (7) where R = E{Y[m, n]YT[m, n]} is the covariance matrix between the samples used for prediction and r = E{y[m, n]YT[m, n]} is the correlation between a sample being predicted and those used for its prediction.

,RR 2R

[1,1] R RR

xxxxxx

xxxxxx

+

+=

]2,0[]0,0[]1,1[2]0,2[]0,0[

R

=

]1,0[]0,1[

xx

xxRR

r

=

0,1

1,0

a

a and a

2.2 Noncausal Linear Vector Prediction There are three methods for multichannel images in linear vector prediction. The methods are intrachannel predictor, contrained intrachannel predictor and interchannel predictor. The first two methods use scalar approach and the third one, vector approach. In intrachannel predictor, intrachannel correlation is exploited, in which a component of a vector signal in one channel is predicted based on the other components in the same channel and likewise, the prediction coefficients of each channel is obtained using Equ. (7). In the case of contrained intrachannel predictor, the same prediction coefficient of a channel is used for all the different channels. Interchannel predictor is a true linear vector predictor that exploits both intrachannel and interchannel correlation. The predicted vector pixel of a L-channel image is expressed as follows:

Ayy

yyy^

],[]),1[],1([

])1,[]1,[(],[

0,1

1,0

nmnmnmA

nmnmAnmTY=++−

+++−=

(8) where each vector pixel has L components, and

Tnmnm

nmnmnm

]}],1[],1([{

]},1,[]1,[[{],[

++−

++−=

yy

yyY,

considering sample vectors at horizontal row and vertical column only in the window, and A = [A0,1, A1,0]T. As before, the expected value of the product of the error and vector sample is zero, i.e. orthogonal, we obtain the predictor coefficients A as follows:

r=AR (9) where R = is the covariance matrix between the vector samples used for prediction and =

is the correlation between a sample being predicted and those used for its prediction.

]},[],[{ nmnmE TYY

r]},[ nmT],n[{ mE Yy

, 2

[1,1]

xxxxxx

xxxxxx

+

+=

]2,0[]0,0[]1,1[2]0,2[]0,0[

RRR

RRRR

=

]1,0[]0,1[

xx

xxR

Rr

=

0,1

1,0

A

AA and

If the difference between the current vector sample and the predicted one is bigger than a pre-defined threshold, the current sample is declared corrupted with impulse noise.

3. Simulation Results The test images are 24-bit color images of Lena, Mandrill, Miramar, Point Loma, Golden Gate and Terrain. All images are of 512 512 size. Impulse noise, generated by the color impulse noise model is used for testing. Impulse noises are artificially injected in these images. The performances are evaluated by the visual observation and in terms of the peak signal to noise ratio (PSNR). In all cases, a window of 3× 3 size is used and the block size is 32 32. The vector median filter (VMF), marginal medial filter (MMF) [37], center-weighted vector median filter (CWVMF) [38] and causal linear vector prediction filters are used for comparison with the noncausal linear vector prediction based filters. Different prediction filters are interchannel predictor filter (LF1), intrachannel predictor filter (LF2), and constrained intrachannel predictor filter (LF3).

×

×

The first set of experiments are conducted to study the efficiency of the proposed filter in removal of impulse noise at different noise ratios. The results are taken using Lena image and Tables 1 and 2 show the results, where the fixed-valued and random-valued impulse noise ratios range from 10% to 60% with p1 = .25 p2 = .25, p3 = .25 and .10 p .60 (p≤ ≤ 1, p2, p3 are intrachannel noise ratios and p, channel factor ). Thresholds θ are set at 30 and 50 for the random-valued and fixed-valued impulse noise ratios respectively. It has been is seen from these tables that the noncausal linear vector prediction filters give better performances (in PSNR). We list more results in Tables 4 and 5 at 20% (p1 = .25 p2 = .25, p3 = .25 and p =.20) impulse noise ratio. Here again, noncausal linear vector prediction filters have better performances. It has been clearly seen

from these tables that noncausal LF1, LF2 and LF3 have almost equal performance, though LF3 has less complexity in terms of computation and storage bits. Fig. 3 shows the efficiency of noncausal LF1 as compared with the standard VMF, applied on Lena image corrupted with 60% impulse noise ( p1 = .25, p2 = .25, p3 = .25 and p = .60). Figs. 7(e) and 7(f) show the difference between these filter outputs and the original image. It has been observed from these images that there are more impulses leftover in the VMF outputs. 4. Conclusions Noncausal linear prediction filter is proposed in this paper. This new technique is an improvement over the commonly used causal linear predictor. It considers both past and present pixels in the neighborhood of the current sample to predict its value, while the causal filters use only past values. Experimental results show better performance than other filters. References [1] N. S. Jayant and P. Noll, Digital

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Switch

Filtered Image

Corrupted Image

Impulse Noise Detector

Linear Predictor

Median Filter

Fig.1 Block Diagram of Linear Predictor Filter.

Fig.2 Noncausal region of support for a first-order linear prediction over a block size of 32 × 32 pixels with 3 × 3 window size. TABLE 1: Comparative Restoration Results of Various Filter at Different fixed-valued impulse noise Ratios on Lena Image.

Causal Noncausal % Noise

VMF MMF CWVMF

LF1 LF2 LF3 LF1 LF2 LF3

10 32.65 32.37 33.65 35.04 35.08 34.99 35.41 35.33 35.39 20 32.13 31.76 32.83 33.86 33.82 33.85 34.03 34.13 34.05 30 31.45 31.04 31.59 32.60 32.76 32.73 32.96 33.01 32.97 40 30.84 30.29 30.55 31.60 31.90 31.75 31.92 31.92 31.82 50 29.79 29.48 29.73 30.92 30.96 30.80 30.87 31.16 31.13 60 28.70 28.89 28.53 29.84 29.74 29.99 30.23 30.21 30.15

TABLE 2: Comparative Restoration Results of Various Filter at Different random-valued impulse noise Ratios on Lena Image.

Causal Noncausal % Noise

VMF MMF CWVMF LF1 LF2 LF3 LF1 LF2 LF3

10 32.63 32.34 33.60 34.19 34.38 34.36 34.60 34.71 34.67 20 32.06 31.74 33.02 33.26 33.23 33.21 33.54 33.46 33.39 30 31.51 31.13 32.23 32.20 32.15 32.30 32.46 32.53 32.57 40 30.87 30.32 31.29 31.43 31.37 31.31 31.80 31.58 31.55 50 30.10 29.52 30.67 30.62 30.37 30.52 30.82 30.83 30.83 60 29.41 28.73 29.71 29.62 29.64 29.63 29.98 30.04 29.90 TABLE 3: Comparative Restoration Results of Various Filters in Removal Fixed-valued Impulse Noise at 20% Noise Ratio from Different Images. Filter Causal Noncausal Image

VMF MMF CWVMFLF1 LF2 LF3 LF1 LF2 LF3

Lena 32.13 31.76 32.83 33.86 33.82 33.85 34.03 34.13 34.05Mandrill 22.56 22.23 23.28 23.19 23.18 23.18 23.29 23.27 23.27Miramar 26.00 25.55 27.10 27.00 26.98 27.02 27.12 27.11 27.11Point Loma

33.16 32.98 36.16 36.40 36.48 36.51 36.33 36.07 36.06

Golden Gate

21.84 21.90 22.14 21.96 22.06 21.96 22.13 22.11 22.10

Terrain 20.73 20.77 21.23 21.20 21.18 21.18 21.22 21.24 21.24 TABLE 4: Comparative Restoration Results of Various Filters in Removal Random-valued Impulse Noise at 20% Noise Ratio from Different Images. Filter Causal Noncausal Image

VMF MMF CWVMF LF1 LF2 LF3 LF1 LF2 LF3

Lena 32.06 31.74 33.02 33.26 33.23 33.21 33.54 33.46 33.39 Mandrill 22.48 22.12 23.42 22.49 22.49 22.48 22.53 22.53 22.53 Miramar 25.93 25.49 26.97 26.51 26.51 26.51 26.59 26.58 25.60 Point Loma

33.20 32.90 33.96 35.46 35.47 35.40 35.63 35.67 35.72

Golden Gate

25.05 25.77 24.91 26.00 25.99 25.80 25.69 25.66 25.96

Terrain 20.71 20.76 21.63 20.87 20.87 20.84 20.90 20.92 20.91

Fig. 3: (a) Original Lena Image, (b) With 60% fixed-valued impulse noise, (c) and (d) are the filtered outputs of VMF and noncausal LF1, (e) and (f) the difference between the original and images (c) and (d) respectively.