Nielsen, Allan Aasbjerg

Based on regression analysis this paper gives a description for simple image filter design. Specifically 3x3 filter implementations of a quadratic surface, residuals from this surface, gradients and the Laplacian are given. For the residual a 5x5 filter is given also. It is shown that the 3x3 filter for the residual gives low values for horizontal and vertical lines and edges as opposed to diagonal ones. Therefore an extension including a rotated version of the filter for the residual to ensure low values for lines and edges in all directions is suggested. It is also shown that the 5x5 filter for the residual does not give low values for lines and edges in any direction. The performance of six noise models including the ones mentioned above are compared. Based on visual inspection of results from an example using a generated image (with all directions and many spatial frequencies represented) it is concluded that if striping is to be considered as a part of the noise, the residual from a 3x3 median filter seems best. If we are interested in a salt-and-pepper noise estimator the proposed extension to the 3x3 filter for the residual from a quadratic surface seems best. Simple statistics and autocorrelations in the estimated noise images support these findings

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An Extension to a Filter Implementation of a Local Quadratic Surface for ImageNoise EstimationAllan Aasbjerg NielsenIMM, Department of Mathematical ModellingTechnical University of Denmark, Building 321DK-2800 Lyngby, Denmarkhttp://www.imm.dtu.dk/aaaa@imm.dtu.dkAbstractBased on regression analysis this paper gives a descrip-tion for simple image filter design. Specifically 33 filterimplementations of a quadratic surface, residuals from thissurface, gradients and the Laplacian are given. For theresidual a 55 filter is given also. It is shown that the33 filter for the residual gives low values for horizontaland vertical lines and edges as opposed to diagonal ones.Therefore an extension including a rotated version of thefilter for the residual to ensure low values for lines andedges in all directions is suggested. It is also shown that the55 filter for the residual does not give low values for linesand edges in any direction. The performance of six noisemodels including the ones mentioned above are compared.Based on visual inspection of results from an example usinga generated image (with all directions and many spatial fre-quencies represented) it is concluded that if striping is to beconsidered as a part of the noise, the residual from a 33median filter seems best. If we are interested in a salt-and-pepper noise estimator the proposed extension to the 33filter for the residual from a quadratic surface seems best.Simple statistics and autocorrelations in the estimated noiseimages support these findings.1. IntroductionAn important task in image analysis is the estimation ofnoise content. This is often done by means of methods thatestimate the noise directly from the data such as residualsfrom mean or median filters. In this paper filter implemen-tations of residuals from a quadratic surface in small (33and 55) windows are given. Also 33 filters for gradientsand the Laplacian are given. A quadratic surface is cho-sen for its expected ability to adapt to both dark and brightlines and edges in all directions. A potential drawback ofthis otherwise desired characteristic is 1) the adaption tothe horizontal (across-track) striping often present in datafrom whisk-broom scanners such as the (spaceborne) Land-sat Thematic Mapper (TM) and the Airborne Visible andInfra-Red Imaging Spectrometer (AVIRIS), and the DigitalAirborne Imaging Spectrometer (DAIS), and 2) the adap-tion to vertical (along-track) striping often present in datafrom push-broom scanners such as the (spaceborne) SPOTHigh Resolution Visible (HRV) and the Compact AirborneSpectrographic Imager (casi), and the (airborne) ReflectiveOptics System Imaging Spectrometer (ROSIS). For a de-scription and comparison of six noise estimators includingthe mean and median filters see [9]. Space limitations al-low neither examples with real world data nor the applica-tion of the noise models to e.g. comparisons between theMAF/MNF transformation, [11, 4, 1, 3, 8, 6, 10, 7], and theprincipal components (PC) transformation.2. Quadratic Surface in a 33 windowIn this section filters based on a quadratic surface in a33 window are deduced. This is done by means of regres-sion analysis.2.1. ResidualsConsider a 33 univariate image Z with the followingnumbering of pixelsZ1Z2Z3Z4Z5Z6Z7Z8Z9where Zidenotes the pixel value at location [Xi; Yi]T. Wewish to predict the pixel values Ziby a quadratic surfacewhich is chosen for its expected ability to adapt to both darkand bright lines and edges in all directions present in the33 imageZi= 0+ 1Xi+ 2Yi+ 3X2i+ 4Y2i+ 5XiYi+ "i;i = 1; : : : ; 9where iare parameters to be estimated. In matrix notationwe getZ =266666666666641 X1Y1X21Y21X1Y11 X2Y2X22Y22X2Y21 X3Y3X23Y23X3Y31 X4Y4X24Y24X4Y41 X5Y5X25Y25X5Y51 X6Y6X26Y26X6Y61 X7Y7X27Y27X7Y71 X8Y8X28Y28X8Y81 X9Y9X29Y29X9Y937777777777775 + " =X + "where Z = [Z1; : : : ; Z9]T,  = [0; : : : ; 5]T and " =["1; : : : ; "9]T. For the norm of the residuals " which havedispersion we getjj"jj2= "T 1" = (Z  X)T 1(Z  X)which is a quadratic function in . To minimise we differ-entiate and set the derivative to zero@@jj"jj2= 2XT 1X   2XT 1Z = 0 = (XT 1X) 1XT 1Z:If we think of the 33 image as a window moving over alarger image to make local estimates of a quadratic surfaceand consider the center pixelZ5as the origin of a coordinatesystem that moves with the window, we getZ5=1 X5Y5X25Y25X5Y5 + "5=1 0 0 0 0 0 + "5= 0+ "5:We see that with fixedwe can implement the quadraticsurface estimator for the center pixel as a filter: multiplythe first row of (XT 1X) 1XT 1 with Z to obtain0. With  = 2I where 2 is the variance of "iand I isthe identity matrix we get (Gaussian or other weights canbe applied through another choice of )X =266666666666641  1 1 1 1  11 0 1 0 1 01 1 1 1 1 11  1 0 1 0 01 0 0 0 0 01 1 0 1 0 01  1  1 1 1 11 0  1 0 1 01 1  1 1 1  137777777777775XTX =266666649 0 0 6 6 00 6 0 0 0 00 0 6 0 0 06 0 0 6 4 06 0 0 4 6 00 0 0 0 0 437777775(XTX) 1=26666664590 0  13 1300160 0 0 00 0160 0 0 130 0120 0 130 0 01200 0 0 0 01437777775:In this case we are interested in 0so we need the firstrow of (XTX) 1XT only (below we shall need morerows)(XTX) 1XT= (1)26666664 1929 19295929 1929 19 16016 16016 160161616160 0 0  16 16 1616 131616 131616 1316161616 13 13 13161616 140140 0 0140  1437777775:Because of the special structure of (XTX) 1 (the threezeros in columns two, three and six of the first row) andXT (rows two, three and six corresponding to the X-, Y -andXY -terms in the estimator) 0corresponding to the firstrow of (XTX) 1XT is independent of the X-, Y - andXY -terms.The estimator for Z5which we call ^Z5is^Z5= 1929 19295929 1929 19Zor written as a filter for Z 1929 19295929 1929 19:For the residual "5= Z5 ^Z5we get0 0 00 1 00 0 0 19 1 2  12 5 2 1 2  1=191  2 1 2 4  21  2 1which is separable. We note that the weights add to zeroand that all rows and columns have weights that add to zero.Hence horizontal and vertical ideal one-pixel wide lines andideal edges have residuals equal to zero. Diagonal ones donot. The filter for the residual is proportional to the filterpostulated in [5]. The filter weights are independent of thechosen coordinate system.2.2. GradientsTo obtain gradients in the column and row directions wefind@Z5@X5= 1+ 23X5+ 5Y5= 1@Z5@Y5= 2+ 24Y5+ 5X5= 2:From equation 1 we get the filters16 1 0 1 1 0 1 1 0 1and 161 1 10 0 0 1  1  1:2.3. The LaplacianTo obtain the Laplacian we find@2Z5@X25= 23;@2Z5@Y25= 24and r2 = 2(3+ 4):From equation 1 we get the filter132  1 2 1  4  12  1 2:3. Quadratic Surface in a 55 windowAs in the 33 case for a 55 window with this pixelnumberingZ1Z2Z3Z4Z5Z6Z7Z8Z9Z10Z11Z12Z13Z14Z15Z16Z17Z18Z19Z20Z21Z22Z23Z24Z25we get for the center pixel Z13= 0+ "13with Z =[Z1; : : : ; Z25]T,  = [0; : : : ; 5]T= (XTX) 1XTZ (Xis now 25 rows by 6 columns), " = ["1; : : : ; "25]T.For the residual "13= Z13 ^Z13we get117513  2  7  2 13 2  17  22  17  2 7  22 148  22  7 2  17  22  17  213  2  7  2 13:We note that the weights add to zero and that no rows orcolumns have weights that add to zero. We see that we donot obtain the same desired filter characteristics as with the33 filter.In a similar fashion we could obtain filters for gradientsetc. with other filter sizes.4. An Extension to the 33 filterIn the 33 case we saw that horizontal and vertical linesand edges have low residuals as opposed to diagonal ones.Therefore the following extension to the simple 33 filteris suggested: first apply the simple filter derived above, thenapply the same filter rotated 45 and use the residual closerto zero. This ensures low residuals for both horizontal, ver-tical and diagonal lines and edges.5. ExampleAn example with a generated image is shown. The gen-erated image features all directions and many spatial fre-quencies. Results from six noise estimators are shown,namely1. simple differencing with the immediate north and eastneighbours corresponding to this 22 filter (with thecenter pixel placed in the lower left corner)12 1 02  1;this is used in [2] which is a commercial softwarepackage that offers an MNF transformation;2. residual from a 33 mean filter19 1  1  1 1 8  1 1  1  1;it is seen that this filter results in high residuals for linesand edges in all directions;3. residual from a 33 median filter;4. residual from the 33 filter for a quadratic surface de-rived above191  2 1 2 4  21  2 1;5. residual from the 33 filter for a quadratic surface de-rived above rotated 4519 2 1  21 4 1 2 1  2;it is seen that this filter results in low residuals for di-agonal lines and edges;6. residual from the extended 33 filter for a quadraticsurface proposed in this paper (i.e., choose the residualcloser to 0 from the two filters mentioned immediatelyabove).The generated 256256 image which features all direc-tions and many spatial frequencies is shown in Figure 1.The left column shows the generated image itself (top)and the generated image with pseudo-random, indepen-dent, zero-mean, Gaussian noise with a signal-to-noise ratioequal to one added (bottom). The right column shows thecorresponding noise images as estimated by means of theresidual from a quadratic surface in a 55 window. Table 1shows simple statistics and autocorrelations between E-W,N-S, SW-NE, SE-NW neighbours and their mean value inthese images.Figures 2 and 3 show noise as estimated from the six 33filters given above stretched linearly between minimum andmaximum, Figure 2 on the image in Figure 1 top-left, andFigure 3 on the image in Figure 1 bottom-left. Tables 2 and3 show simple statistics and autocorrelations between E-W,N-S, SW-NE, SE-NW neighbours and their mean value inthe images in Figures 2 and 3, respectively.6. Discussion and ConclusionsFigure 1 shows that the 55 quadratic surface residualfilter does not give low values in any direction for interme-diate and high spatial frequencies. Figures 2 and 3 showthat 1) noise model 1 leaves more visual structure than theother noise models, noise estimated from models 1, 2, 3 and5 contain horizontal and vertical striping (which for model5 complies with the filter weights), noise estimated frommodel 4 contains diagonal structure for intermediate andhigh spatial frequencies, noise estimated from model 5 con-tains low and intermediate spatial frequency diagonal struc-ture, and noise estimated from model 6 contains intermedi-ate spatial frequency diagonal structure; 2) for noise model4 we get low estimates in the E-W and N-S directions forall spatial frequencies also with severe noise present (thesignal-to-noise ratio is one); 3) for noise model 5 we getlow estimates in the SW-NE and SE-NW directions for highspatial frequencies; and 4) for noise model 6 we get a com-bination of the characteristics of models 4 and 5.Remarks based on inspection are supported by the sim-ple statistics and autocorrelations calculated. If striping isto be considered as a part of the noise, model 3 seems tobe the best noise detector since it picks up both salt-and-pepper noise and striping in all directions at many spatialfrequencies. The suggested noise model 6 seems to be thebest salt-and-pepper noise detector.7. AcknowledgmentsThe work reported here is funded partly by the DanishNational Research Councils under the Earth ObservationProgramme.References[1] K. Conradsen, B. K. Nielsen, and T. Thyrsted. A compari-son of min/max autocorrelation factor analysis and ordinairyfactor analysis. In Proceedings from the Nordic Symposiumon Applied Statistics, pages 47–56, Lyngby, Denmark, Jan-uary 1985.[2] ENVI User’s Guide, the Environment for Visualizing Images,Version 2.6. Research Systems, Inc., and Better SolutionsConsulting, 1997. Internet http://www.rsinc.com/.[3] B. K. Ersbøll. Transformations and Classifications of Re-motely Sensed Data: Theory and Geological Cases. PhDthesis, Institute of Mathematical Statistics and OperationsResearch, Technical University of Denmark, Lyngby, 1989.297 pp.[4] A. A. Green, M. Berman, P. Switzer, and M. D. Craig. Atransformation for ordering multispectral data in terms ofimage quality with implications for noise removal. IEEETransactions on Geoscience and Remote Sensing, 26(1):65–74, Jan. 1988.[5] J. Immerkær. Fast noise variance estimation. Computer Vi-sion and Image Understanding, 64(2):300–302, 1996.[6] A. A. Nielsen. Analysis of Regularly and IrregularlySampled Spatial, Multivariate, and Multi-temporal Data.PhD thesis, Department of Mathematical Modelling, Tech-nical University of Denmark, Lyngby, 1994. Internethttp://www.imm.dtu.dk/aa/phd/.[7] A. A. Nielsen, K. Conradsen, and J. J. Simpson. Multivari-ate alteration detection (MAD) and MAF post-processing inmultispectral, bi-temporal image data: New approaches tochange detection studies. Remote Sensing of Environment,64:1–19, 1998.[8] A. A. Nielsen and R. Larsen. Restoration of GERIS datausing the maximum noise fractions transform. In ERIM, ed-itor, Proceedings from the First International Airborne Re-mote Sensing Conference and Exhibition, Volume II, pages557–568, Strasbourg, France, 1994.[9] S. I. Olsen. Estimation of noise in images: An evaluation.Graphical Models and Image Processing, 55(4):319–323,1993.[10] P. Strobl, R. Richter, A. Mu¨ller, F. Lehmann, D. Oertel,S. Tischler, and A. Nielsen. DAIS system performance, firstresults from the 1995 evaluation campaigns. In ERIM, edi-tor, Proceedings from the Second International Airborne Re-mote Sensing Conference and Exhibition, Volume II, pages325–334, San Francisco, California, USA, June 1996.[11] P. Switzer and A. A. Green. Min/max autocorrelation fac-tors for multivariate spatial imagery. Technical Report 6,Stanford University, 1984.Table 1. Generated image, top: simple statis-tics, bottom: autocorrelations between E-W, N-S, SW-NE, SE-NW neighbours and theirmean value.Image Mean Stddev Min Maxtop-left –0.00 0.71 –1.00 1.00top-right –0.00 0.56 –1.32 1.31bot-left –0.00 1.00 –3.69 3.65bot-right –0.00 0.85 –3.58 3.70Image E-W N-S SW-NE SE-NW Meantop-left 0.37 0.37 0.14 0.14 0.25top-right 0.07 0.07 –0.06 –0.06 0.01bot-left 0.18 0.19 0.07 0.07 0.13bot-right –0.08 –0.07 –0.11 –0.11 –0.09Table 2. Generated image noise estimates,top: simple statistics, bottom: autocorre-lations between E-W, N-S, SW-NE, SE-NWneighbours and their mean value.Model Mean Stddev Min Max1 0.00 0.64 –1.76 1.762 –0.00 0.51 –1.05 1.053 –0.00 0.47 –1.51 1.404 –0.00 0.20 –0.97 1.055 0.00 0.38 –1.03 1.036 0.00 0.14 –0.50 0.49Model E-W N-S SW-NE SE-NW Mean1 0.12 0.12 0.21 –0.23 0.062 0.15 0.15 –0.08 –0.08 0.043 0.12 0.12 –0.10 –0.10 0.014 –0.19 –0.19 0.04 0.04 –0.085 0.29 0.29 –0.12 –0.12 0.096 0.00 0.00 –0.06 –0.06 –0.03Table 3. Generated image (with noise) noiseestimates, top: simple statistics, bottom: au-tocorrelations between E-W, N-S, SW-NE, SE-NW neighbours and their mean value.Model Mean Stddev Min Max1 0.00 1.08 –4.59 4.462 –0.00 0.84 –3.47 3.703 0.00 0.85 –3.83 4.114 0.00 0.51 –2.28 2.175 0.00 0.60 –2.53 2.226 0.00 0.33 –1.65 1.80Model E-W N-S SW-NE SE-NW Mean1 –0.17 –0.17 0.19 –0.08 –0.062 –0.05 –0.04 –0.15 –0.15 –0.103 –0.03 –0.03 –0.12 –0.12 –0.074 –0.59 –0.59 0.38 0.38 –0.115 0.11 0.12 –0.27 –0.28 –0.086 –0.11 –0.10 0.02 0.02 –0.04Figure 1. Generated image (top-left), corre-sponding 55 noise image (top-right), gen-erated image with noise added (bottom-left),corresponding 55 noise image (bottom-right), all with linear stretching between min-imum and maximum.Figure 2. Generated - row-wise, noise as estimated from the six models mentioned above, linearstretching between minimum and maximum.Figure 3. Generated with noise - row-wise, noise as estimated from the six models mentioned above,linear stretching between minimum and maximum.

An Extension to a Filter Implementation of Local Quadratic Surface for Image Noise Estimation

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An Extension to a Filter Implementation of Local Quadratic Surface for Image Noise Estimation

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