# Multiscale Image Fusion Using the Undecimated Wavelet Transform With Non-Orthogonal Filter Banks

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4 Fig. 3. Backprojection of a single wavelet coefficient at different scales and directions for the filter bank given in eq. (9). From top to bottom, the coefficient belongs to the horizontal, vertical and diagonal bands. From left to right, the scale increases from one to four. Each scale and direction has been normalized such that it occupies the full dynamic range. fused image is obtained by a simple co-addition of all detail images and the approximation image, a very fast reconstruction is possible. As for the choice of h we use a B-spline filter. Consequently, the overall filter bank may be defined such that [1,, 1] h[n] g[n] δ[n] h[n] h[n] g[n] [0, 1, 0], (8) where h is derived from the B 1 -spline function. In general B- spline functions have some remarkable properties which make them very good choices for wavelet analysis. For example, if we recall that the low-pass filter h in the z-transform domain can be factorized as H(z) (1+z 1 ) L P (z), it can be shown that the B-spline function of degree n L 1 is the shortest and most regular scaling function of order L, with P (z) 1 [11]. However, the approach presented above comes with one disadvantage. More specifically, due to the employed allpass filters, distortions introduced during the fusion process remain unfiltered in the reconstructed image. Hence, in order to improve the overall performance, we choose h and g as in eq. (8) but define the synthesis low-pass filter h as h. In compliance with the perfect reconstruction condition of eq. (6) this yields that g δ + h, resulting in the filter bank h[n] h[n] [1,, 1] g[n] δ[n] h[n]. (9) [1, 6, 1] g[n] δ[n] + h[n] In this scenario g consists entirely of positive coefficients, being thus no longer related to a wavelet function. On the other hand, such a lack of oscillations provides a reconstruction less prone to ringing artifacts. Additionally, unlike eq. (8), distortions introduced during the fusion stage are not transferred unprocessed to the reconstructed image. Fig. 3 shows the backprojection of a wavelet coefficient at different scales and directions for the filter bank given in eq. (9). Note that all images only exhibit positive values. In Section II we discussed how different filter lengths can influence the fusion result. More specifically, we argued that filters with long support sizes lead to an unwanted spreading of coefficient values around image singularities, complicating the feature selection process. Following this reasoning, we propose a final modification of the filter bank given in eq. (9). Note that the analysis filters h and g can be factorized such that h[n] g[n] [1,, 1] [1, 1] [1, 1] [1, 1] [ 1, 1]. (10) Hence, we can move the nd factor of h and g, respectively, to the synthesis filter pair without corrupting the perfect reconstruction condition. This yields the following filter bank: h[n] g[n] [1, 1] [1, 1] h[n] g[n] [1, 3, 3, 1] 8 [ 1, 5, 5, 1] 8. (11) In this setup the analysis filter bank (h, g) consists of the very short -tap Haar filter pair whereas the synthesis filters correspond to a filter pair with larger support size and higher regularity. In other words, by employing this filter bank, the coefficient spreading problem (related to the support of the analysis filters) as well as the introduction of blocking artifacts during reconstruction (related to the regularity of the synthesis filters) is successfully reduced. We will see in the next section that this filter bank is particularly well suited for the fusion of infrared-visible and medical images, which tend to exhibit a high degree of overlapping salient information. IV. RESULTS The performance of the proposed image fusion scheme was compared to the fusion results obtained by applying the Nonsubsampled Contourlet Transform (NSCT) and the Dual-Tree Complex Wavelet Transform (DTCWT). As for the NSCT and the DTCWT, we followed the recommendations published in [] regarding the filter choices and (in case of the NSCT) number of directions. In the case of the UWT-based image fusion scheme, we utilized the filter banks of Section III-A. Hence, in our experiments we used the non-orthogonal filter banks of eqs. (8), (9) and (11). In order to avoid referring to filter banks by their respective equation numbers, we will associate names to them. Henceforth, the filter banks presented in eqs. (8), (9) and (11) will be referred to as Spline 1, Spline and Spline 3 filter banks, respectively. Four decomposition levels were chosen for all transforms. We performed simulations for 10 infrared-visible image pairs using the fusion rules given in eqs. () and (5). As for the objective evaluation of the achieved results, we use in this work three of the most widely used fusion metrics, namely, the performance measures proposed by Xydeas and Petrović Q AB/F [1], Piella Q P [] as well as the Mutual Information (MI), first introduced by Qu et al. [13] in the context of image fusion. Tables I and II list the average results for all infrared-visible and medical image pairs, respectively. It can be noticed that the UWT in combination with the proposed new class of nonorthogonal filter banks significantly outperforms traditional fusion methods based on the NSCT and the DTCWT for all tested fusion metrics. More specifically, by looking at Tables I and II it can be noted that the best performance is achieved for the Spline 3 filter bank, followed by the Spline filter bank (note that, for example, the results for the Spline 3

5 TABLE I FUSION RESULTS FOR INFRARED-VISIBLE IMAGE PAIRS. Transform/Filter Bank Q AB/F MI Q P DTCWT NSCT Spline Spline Spline TABLE II FUSION RESULTS FOR MEDICAL IMAGE PAIRS. Transform/Filter Bank Q AB/F MI Q P DTCWT NSCT Spline Spline Spline Fig.. Magnified fusion results of a medical image pair. (Top-left) DTCWT fused. (Top-right) NSCT fused. (Bottom-left) UWT fused with Spline filter bank. (Bottom-right) UWT fused with Spline 3 filter bank. filter bank are in all cases superior to the ones of DTCWT and NSCT). This is most evident when looking at the magnified fusion results of a medical image pair, depicted in Fig.. It can be seen that the DTCWT- and NSCT-based fusion schemes suffer from a significant loss of edge information, particularly noticeable at the outermost borders of the fused images (top-left and top-right corner of Fig., respectively). There, information belonging to the skull bone (white stripe enclosed within the gray, tube-like structure) partially disappeared. This is due to the superposition of the skull bones, originating from the pair of source images, resulting in coefficient overlaps in the DTCWT and NSCT transform domain, which cannot be resolved by the fusion algorithm. As for the fusion results obtained by the UWT, this effect is reduced and the edge information is preserved to a much higher degree. Moreover, in case of the Spline 3 filter bank, the edges appear to be more accentuated than in the fusion scenario using the Spline filter bank, further indicating the perceptual superiority of the proposed fusion approach. Please note that no performance gains could be achieved for the Spline 1 filter bank. As mentioned in Section III-A, this is mainly due to the absence of filtering operations during reconstruction, since in this case distortions introduced during the fusion process remain unaltered in the fused image. The MATLAB implementation of the proposed UWT-based image fusion framework with non-orthogonal filter banks, together with the fusion results for all tested images, is available for download at fusion/sbrt 01. V. CONCLUSIONS In this paper a novel UWT-based image fusion framework is introduced. The proposed approach takes advantage of the increased filter design freedom of the UWT which allows for the development of a new class of non-orthogonal filter banks. The filters were especially designed for the purpose of image fusion and exhibit useful properties such as a short support size of the analysis filter pair whilst preserving the regularity of the synthesis filters. Thereby, the unwanted spreading of coefficient values around overlapping image singularities is successfully reduced and the introduction of artifacts in the fused reconstruction is avoided. Moreover, the nonsubsampled nature of the UWT permits the design of filter banks where both synthesis filters exhibit only positive coefficients. Such filters provide a reconstructed, fused image less vulnerable to ringing artifacts. The results obtained using objective metrics indicate that our solution leads to a fusion framework which provides clear advantages over traditional multiscale fusion approaches based on state-of-the-art transforms such as the DTCWT and the NSCT. Additionally, the perceptual superiority of the proposed framework was suggested by informal visual inspection of a fused medical image pair. REFERENCES [1] Z. Zhang and R. S. Blum, A categorization of multiscaledecomposition-based image fusion schemes with a performance study for a digital camera application, Proceedings of the IEEE, vol. 87, no. 8, pp , August [] G. Piella, Adaptive Wavelets and their Applications to Image Fusion and Compression, Ph.D Thesis, University of Amsterdam, Amsterdam, Netherlands, 003. [3] A. Ellmauthaler, E. A. B. da Silva, C. L. Pagliari, and S. R. Neves, Infrared-Visible Image Fusion Using the Undecimated Wavelet Transform with Spectral Factorization and Target Extraction, submitted for publication in Proceedings of the 01 IEEE International Conference on Image Processing. [] S. Li, B. Yang, and J. Hu, Performance comparison of different multiresolution transforms for image fusion, Information Fusion, vol. 1, no., pp. 7 8, 011. [5] A. Ellmauthaler, E. A. B. da Silva, and C. L. Pagliari, Multiscale Image Fusion Using the Undecimated Wavelet Transform With Spectral Factorization and Non-Orthogonal Filter Banks, submitted for publication in IEEE Transactions on Image Processing. [6] D. J. Field, Scale-invariance and self-similar wavelet transforms: an analysis of natural scenes and mammalian visual systems, in Wavelets, Fractals and Fourier Transforms: New Developments and New Applications, pp Oxford University Press, [7] A. L. da Cunha, J. Zhou, and M. N. Do, The nonsubsampled contourlet transform: Theory, design, and applications, IEEE Transactions on Image Processing, vol. 15, no. 10, pp , October 006. [8] I. W. Selesnick, R. G. Baraniuk, and N. C. Kingsbury, The dual-tree complex wavelet transform, IEEE Signal Processing Magazine, vol., no. 6, pp , November 005. [9] G. Pajares and J. M. de la Cruz, A wavelet-based image fusion tutorial, Pattern Recognition, vol. 37, no. 9, pp , 00. [10] M. J. Shensa, The discrete wavelet transform: Wedding the à trous and Mallat algorithms, IEEE Transactions on Signal Processing, vol. 0, no. 10, pp. 6 8, October 199. [11] M. Unser, Ten good reasons for using spline wavelets, in Proceedings of the 5th SPIE Conference on Wavelet Applications in Signal and Image Processing, 1997, pp. 31. [1] C. S. Xydeas and V. Petrovic, Objective image fusion performance measure, Electronics Letters, vol. 36, no., pp , February 000. [13] G. Qu, D. Zhang, and P. Yan, Information measure for performance of image fusion, Electronics Letters, vol. 38, no. 7, pp , March 00.

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