Abstract
Recursive filters are commonly used in scale space construction for their efficiency and simple implementation. However these filters have an initialisation problem which either produces unusable results near the image boundaries or requires costly approximate solutions such as extending the boundary manually.
In this paper, we describe a method for the recursive filtering of reflectively extended images for filters with symmetric denominator. We begin with an analysis of reflective extensions and their effect on non-recursive filtering operators. Based on the non-recursive case, we derive a formulation of recursive filtering on reflective domains as a linear but time-varying implicit operator. We then give an efficient method for decomposing and solving the linear implicit system. This decomposition needs to be performed only once for each dimension of the image. This yields a filtering which is both stable and consistent with the ideal infinite extension. The filter is efficient, requiring the same order of computation as the standard recursive filtering.
We give experimental evidence to verify these claims.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Deriche, R.: Fast algorithms for low-level vision. IEEE Tr. on Pattern Analysis and Machine Intelligence 12 (1990) 78–87
Martucci, S.A.: Symmetric convolution and the discrete sine and cosine transforms. IEEE Transactions on Signal Processing 42 (1994) 1038–1051
Deriche, R.: Recursively implementing the gaussian and its derivatives. Technical Report 1893, Programme 4-Robotique, Image et Vision, INRIA-Institut National en Informatique et en Automatique (1993)
da Cunha, A.M.: Espaços de escala e detecção de arestas. Master’s thesis, IMPA, Rio de Janeiro (2000) http://www.visgraf.impa.br/escala.html.
Smith, M.J.T., Eddins, S.L.: Analysis/synthesis techniques for subband image coding. IEEE Transactions on Acoustics, Speech, and Signal Processing 38 (1990) 1446–1456
Weickert, J., ter Haar Romeny, B.M., Viergever, M.A.: Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing 7 (1998) 398–410
Oppenheim, A.V., with John R. Buck, R.W. S.: Discrete-Time Signal Processing. second edn. Prentice-Hall (1999)
Golub, G.H., Loan, C.F.V.: Matrix computations. Third edn. Johns Hopkins University Press (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Appleton, B., Talbot, H. (2003). Efficient and Consistent Recursive Filtering of Images with Reflective Extension. In: Griffin, L.D., Lillholm, M. (eds) Scale Space Methods in Computer Vision. Scale-Space 2003. Lecture Notes in Computer Science, vol 2695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44935-3_49
Download citation
DOI: https://doi.org/10.1007/3-540-44935-3_49
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40368-5
Online ISBN: 978-3-540-44935-5
eBook Packages: Springer Book Archive