WO2009013696A2

WO2009013696A2 - Framework system and method for low-frequency preservation in multiresolution nonlinear adaptive filtering

Info

Publication number: WO2009013696A2
Application number: PCT/IB2008/052914
Authority: WO
Inventors: Gaspar Delso; Raoul Florent
Original assignee: Koninklijke Philips Electronics N. V.
Priority date: 2007-07-24
Filing date: 2008-07-21
Publication date: 2009-01-29
Also published as: WO2009013696A3

Abstract

The present invention describes a multiresolution framework system and method for low frequency preservation in multiresolution nonlinear adaptive filtering, wherein said method uses filtering techniques based on a Laplacian pyramid decomposition (LPD) procedure and a corresponding image reconstruction algorithm inverse to said decomposition procedure. According to a first exemplary embodiment of the adaptive filtering method as proposed by the present invention, low-frequency alterations are corrected after pyramid reconstruction has been performed. A simple way to accomplish this is to decompose the resultant output image (xr1) once again by using the same decomposition procedure which has been used to build the Laplacian pyramid for the first time and then replace the obtained coarse image representation (cN') on the lowest frequency subband (N) by that (cN) of the original image (x1). Thus, after having performed a second reconstruction process, a corrected output image (xr1) would be obtained. In a second exemplary embodiment of this adaptive filtering method as pro- posed by the present invention, it is tried to correct low-frequency alteration by decomposing the unprocessed ({dn}n ∈ {1, 2,..., N}) and processed detail images ({dn}n ∈ {1, 2,..., N}) on each subband (n), respectively, into high ({en}n ∈ 1, 2,..., N}) and low-frequency components ({en}n ∈ {1, 2,..., N}). If this is done with the same filter which is used to create the Laplacian pyramid. The accomplished effect is that of recovering most of the low-frequency content that leaked into each subband. For this purpose, the original Iow-frequency content of each subband (n) is set aside and, after filtering, used to replace undesired altered low frequencies.

Description

Framework System and Method for Low-Frequency Preservation in Multiresolution Nonlinear Adaptive Filtering

The present invention describes a multiresolution framework system and method for low- frequency preservation in multiresolution nonlinear adaptive filtering, wherein said method uses filtering techniques based on a Laplacian pyramid decomposition (LPD) procedure and a corresponding image reconstruction algorithm inverse to said decomposition procedure.

BACKGROUND OF THE INVENTION

An often undesired side effect of image processing, which especially occurs in the case when multiresolution image enhancement filters are applied for encoding and decod- ing digital images, respectively, is the alteration of original low- frequency content, which means the alteration of the local mean of an original image. This effect is unacceptable in certain applications, such as e.g. in medical image inspection and quantization, where the local mean intensity provides useful diagnostic information to a physician.

Multiresolution image representation approaches are widely used in several medical imaging systems, in particular for signal enhancement and noise reduction purposes, which is due to the fact that inspection of noisy images for diagnostic purposes is a common need in medical practice. The main idea behind these approaches is to partition the original image data into a set of several subbands, each containing a different part of the image frequency spectrum. These techniques are very appealing as they allow a physician to discrimi- nate the image structures at various scales and to adapt filtering consequently, yet remaining computationally efficient. This is true for several anisotropic filtering techniques, such as e.g. for those filtering techniques which are implemented in the Xres™ adaptive filter - an important feature in the Allura X-ray products developed by Philips.

The Laplacian pyramid decomposition (LPD) algorithm as proposed by P. J. Burt and E. H. Adelson in their article "The Laplacian Pyramid as a Compact Image Code" (IEEE Transactions on Communications, Vol. 31 (4), pp. 352-540), which has been published in 1983, is an early example of multiresolution image representation. Basic motivation thereby was to find an image representation which decorrelates the image pixels, such as to design an efficient compressed code. In the LP approach, an image to be visualized is represented as a hierarchy of different versions of an image that are provided in different grades of resolution. First, a coarse approximation of the original image is used to generate a prediction, which is then subtracted from the original image to obtain a detail image (also referred to as "difference image" or "subband representation"). The process is iterated on the coarse image for further levels of decomposition. Each detail image is reduced in both resolution and sample density, hence the pyramid structure. A down-sampled, coarse approximation of the original image is added at the top of the pyramid. Each pyramid level corresponds to a different subband of image frequencies. For effective compression, all the subbands are quantized and entropy-coded before transmission or storage. Then, a simple reconstruction filtering method is to regenerate the prediction based on the coarse image and add it back to the detail image.

The need for computational efficiency imposes serious constraints on the filters to be used for the construction of a Laplacian pyramid. The use of short finite impulse response antialiasing and reconstruction filters to respectively perform the down- and up- sampling operations to be executed by the Laplacian pyramid filtering structure leads to a suboptimal partitioning of the frequency spectrum between the individual pyramid subbands. This unavoidable imperfection in the pyramidal decomposition procedure is not a serious problem by itself, which is due to the fact that the pyramidal recomposition operation is by definition capable of compensating all frequency misplacements. However, this capability is limited to those cases where the subbands have not been altered, whereas subband alteration is after all the purpose of multiresolution filtering.

As already mentioned above, undesired alteration of the original image low- frequency content is a serious problem. It is caused by low- frequency content data which, due to the limitations of pyramid decomposition, leaks into high-frequency subbands. These low frequencies will be modified by the filtering, showing, once the image has been recomposed, as local alterations of the image mean value that can be misinterpreted by a physician.

SUMMARY OF THE INVENTION

In view of the problem explained above, it may be an object of the present invention to either prevent or correct undesired alterations of low- frequency image data which occur due to multiresolution image representation and adaptive filtering of coarse image subbands in a Laplacian pyramid decomposition and reconstruction procedure. To address this object, a first aspect of the present invention is dedicated to a method for low- frequency preservation in multiresolution nonlinear adaptive filtering, wherein said method uses filtering techniques which are based on a non-orthogonal image decomposition procedure and a corresponding image reconstruction procedure inverse to the non-orthogonal image decomposition procedure. According to the proposed method, low- frequency alterations which occur when reconstructing an original image from a set of coarse images representing different levels of resolution, said coarse images resulting from a non- orthogonal image decomposition procedure applied to said original image, are corrected by means of a baseband or subband low- frequency reinjection procedure. In this connection, the non-orthogonal image decomposition procedure may e.g. be implemented by a Laplacian pyramid decomposition procedure, and the corresponding image reconstruction procedure inverse to the non-orthogonal image decomposition procedure may e.g. be implemented by a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure mentioned above. According to a first exemplary embodiment of the method as proposed by the present invention, low-frequency alterations are corrected after pyramid reconstruction has been performed. This is done in that the output image of a filtering process applied for reconstructing an original image which has been submitted to a Laplacian pyramid decomposition procedure and a post-processing procedure is re-decomposed with a Laplacian pyramid de- composition procedure and, after having replaced the coarse image representation of reconstructed output image on the lowest frequency subband of the Laplacian pyramid when being re-decomposed by that of the original image, recomposed with a Laplacian pyramid recomposition procedure. It should be noted that this approach will only correct those errors occurring due to low frequencies that were initially misplaced in high-frequency bands and, due to the alterations introduced by the filtering, are placed in the baseband by performing this second decomposition procedure. Nevertheless, there are still those errors due to low frequencies misplaced in both the first and the second decomposition procedure. To correct these errors, a second approach according to a second exemplary embodiment of the method as described above, said second approach showing slightly better performance though at a higher memory cost, is proposed by the present invention.

According to the aforementioned second exemplary embodiment of claimed method, low-pass filtered versions of the subband representations obtained when decomposing the original image are respectively reinjected to a post-processed and high-pass-filtered version of said original image for each subband of the Laplacian pyramid before image representations of different resolutions consisting of reinjected low- frequency images additively combined with the corresponding post-processed and high-pass-fϊltered versions of said original image are submitted to a Laplacian pyramid recomposition procedure inverse to the ap- plied Laplacian pyramid decomposition procedure.

According to the present invention, said post-processing may e.g. be given by some kind of adaptive filtering for executing a combined multiscale enhancement procedure which may e.g. include a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure. The Laplacian pyramid decomposition procedure which is applied for re- decomposing reconstructed output image may be the same Laplacian pyramid decomposition procedure as the one which has been used for decomposing the original image. Likewise, the Laplacian pyramid recomposition procedure used for re-composing re-decomposed image representations of reconstructed output image may be the same Laplacian pyramid recomposi- tion procedure as the one that has been used for reconstructing decomposed image representations of the original image.

According to a second aspect, the present invention is directed to a multiresolu- tion framework system for low- frequency preservation in multiresolution nonlinear adaptive filtering, wherein said multiresolution framework system comprises a nonlinear adaptive fil- tering structure implementing a non-orthogonal image decomposition procedure as well as a corresponding image reconstruction procedure inverse to said non-orthogonal image decomposition procedure. Thereby, said filtering structure is adapted for performing a baseband or subband low-frequency reinjection procedure for correcting low- frequency alterations that occur when reconstructing an original image from a set of coarse images representing differ- ent levels of resolution, said coarse images resulting from a non-orthogonal image decomposition procedure applied to said original image. In this connection, the non-orthogonal image decomposition procedure implemented by said filtering structure may e.g. be given by a Laplacian pyramid decomposition procedure, and the corresponding image reconstruction procedure inverse to the non-orthogonal image decomposition procedure implemented by said fil- tering structure may e.g. be given by a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure mentioned above.

According to a first exemplary embodiment, the claimed multiresolution framework system may comprise a first filtering block, a second filtering block and a connec- tion line. Thereby, said first filtering block may be equipped with a first Laplacian pyramid decomposition module followed by a processing module and a first Laplacian pyramid re- composition module. Said second filtering block may comprise a second Laplacian pyramid decomposition module followed by a second Laplacian pyramid recomposition module. The connection line thereby connects the input port of an interpolation filter on the baseband level of said second filtering block's recomposition module with the output port of a decimation filter on the baseband level of said first filtering block's decomposition module. As foreseen by the present invention, said first filtering block may be adapted for reconstructing an original image which has been submitted to a Laplacian pyramid decomposition procedure and a post-processing procedure, and said second filtering block may be adapted for re- decomposing the resultant reconstructed image with a Laplacian pyramid decomposition procedure and recomposing the re-decomposed image representations with a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure performed by said second Laplacian pyramid decomposition module. Aforementioned recompo- sition module of said second filtering block thereby differs from the recomposition module contained in the Laplacian pyramid filtering structure as proposed by Burt and Adelson in that the recomposition module on the baseband level of the Laplacian pyramid filtering structure implemented by said second filtering block is fed with the coarse image representation on the baseband level of the Laplacian pyramid filtering structure implemented by said first filtering block, said coarse image representation being supplied via said connection line.

According to a second exemplary embodiment, the claimed multiresolution framework system may comprise a filtering block which is equipped with a Laplacian pyramid decomposition module followed by a processing module and a Laplacian pyramid recomposition module, wherein said processing module is adapted for performing a method whereupon a low-pass filtered version of the subband representations obtained when decomposing the original image as done by said Laplacian pyramid decomposition module is reinjected to post-processed and high-pass-filtered versions of the original image for each sub- band of the Laplacian pyramid. In addition thereto, said Laplacian pyramid recomposition module is adapted for performing a Laplacian pyramid recomposition procedure inverse to the applied Laplacian pyramid decomposition procedure on a set of image representations of different resolutions consisting of reinjected low- frequency images additively combined with the corresponding post-processed and high-pass-filtered versions of said original image. Said processing module may be adapted for performing a post-processing procedure that may e.g. be given by some kind of adaptive filtering for executing a combined multiscale enhancement procedure that may e.g. include a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure. Finally, a still further aspect of the invention is dedicated to a computer software product being programmed for performing a method as described above.

BRIEF DESCRIPTION OF THE DRAWINGS

Advantageous features, aspects, and advantages of the invention will become evident from the following description, the appended claims and the accompanying drawings.

Thereby,

Fig. 1 shows a schematic block diagram of the decomposition module, the processing module and the recomposition module of a conventional TV-stage Laplacian pyramid filtering structure,

Fig. 2 shows a schematic block diagram of a filtering structure according to a first exemplary embodiment of the present invention which is used for correcting undesired low- frequency alteration caused by multiresolution image representation by means of baseband reinjection,

Fig. 3 shows a schematic block diagram of a filtering structure according to a second exemplary embodiment of the present invention that is used for correcting undesired low- frequency alteration caused by multiresolution image representation by means of subband low- frequency reinjection,

Fig. 4 shows a flow chart that illustrates the filtering procedure implemented by the filtering structure as described with reference to Fig. 2, and

Fig. 5 shows a flow chart that illustrates the filtering procedure implemented by the filtering structure as described with reference to Fig. 3. DETAILED DESCRIPTION OF THE PRESENT INVENTION

In the following sections, exemplary embodiments of the claimed multiresolu- tion framework system as well as exemplary embodiments of the claimed method according to the present invention will be explained in more detail with respect to exemplary refinements and referring to the accompanying drawings.

In Fig. 1, a block diagram of a multiresolution filter 100 based on a Laplacian pyramid filtering structure as proposed by Burt and Adelson is shown which can be used for spectral decomposition and reconstruction of an input image signal. As mentioned above, the idea is to first derive a coarse approximation of the original image by down-sampling and low-pass filtering the image data, in the literature referred to as "decimation step". The obtained coarse approximation is then up-sampled and interpolated, also known as "interpolation step", which yields a prediction of the original image. The prediction image thereby represents redundant information that is to be subtracted from the original image. The ob- tained prediction contains highly decorrelated pixels and has low variance. The coarse approximation and the prediction can then be encoded at a lower bit rate than the original image. By iterating this process on the coarse image, another resolution level and finally a pyramid of detail images representing different grades of resolution is generated. To achieve data compression, all subbands are quantized and entropy-coded, whereby quantization is modeled as an independent additive noise process. As shown in Fig. 1, the decimation and interpolation step are implemented by a decomposition module 101. For image reconstruction, which is performed in a subsequent recomposition module 103 post-connected to a processing module 102 which may e.g. perform a combined multiscale enhancement procedure, the latter including e.g. a contrast equalization, edge enhancement, latitude reduction and/or any other filter- ing procedure to be applied to each subband n of the decomposed image signal, a given coarse image is up-sampled and interpolated, and the resultant prediction is added back to the detail image. For an M-dimensional case where a digital image signal (also referred to as "original image" or "input image") is given by a discrete sequence {x_m \ m e {1, 2, 3, ..., M) } of real- valued intensity values for each pixel of said image, said intensity values constituting the components of an M-dimensional vector X_n := [

x_n\ , x_n2, ..., e ^M (with n e {1, 2, ..., N]), the analysis equations for describing the filtering procedure performed by the decomposition module 101 on the left side of said filtering structure for each subband n of the Laplacian pyramid are given by H y. ^'■= X_n VnG {1,2, ...,N], (1) d. I -G- H

= A and the corresponding synthesis equations for describing the functionality of the recomposition module 103 performing the above-mentioned image reconstruction procedure, said recomposition module 103 being subsequently arranged to the processing module 102 following said decomposition module 101, are given by

*_N = [ ^L=<±, = F] ^{J ■} t y—N = = G- c_N + F- d_N and (2a) v

= S

χ_n = [G, F]- = G-x^r _n+l+F-d_n VnG {1,2,...,N-I], (2b) d_r where X_n denotes the above-defined input image vector for the n-th subband of the Laplacian pyramid, y (with y G ^2M) denotes a decomposed version of the input image vector of this subband and X_n (with X_n G ^M) is the reconstructed input image vector of this subband (also referred to as "output vector"), c__n and d__n (with c_n, d__n G ^M) respectively denote the coarse signal and the prediction error representing the detail image on the n-th level of the Laplacian pyramid, F_ (with F_ G ^MxM) is a processing matrix (or, in a nonlinear case, a vector-valued processing function of argument vectors [(I_n] _{n e} {i, 2, ..., _N\) for describing the aforementioned combined multiscale enhancement, H_ and G_ (with G_, H_ G ^MxM) denote the decimation and interpolation matrices, respectively,

/ :=diag (1,1, ..., 1) = (δj ye {1,2, ,M] -with

M matrix elements

1, for i = j δ,, := with 1,7 e {1,2, ...,M] (2c)

0, for i ≠ j being the Kronecker delta - is the MxM-dimensional identity matrix, A (with A 2MYM )- represents the so-called analysis matrix and S (with S MxIM-, is the corresponding synthesis matrix. In a more practical setting where the subbands are quantized, the analysis equations are given by y. V « € {1, 2, ..., N-I), (3)

and the corresponding synthesis equations are given by x_N = [G, F ] ^■ y = G ^■ c_N + F ^■ d_N and (4a)

= cl ,, '

£ = [G, F ] ^. = G - x^r _n+l + F - d_n V Π G {1, 2, ..., N-I), (4b) d_r where η (with η e ^M) denotes a quantization noise vector which is to be added to decomposed input vector y . This leads to the following reconstruction error:

Δi = [G, F ] - Ay_^ = G η ^c + F V (5a)

= s

with Av := γ — v = (5b)

Α_^d v —

= η

There are such applications, as in the case of medical diagnostics of thoracic images, where noise reduction and feature enhancement algorithms (both concerning the medium to high-frequency spectrum) would be desirable, whereas an alteration of the image data's low-frequency content could lead to erroneous medical decisions. In these cases, multire- solution filtering using adaptive filtering techniques in the high-frequency subbands, said filtering techniques leaving the baseband untouched, has been successfully applied. In the fol- lowing, however, the influence of said quantization noise on the reconstructed image shall be ignored for the sake of keeping the equations simple.

In Fig. 2, a schematic block diagram of a filtering structure 200 according to a first exemplary embodiment of the present invention is shown which is used for correcting undesired low- frequency alteration due to multiresolution image representation by means of baseband reinjection. Aside from a processing module 202, which may e.g. describe a combined multiscale enhancement, e.g. including a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure to be applied to each subband n of the decomposed image signal, the depicted filtering structure 200 consists of a first filtering block 200' and a second filtering block 200", each comprising two subsequently arranged nonlinear filtering blocks 201 and 203 or 204 and 205, respectively, for implementing two separate Lap- lacian pyramid decomposition and recomposition (reconstruction) procedures.

For the sake of avoiding unnecessary redundancies in the description of this application, said block diagram shall be described in conjunction with a flow chart as shown in Fig. 4, which illustrates the filtering procedure that is implemented by the filtering structure as described with reference to Fig. 2. In the following, reference signs in brackets refer to the procedure steps of said flow chart. A digital image signal, given in the form of the M- dimensional vector ΛJ_V as defined above, which is inputted (SO) to said filtering structure is first decomposed (Sl) by means of a Laplacian pyramid decomposition procedure as ex- pressed by equation (1), thus yielding a set of coarse images given by vectors

{ c_n I c_B := H_ - X_n V n e {1, 2, ..., N) } and a set of detail images, said detail images being represented by prediction error vectors { d__n \ d__n := (/ - G_ ^■ H_) ^■ x_n V « e {l, 2, ..., N) ). After that, processing matrix F (or, in a nonlinear case, a vector-valued processing function of argument vectors Id_n) „ _e {i, 2, ..., _N}, respectively) is applied (S2) to the resultant detail image of each coarse signal's subband n. Hence, a set of processed detail images, the latter being denoted by processed prediction error vectors { d__n \ d__n := F_ - d__n V « e {l, 2, ..., N) ) is obtained, which are then recomposed (S3) by a Laplacian pyramid recomposition procedure as described by synthesis equations (2a) and (2b), thus yielding a set of reconstructed output images, denoted by vectors {χ^r _n ) _{n ≡} {1, 2, ..., N}, which are characterized by an undesired low- frequency alteration.

So far, proposed filtering structure as depicted in Fig. 2 corresponds to the Laplacian pyramid-based filtering structure of the multiresolution filter (in the following referred to as "first filtering block") which is described with reference to Fig. 1. As shown in Fig. 2, a second filtering block 200", consisting of a Laplacian pyramid decomposition module 204 followed by a corresponding Laplacian pyramid recomposition module 205 is post-connected to the aforementioned first filtering block 200'. Thus, reconstructed output vector x[ , which corresponds to input vector X₁ ' of said second filtering block, is decomposed (S4) into a Laplacian pyramid structure as shown in the corresponding flow chart depicted in Fig. 4, thereby using the same decomposition procedure as described with reference to Fig. 1. Given M-dimensional vector X_n' := x_n2 , ..., x_HM']^r (with x_n ' e ^M, n e

{1, 2, ..., N)) as an input vector for subband n of the Laplacian pyramid structure forming said second filtering block, the analysis equations for describing the filtering procedure which is performed by the decomposition module of said second filtering block for each subband n of the Laplacian pyramid are given by

C — n H y ^■= X_n ' V Π G {1, 2, ..., N] . (6) d " / - G H

= A After having replaced (S5) the coarse image represented by vector c_N ' on pyramid level N of the lowest frequency subband (baseband) in the Laplacian pyramid resultant from the decomposition procedure realized by the decomposition module 204 of said second filtering block 200" by the coarse image represented by vector c_N on the same pyramid level in the Laplacian pyramid resultant from the decomposition procedure as realized by the de- composition module 201 of said first filtering block 200', re-decomposed images represented by vectors { d__n " } „ _e {i, 2, ..., _N\ are reconstructed (S6) ("recomposed") in a recomposition module 205 of the second filtering block 200". Thereby, the same reconstruction algorithm as applied for performing the reconstruction algorithm performed in the recomposition module 203 of the first filtering block 200' preconnected to said second filtering block 200" is used. The corresponding synthesis equations for describing the functionality of the recomposition module 205 subsequently arranged to the decomposition module 204 of said second filtering block 200", said recomposition module 205 performing the above-mentioned second image reconstruction procedure, are given by

*_N = [ ^L =G,' l = \ ^J -y_iV ' _, = G - c_N + d_N " and (7a) v

= 5"

^ = W, _ ! ] ^■ = G - x^r _tt+; + d_tt " V « € {1, 2, ..., N-I], (7b)

where y ' (with y ' e 2M )-, denotes a decomposed version of the input image vector x_N ' for subband N, x[ ' (with x[ ' e ^M) is the reconstructed input image vector of subband n (also referred to as "output vector" of said second filtering block on pyramid level n), and c__n ' and d__n " (with c__n ' , d__n " e ^M) respectively denote the coarse signal and the prediction error representing the detail image on the n-th level of the Laplacian pyramid.

From the block diagram depicted in Fig. 2 it is apparent that the aforementioned replacement of prediction vector c_N' by the prediction vector c_N on the pyramid level of the lowest fre- quency subband N of the preceding first filtering block is implemented by a connection line LI that connects the input port of the interpolation filter described by interpolation matrix G_ on the pyramid level N of said second filtering block's recomposition module 205 with the output port of the decimation filter described by decimation matrix H on pyramid level N of said first filtering block's decomposition module 201.

In Fig. 3, a schematic block diagram of a filtering structure 300 according to a second exemplary embodiment of the present invention which is used for correcting undesired low- frequency alteration caused by multiresolution image representation by means of subband low- frequency reinjection is shown. Aside from an image decomposition module 301 (not shown) for implementing a non-orthogonal image decomposition procedure, which may e.g. perform a Laplacian pyramid decomposition procedure such as implemented by the Laplacian pyramid decomposition modules 101 and 201 in the filtering structures as described with reference to Figs. 1 and 2, respectively, filtering structure 300 as depicted in Fig. 3 further comprises an extended processing module 302 and a recomposition module 303 for implementing a non-orthogonal image recomposition procedure inverse to said image decomposition procedure so as to reconstruct an original image inputted to the non-orthogonal image decomposition module 301 of said filtering structure.

Analogously, said block diagram shall be described in conjunction with a flow chart as shown in Fig. 5, which illustrates the filtering procedure that is implemented by the filtering structure as described with reference to Fig. 3. Although this flow chart is exempla- rily described with regard to a Laplacian pyramid decomposition and corresponding Laplacian pyramid recomposition procedure, it should be noted that the invention is not restricted thereto but may generally be directed to non-orthogonal image decomposition and recomposition. In the following, reference signs in brackets refer to the procedure steps of said flow chart. A digital image signal, given in the form of the M-dimensional vector xw as defined above, which is inputted (SO') to filtering structure 300 is first decomposed (Sl ') by means of a Laplacian pyramid decomposition procedure (cf. equation (I)) yielding a set of coarse images given by vectors {c_n \ c_n := H_ ^■ X_n V » e {l, 2, ..., N) } and a set of detail images, said detail images being represented by prediction error vectors { d__n I d__n := (/ - G_ ^■ H_) ^■ x_n V « e {l, 2, ..., N) ). The obtained detail images for each coarse signal's subband n are then submitted to a processing which may e.g. be some kind of adaptive filtering for executing a combined multiscale enhancement procedure including a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure (S2'). This leads to a set of processed detail images, given by processed prediction error vectors { d__n' | d__n' := F_ ^■ d__n V BE {1,2, ..., N))An addition, a set of low- frequency components {e_n I e_n := d_n,LP V n e {1, 2, ..., N) ) (8a) with d_ntLP = [l-σ(v -v_co)]-d_n yn (8b) is extracted from the unprocessed detail images by means of a filtering operation and stored for later use, and a set of complementary high-frequency components

{_e;\ _ e; := d;-d_n,LP' = d_n,HP ^> V n e {1, 2, ..., N) ) with (9a) </„_,„,' = σ(v -v_∞ )^■</„' Vn and </„_,„' = [l - σ (v -v_∞ )] ^• d; Vn

(9b, c) is extracted from the processed detail images by means of a further filtering operation (S3'). Thereby,

0 for v < 0 σ:v H σ(v ) := --[sgn(y) + l] = 0.5 for v = 0 (10)

1 for v > 0 denotes the Heaviside function and V _co is the cut-off frequency of said low- pass filter in the domain of spatial frequency V . This separation can be accomplished by a number of different methods. Preferably, the same low-pass filter which is used to create the Laplacian pyramid in the decomposition procedure is applied here. As shown above in equation (9a), high-pass filtered version {e^)_{n e} {i, 2, ..., _N} of processed prediction error vectors {dj_l ^y)_{n e} {i, 2, ...,N) is calculated by subtracting (S4') low-pass-fϊltered versions of said processed prediction error vectors from the processed prediction error vectors for each sub- band n. This can also be achieved by filtering processed prediction error vectors {d^) „ _e {i, 2, ...,w} with a high-pass finite impulse response kernel σ (v - V_∞ ) which is equivalent to said subtraction as illustrated by equation (9b) in conjunction with equation (9a). After that, the aforementioned original low-frequency components {§„) „ _e {i, 2, ..., _N}, which have been kept for further processing, are reinjected (S5') into the respective subband n, thus yielding vectors {/ I / := e_B+e_B' = [l-σ(v -v_∞ )]^•</„ + σ(v -v_∞ )^•</„' = [/ - σ (v -vj • (/ - £ )] • </„ V « . (1, 2, ..., JV) ). (11)

= i_ai

Vectors {/„} „ _e {i, 2, ..., w} are then recomposed (S6') by a Laplacian pyramid recomposition procedure. By using equation (11) and analysis equation (1), the synthesis equations for describing the functionality of recomposition module 303, said recomposition module performing the above-mentioned second image reconstruction procedure, can be written as

— x_N ™ = \ L =G,' = Iext ] J - --y⁷_7WV = ==G - c —_N ™ + f • ->_7 JVj and ( V12a) '

= S"

*_{B +}1 i ⁼ [≤' d - = G - x_n+l + f V /i e (1, 2, ..., JV-I), (12b)

thus yielding a set of reconstructed output images, represented by vectors {x[ } n _≡ {1, 2, ...,N)- Thereby, y (with j e ^2M) denotes a decomposed version of the input image vector x_w for subband TV, x[ (with x^ e ^M) is the reconstructed input image vector of sub- band n (also referred to as "output vector" of said filtering structure on pyramid level ή), and c_κ and d__n (with c_n , d__n e ^M) respectively denote the coarse signal and the prediction error representing the detail image on the n-th level of the Laplacian pyramid. It should be noted that the two embodiments as given by the block diagrams of the filtering structures depicted in block diagrams of Fig. 2 and 3 are not mutually exclusive, as steps S4, S5 and S6 from the method implemented by the filtering structure of Fig. 2, which is described with reference to Fig. 4, can be used to refine the results obtained with the method implemented by the filtering structure of Fig. 3, which is described with reference to Fig. 5.

APPLICATIONS OF THE INVENTION

Typical examples of applications where the preservation of mean intensity is critical would be the whole range of bone mineral density (BMD) measurement techniques, such as e.g. dual X-ray absorptiometry (DXA) and quantitative computed tomography (QCT). Classical filtering techniques would therefore be inadequate, because mean intensity alterations they induce could bias the measures. The proposed algorithms are intended to be a simple and efficient solution to this undesirable effect, being both easily introduced in the existing filter implementations and easily adapted to most common adaptive filter approaches.

An immediate application of this approach would thus be the improvement of denoising or noise enhancement filters which are currently used on X-ray and 3D rotational angiography scans where a local alteration of the image mean value can be perceived as pathological. Furthermore, many other medical protocols in the scope of which denoising filters are applied involve (or would benefit from) multiresolution adaptive filtering.

While the present invention has been illustrated and described in detail in the drawings and in the foregoing description, such illustration and description are to be considered illustrative or exemplary and not restrictive, which means that the invention is not limited to the disclosed embodiments. Other variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure and the appended claims. In the claims, the word "com- prising" does not exclude other elements or steps, and the indefinite article "a" or "an" does not exclude a plurality. A single processor or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures can not be used to advantage. A computer program may be stored/distributed on a suitable medium, such as e.g. an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as e.g. via the Internet or other wired or wireless telecommunication systems. Any reference signs in the claims should not be construed as limiting the scope of the invention.

Claims

CLAIMS:

1. A method for low- frequency preservation in multiresolution nonlinear adaptive filtering, said method using nonlinear filtering techniques based on a non-orthogonal image decomposition procedure and a corresponding image reconstruction procedure inverse to said non-orthogonal image decomposition procedure, wherein low-frequency alterations which occur when reconstructing (S3) an original image (X₁) from a set of coarse images ({c_n} _{n ≡} {1, 2, ..., _N}) representing different levels of resolution, said coarse images resulting from a non-orthogonal image decomposition procedure (Sl) applied to said original image (X₁), are corrected by means of a baseband or subband low- frequency reinjec- tion procedure (S5 or S5', respectively).

2. A method according to claim 1, wherein the non-orthogonal image decomposition procedure (Sl) is implemented by a Laplacian pyramid decomposition procedure and the corresponding image reconstruction procedure (S3) inverse to the non-orthogonal image decomposition procedure (Sl) is implemented by a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure.

3. A method according to claim 2, wherein the output image (x[ ) of a filtering process used for reconstructing (S3) an original image (X₁) which has been submitted to a Laplacian pyramid decomposition procedure (Sl) and a post- processing procedure (S2) is re-decomposed (S4) with a Laplacian pyramid decomposition procedure and, after having replaced (S5) the obtained coarse image representation (C_N ) of reconstructed output image (x[ ) on the lowest frequency subband (N) of the Laplacian pyramid when being re-decomposed by that (c#) of the original image (X₁), recomposed (S6) with a Laplacian pyramid recomposition procedure.

4. A method according to claim 2, said method comprising the step of respectively reinjecting (S5') low-pass filtered versions of the subband representations ({d_n} n e {l, 2, ..., N}) obtained when decomposing the original image (X₁) to a post-processed and high-pass-filtered version of said original image for each subband (n) of the Laplacian pyramid before submitting image representations ({/„} „ _e {i, 2, ..., _N}) of different resolutions consisting of reinjected low-frequency images ({e_n} „ _e {i, 2, ..., _N}) additively combined with the corresponding post-processed and high-pass-filtered versions ({e_n'} n _≡ {1, 2, ..., N}) of said original image (X₁) to a Laplacian pyramid recomposition procedure (S6') inverse to the applied Lap- lacian pyramid decomposition procedure (Sl ').

5. A method according to claim 4, wherein said post-processing (S2, S2') is given by some kind of adaptive filtering for executing a combined multiscale enhancement procedure which may e.g. include a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure.

6. A method according to anyone of claims 3 and 5, wherein the Laplacian pyramid decomposition procedure (S4) used for re-decomposing re- constructed output image (x[ ) is the same Laplacian pyramid decomposition procedure (Sl) as the one which has been used for decomposing the original image (x[ ).

7. A method according to anyone of claims 4 or 5, wherein the Laplacian pyramid recomposition procedure (S6) used for re-composing re- decomposed image representations ({y_H'} « e {1, 2, ..., N}) of reconstructed output image (x[ ) is the same Laplacian pyramid recomposition procedure (S3) as the one which has been used for reconstructing decomposed image representations ({y_n} _{n ≡} {1, 2, ..., N}) of the original image

(X₁).

8. A multiresolution framework system for low-frequency preservation in multi- resolution nonlinear adaptive filtering, said multiresolution framework system comprising a nonlinear adaptive filtering structure (200 or 300, respectively) implementing a non- orthogonal image decomposition procedure and a corresponding image reconstruction procedure inverse to said non-orthogonal image decomposition procedure, wherein said filtering structure (200 or 300, respectively) is adapted for performing a baseband or sub- band low-frequency reinjection procedure for correcting low-frequency alterations which occur when reconstructing an original image (X₁) from a set of coarse images ({c_n} „ _e {i, 2, ..., _N\) representing different levels of resolution, said coarse images resulting from a non-orthogonal image decomposition procedure applied to said original image (X₁).

9. A multiresolution framework system according to claim 8, wherein the non-orthogonal image decomposition procedure (Sl) implemented by the nonlinear adaptive filtering structure (200 or 300, respectively) is given by a Laplacian pyramid decomposition procedure and the corresponding image reconstruction procedure (S3) inverse to the non-orthogonal image decomposition procedure (Sl) implemented by the nonlinear adaptive filtering structure (200 or 300, respectively) is given by a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure.

10. A multiresolution framework system according to claim 9, comprising

- a first filtering block (200') equipped with a first Laplacian pyramid decomposition module (201) followed by a processing module (202) and first Laplacian pyramid recomposition module (203),

- a second filtering block (200") comprising a second Laplacian pyramid de- composition module (201) followed by a second Laplacian pyramid recomposition module (203) and

- a connection line (LI) which connects the input port of an interpolation filter on the baseband level of said second filtering block's recomposition module (205) with the output port of a decimation filter on the baseband level of said first filtering block's de- composition module (201), wherein said first filtering block (200') is adapted for reconstructing an original image (X₁) which has been submitted to a Laplacian pyramid decomposition procedure (Sl) and a post-processing procedure (S2), and said second filtering block (200") is adapted for re-decomposing the resultant reconstructed image with a Laplacian pyramid decomposition procedure and recomposing (S6) the re- decomposed image representations ({c_n} „ _e {i, 2, ..., _N\) with a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure performed by said second Laplacian pyramid decomposition module (201), wherein the recomposition module (205) of said second filtering block (200") differs from the recomposition module contained in the Laplacian pyramid filtering structure as proposed by Burt and Adelson in that the recomposition module (205) on the baseband level of the Laplacian pyramid filtering structure implemented by said second filtering block (200") is fed with the coarse image representation (cjv) on the baseband level of the Laplacian pyramid filtering structure implemented by said first filtering block (200'), said coarse image representation (c#) being supplied via said con- nection line (LI).

11. A multiresolution framework system according to claim 9, comprising a filtering block (300) which is equipped with a Laplacian pyramid decomposition module (301) followed by a processing module (302) and a Laplacian pyramid recomposition module (303), wherein said processing module (302) is adapted for performing a method whereupon low-pass filtered versions of the subband representations ({d_n} „ _e {i, 2, ..., _N\) obtained when decomposing the original image (X₁) as done by said Laplacian pyramid decomposition module (301) are respectively reinjected (S5') to a post-processed and high-pass-filtered version of the original image for each subband (n) of the Laplacian pyramid, and said Laplacian pyramid recomposition module (303) is adapted for performing a Laplacian pyramid recomposition procedure (S6') inverse to the applied Laplacian pyramid decomposition procedure (Sl ') on a set of image representations ({f_n} „ _e {i, 2, ..., _N}) of different resolu- tions consisting of reinjected low-frequency images ({e_n} „ _e {i, 2, ..., _N}) additively combined with the corresponding post-processed and high-pass-filtered versions ({e_n'} n _≡ {1, 2, ..., N}) of said original image (X₁).

12. A multiresolution framework system according to claim 11, wherein said processing module (302) is adapted for performing a post-processing procedure which is given by some kind of adaptive filtering for executing a combined multiscale enhancement procedure which may e.g. include a contrast equalization, edge enhancement, lati- tude reduction and/or any other filtering procedure.

13. A computer software product being programmed for performing a method for low- frequency preservation in multiresolution nonlinear adaptive filtering, said method using nonlinear filtering techniques based on a non-orthogonal image decomposition procedure and a corresponding image reconstruction procedure inverse to said non-orthogonal image decomposition procedure, wherein low-frequency alterations which occur when reconstructing (S3) an original image (xi) from a set of coarse images ({c_n} _{n ≡} {1, 2, ..., _N}) representing different levels of resolution, said coarse images resulting from a non-orthogonal image decomposition procedure (Sl) applied to said original image (xi), are corrected by means of a baseband or subband low- frequency reinjec- tion procedure (S5 or S5', respectively).

14. A computer software product according to claim 13, wherein the non-orthogonal image decomposition procedure (Sl) is implemented by a Laplacian pyramid decomposition procedure and the corresponding image reconstruction procedure (S3) inverse to the non-orthogonal image decomposition procedure (Sl) is implemented by a Laplacian pyramid recomposition procedure inverse to the Laplacian pyramid decomposition procedure.

15. A computer software product according to claim 14, wherein the output image (x[ ) of a filtering process used for reconstructing (S3) an original image (X₁) which has been submitted to a Laplacian pyramid decomposition procedure (Sl) and a postprocessing procedure (S2) is re-decomposed (S4) with a Laplacian pyramid decomposition procedure and, after having replaced (S5) the obtained coarse image representation (c#') of reconstructed output image (x[ ) on the lowest frequency subband (N) of the Laplacian pyra- mid when being re-decomposed by that (c#) of the original image (X₁), recomposed (S6) with a Laplacian pyramid recomposition procedure.

16. A computer software product according to claim 14, said method comprising the step of respectively reinjecting (S5') low-pass filtered versions of the subband representations ({dn} n e {l, 2, ...,N)) obtained when decomposing the original image (X₁) to a post-processed and high-pass-fϊltered version of said original image for each subband (n) of the Laplacian pyramid before submitting image representations ({/„} „ _e {i, 2, ..., N)) of different resolutions consist- ing of reinjected low-frequency images ({e_n} „ _e {i, 2, ..., N)) additively combined with the corresponding post-processed and high-pass-filtered versions ({e_n'} _{n e} {1, 2, ...,N)) of said original image (X₁) to a Laplacian pyramid recomposition procedure (S6') inverse to the applied Laplacian pyramid decomposition procedure (Sl ').

17. A computer software product according to claim 15, wherein said post-processing (S2, S2') is given by some kind of adaptive filtering for executing a combined multiscale enhancement procedure which may e.g. include a contrast equalization, edge enhancement, latitude reduction and/or any other filtering procedure.

18. A computer software product according to anyone of claims 15 and 17, wherein the Laplacian pyramid decomposition procedure (S4) used for re-decomposing reconstructed output image (x[ ) is the same Laplacian pyramid decomposition procedure (Sl) as the one which has been used for decomposing the original image (x[ ).

19. A computer software product according to anyone of claims 16 or 17, wherein the Laplacian pyramid recomposition procedure (S6) used for re-composing re- decomposed image representations ({jV} « e {1, 2, ...,N)) of reconstructed output image (x[ ) is the same Laplacian pyramid recomposition procedure (S3) as the one which has been used for reconstructing decomposed image representations ({y_n} n _≡ {1, 2, ..., N)) of the original image

(X₁).