Ulllted States Patent [19]
`Bernard et al.
`
`US005911012A
`[11] Patent Number:
`[45] Date of Patent:
`
`5,911,012
`Jun. 8, 1999
`
`[54] METHOD FOR THE TEMPORAL FILTERING
`()1? THE NOISE IN AN IMAGE ()F A
`SEQUENCE OF DIGITAL IMAGES, AND
`DEVICE FOR CARRYING OUT THE
`METHOD
`
`[75] Inventors: Franck Bernard, Paris; Raoul Florent,
`Valenton, both of France
`
`[73] Assignee: U.S. Philips Corporation, NeW York,
`NY'
`
`.... .. 382/260
`5,689,591 11/1997 Balram et al.
`5,715,335
`2/1998 De Haan et al. ..................... .. 382/265
`OTHER PUBLICATIONS
`
`“A NeW Approach to Linear Filtering and Prediction Prob
`lems”, by RE. Kalman, Transactions of the ASME, Journal
`of Basic Engineering, Series 82D, pp, 35-45, 1960.
`
`Primary Examiner—Thomas D. Lee
`Attorney, Agent, or Firm—DWight H- Renfrew, Jr
`[57]
`ABSTRACT
`
`[21] Appl. No.: 08/674,061
`[22] F1 d
`J l 1 1996
`1 e I
`u ‘
`’
`Foreign Application Priority Data
`
`[30]
`
`France ................................. .. 95 07889
`Jun. 30, 1995 [FR]
`[51]
`Int. Cl? ..................................................... .. G06K 9/40
`[52] US. Cl. ........................ .. 382/260; 382/265; 382/275;
`382/132
`
`
`[58] Field of searggzb'ég"56636535; 257435582’ 113526’ ’ ’ ’ ’ ' ’
`
`
`[56]
`References Cited
`
`
`
`
`
`
`
`A method for the temporal ?ltering of a sequence of digi
`tiZed noisy images includes the evaluation of an anti-causal
`?ltered sample (P?) in order to reconstruct a present noisy
`sample (I,P)'of a given pixel in a present image by an
`anti-causal linear combination of a causal ?ltered sample
`(PIC) Obtained by preliminary Causal linear temporal ?ltering
`in associatiof With Coefficients (bl-Z) andAan anti-Causal Poisy
`Sample (1H1 )' To the sampleswr ’Ir+1 “he? are asslgngd
`Weights calculated as functions of a causal gain factor (K, ),
`
`cient (09A), the Weight of the anti-causal sample (I,+1A) being
`equal to the probability ‘of intensity continuity betWeen the
`anti-causal sample (I,+1 ) and a previous ?ltered sample
`(Ptc, P,_1C) in the sequence. A device for carrying out the
`method is con?gured for calculating the anti-causal linear
`Combination
`'
`
`20 Claims, 7 Drawing Sheets
`
`U.S. PATENT DOCUMENTS
`_
`1/1994 Chiu et al. ............................ .. 378/156
`5,278,887
`9/1994 Nonnweiler et al. ................. .. 382/260
`5,347,590
`5,467,380 11/1995 De Jonge et al. ................... .. 378/98.2
`5,600,731
`2/1997 SeZan et al. .......................... .. 382/107
`
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`SAMSUNG EXHIBITS 1020
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`U.S. Patent
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`Jun. 8, 1999
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`Sheet 1 of7
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`5,911,012
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`Fl 6.1
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`U.S. Patent
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`Jun. 8, 1999
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`5,911,012
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`U.S. Patent
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`U.S. Patent
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`Jun. 8, 1999
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`Sheet 5 of7
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`5,911,012
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`U.S. Patent
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`U.S. Patent
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`1
`METHOD FOR THE TEMPORAL FILTERING
`OF THE NOISE IN AN IMAGE OFA
`SEQUENCE OF DIGITAL IMAGES, AND
`DEVICE FOR CARRYING OUT THE
`METHOD
`
`FIELD OF THE INVENTION
`
`The invention relates to a method for the temporal ?lter
`ing of the noise in an image in a sequence of digital images,
`as Well as to a device for carrying out this method.
`The invention can be used notably for the processing of
`medical images formed in an X-ray ?uoroscopy mode by
`means of a system in Which the X-ray beam has a loW
`intensity and produces a sequence of noisy, loW-contrast
`images Which must be ?ltered in order to remove the noise
`therefrom, Without degrading the details.
`Temporal ?ltering consists of the smoothing as a function
`of time of a monodimensional, so-called temporal signal
`formed by the intensity values of a pixel having a given
`location Within the images of the sequence.
`
`10
`
`15
`
`2
`SUMMARY OF THE INVENTION
`
`A problem consists in that, due to the very loW intensity
`of the ?uoroscopic X-ray beam, the images of the sequence
`are extremely noisy and frequently contain noise peaks.
`It is a further problem that due to the fact that each image
`of the sequence is separated from the next image by a small
`time gap, an important event such as the displacement of a
`small tool such as a catheter may take place from one image
`to another. The ?ltering of the image in Which the movement
`of said small object appears may not distort or erase the
`object.
`It is an object of the present invention to provide a
`temporal ?ltering method for the reduction of the noise in
`the successive images of a sequence of digital images, Which
`method
`is carried out substantially in real time, i.e. With a very
`small delay Which is not perceptible to an operator
`observing the sequence of images, taking into account
`the rate at Which the images of the sequence are
`formed,
`reduces the residual noise behind a discontinuity edge of
`the temporal intensity signal, Without attenuating the
`discontinuity edge;
`is capable of distinguishing the noise peaks from the
`temporal signal variations Which are due to real
`movements, and reduces the noise peaks;
`does not erase or distort the moving small objects.
`These objects are achieved by means of a method for the
`temporal ?ltering of the noise in an image Which is referred
`to as the present image and forms part of a sequence of
`images in the form of a tWo-dimensional matrix of pixels
`Which have digitiZed noisy intensity values Which are
`referred to as samples,
`Which method comprises the evaluation of a ?ltered
`sample, referred to as a present anti-causal sample, in
`order to reconstruct a noisy sample corresponding to a
`pixel in a given location (x,y) in the present image by
`a linear combination, referred to as an anti-causal
`combination, of a present ?ltered sample, referred to as
`a present ?ltered causal sample, obtained by prelimi
`nary temporal ?ltering, referred to as causal and asso
`ciated With coef?cients, and of at least one noisy
`sample Which is later than said present noisy sample
`and is referred to as an anti-causal noisy sample, said
`samples being Weighted by Weights calculated respec
`tively as a function of a so-called causal gain factor,
`evaluated as the inverse of the sum of the coef?cients
`associated With the causal ?ltering, and of a so-called
`anti-causal continuity coef?cient, associated With the
`anti-causal sample evaluated as a probability of inten
`sity continuity betWeen said anti-causal sample and a
`preceding ?ltered sample in the sequence.
`A device for carrying out the above method comprises:
`an image processing system for supplying a noisy digi
`tiZed intensity, referred to as a noisy present sample, of
`a pixel having a given location (x,y) in an image in the
`form of a matrix of pixels arriving at said present
`instant (t), and the noisy intensity of the pixel Which is
`later than the present pixel, referred to as an anti-causal
`sample, of the same location (x,y) in the matrix of the
`later image,
`a ?rst sub-assembly Which is referred to as a causal
`sub-assembly, Whose input receives the present noisy
`sample, and comprises linear ?ltering means With
`Weights for evaluation and for delivering on one output
`
`25
`
`DESCRIPTION OF THE RELATED ART
`A temporal ?ltering method is already knoWn from the
`publication by R. E. KALMAN: “A neW approach to linear
`?ltering and prediction problems” in “Transactions of the
`ASME”, Journal of Basic Engineering, Series 82D, pp.
`35—45, 1960.
`Kalman ?ltering is de?ned by a recursive equation pro
`ducing the ?ltered intensity of an instantaneous pixel of an
`image of the sequence as a function of hypotheses made a
`priori, as a function of the intensity of the pixel in the same
`location in the preceding image of the sequence, and as a
`function of a factor Which is referred to as a Kalman gain.
`35
`This equation can produce tWo recursive algorithms. A
`problem exists in that as soon as a slight movement occurs
`betWeen tWo images, this movement causes an ascending or
`descending edge, referred to as an intensity discontinuity,
`Which appears on the curve of said temporal signal to be
`smoothed.
`In the ?rst algorithm the Kalman gain is chosen to be
`completely constant: this results in exponential streaking
`Which affects said intensity discontinuity edge caused by
`movement. Thus, a small object in the noisy original image,
`for example a catheter Which could be quickly moved, thus
`giving rise to a step in the intensity signal, could have
`disappeared from the ?ltered image because the ?anks of the
`step are deformed by ?ltering. This algorithm erases the
`small objects.
`In the second algorithm the Kalman gain is a function of
`the difference betWeen the noisy intensity observed for a
`pixel having a given location at a given instant. As a result,
`the temporal signal is smoothed before the discontinuity;
`hoWever, it is no longer ?ltered behind the discontinuity, so
`that residual noise exists behind intensity discontinuity edge.
`The knoWn temporal ?ltering method, therefore, has the
`draWback that it cannot be effectively applied to a series of
`very noisy images representing animated small objects.
`Thus, the knoWn temporal ?ltering method does not solve
`some major problems encountered When the temporal ?l
`tering method is applied to a sequence of images acquired in
`the X-ray ?uoroscopy mode as performed, for example for
`real-time folloWing of a medical operation during Which a
`tool of extremely small diameter, such as a catheter, is
`introduced into or displaced through the Zone being
`observed.
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`3
`a ?rst ?ltered value, referred to as a ?ltered causal
`sample, of the present sample, a ?ltered sample of the
`preceding instant on another output, and a causal gain
`factor equal to the inverse of the Weights of said linear
`?ltering on another output, and
`a second sub-assembly, referred to as an anti-causal
`sub-assembly, an input of Which receives the ?ltered
`causal sample of the preceding instant, a further input
`of Which receives the present ?ltered causal sample,
`and another input of Which receives the noisy anti
`causal sample, and comprises calculation means for
`evaluating the anti-causal integration relation With the
`causal gain factor and for delivering on its output a
`second value Which is referred to as a ?ltered anti
`causal sample and constitutes the ?ltered sample for the
`reconstruction of the noisy present sample.
`This device offers the advantage that it can be readily put
`into operation and that it provides effective, substantially
`real-time temporal ?ltering of the noise Without deteriorat
`ing the details of the image.
`
`BRIEF DESCRIPTION OF THE DRAWINGS
`
`The invention Will be described in detail hereinafter With
`reference to the accompanying diagrammatic draWings;
`therein:
`FIG. 1 shoWs an X-ray imaging device;
`FIG. 2 shoWs a sequence of digital images;
`FIG. 3A shoWs a noisy temporal signal With an intensity
`discontinuity due to a movement;
`FIG. 3B shoWs another noisy temporal signal With a noise
`peak;
`FIG. 3C illustrates the determination of the standard
`deviation of the noise;
`FIG. 4A shoWs the ?ltered temporal signal corresponding
`to the noisy temporal signal of FIG. 3A in a ?rst embodiment
`of the invention;
`FIG. 4B shoWs the ?ltered temporal signal corresponding
`to the noisy signal of FIG. 3B in the ?rst embodiment of the
`invention;
`FIG. 4C shoWs the temporal signal of FIG. 3A ?ltered in
`a further version of said ?rst embodiment;
`FIG. 5A shoWs the ?ltered temporal signal corresponding
`to the noisy temporal signal of FIG. 3A in a second embodi
`ment of the invention;
`FIG. 5B shoWs the ?ltered temporal signal corresponding
`to the noisy temporal signal of FIG. 3B in the second
`embodiment of the invention;
`FIG. 5C shoWs the temporal signal of FIG. 3B ?ltered in
`a further version of said second embodiment;
`FIGS. 6A to 6D shoW various feasible examples of the
`function F;
`FIGS. 7A and 7B illustrate, in the form of functional
`blocks, a device for carrying out the temporal ?ltering
`method With a causal component and an anti-causal com
`ponent.
`
`DESCRIPTION OF PREFERRED
`EMBODIMENTS
`
`I/X-ray device
`FIG. 1 shoWs a digital radiography system, comprising an
`X-ray source, a mobile table 2 for a patient, and an image
`intensi?er device 3 Which is coupled to a video tube 4 Which
`applies data to a digital image processing system 5 which
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`comprises a microprocessor. The latter comprises a plurality
`of outputs, an output 6 of Which is coupled to a monitor 7
`for the display of the sequence of radiographic images or
`intensity images.
`The digital radiographic image can comprise 512x512 or
`1024x1024 pixels Which are encoded on 8 bits or 10 bits.
`Each pixel can thus have one of 256 or 1024 intensity levels.
`For example, the dark regions have a loW intensity level and
`the bright regions of the image have a high intensity level.
`The digital image can be acquired in the ?uoroscopy
`mode. The invention can be used notably for the ?ltering of
`angiographic images.
`In any case, the invention takes into account neither the
`method used to acquire the sequence of digital images nor
`the nature of the objects represented therein, but concerns
`exclusively the ?ltering of these images in this sequence in
`order to eliminate the noise.
`II/Temporal noise ?ltering method
`The present invention proposes a method for the temporal
`?ltering of the noise successively in each image of a
`sequence of noisy digital images. The method executes the
`?ltering operation from the ?rst to the last noisy image
`observed. Therefore, this method is carried out substantially
`in real time.
`The method is capable of detecting the movement of
`objects recorded in the images of the sequence. It is based
`partly on adaptive recursive ?ltering steps.
`Referring to FIG. 2, the method for the temporal ?ltering
`of the noise comprises ?rst of all the acquisition and
`digitiZation of a sequence of noisy images formed at a given
`rate.
`From the instant Which is most remote in time until the
`most recent instant this sequence is composed of:
`a number of past intensity images, denoted by JJ-C and
`referred to as “causal images”, formed after the ?rst
`instant j=t—k0, Where k0 is an integer equal to the
`number of images of the sequence minus tWo, until the
`instant j=t—1, Where t is the instant of arrival of the
`image being ?ltered;
`an image JtP being ?ltered, referred to as the present
`image, Which arrives at the instant j=t, and
`a supplementary image JIHA, referred to as anti-causal or
`future image, Which arrives at a later instant j=t+1.
`The present image J [P is in reality ?ltered just after said
`later instant t+1, i.e. With a small delay relative to the instant
`of its arrival.
`Each noisy digital image J 1- consists of a tWo-dimensional
`matrix
`y) of pixels, each of Which is characteriZed by
`its coordinates x, y in the matrix and by an intensity level
`lj-(x, y) of an intensity level scale, Where j is the index
`corresponding to that of the image.
`A so-called noisy temporal signal is formed of the differ
`ent intensity levels lj-(x, y) of the pixels
`y) as a function
`of the time "c as shoWn in the FIGS. 3A, 3B and 3C, in Which
`lj-(x, y) is plotted on the ordinate and "c on the abscissa. The
`temporal ?ltering method in accordance With the invention
`aims to ?lter or smooth the noise affecting said temporal
`signal I('c) in order to obtain a ?ltered temporal signal P("c)
`as shoWn in FIG. 4 or 5. Hereinafter, the intensities consti
`tuting the points of the temporal signal I('c) Will be referred
`to as “samples”.
`The method in accordance With the invention refers to the
`case Where the principal movements occurring in the image
`sequence have already been compensated for by movement
`compensation methods knoWn from the state of the art.
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`6
`The temporal ?ltering method is based on a linear com
`bination of causal and anti-causal samples With the present
`sample in Which the Weights bJ-C et bt+1A attached to the
`causal and anti-causal samples have a speci?c form.
`The Weight attached to a given causal sample is a prob
`ability of intensity continuity betWeen said given causal
`sample and the present sample.
`Generally speaking, the formule of the causal Weight
`bt_kC to be applied to a causal sample It_kC is given by
`relation (2):
`
`5
`This method performs a ?ltering operation on the noise of
`the temporal signal I('c) by treating the small residual or
`non-compensatable local movements of the objects in the
`image sequence.
`This method performs a ?ltering operation on the noisy
`sample If arriving at the present instant t, and produces a
`?ltered sample PtA by utiliZing the noisy sample lip of the
`present instant t, the earlier samples, referred to as noisy
`causal samples It_kOC, .
`.
`. It_3C,I,_2C, It_1C, observed at the
`past instants from t—kO to t—1, and the next sample referred
`to as the noisy anti-causal sample It“A observed at the later
`instant t+1, referred to as future, Which is the instant after
`Which the processing of If actually takes place.
`The temporal ?ltering is preferably performed individu
`ally for each pixel
`y) at the different coordinates x, y
`of the tWo-dimensional matrix.
`FIG. 3A shoWs a noisy temporal signal I('c) Which
`comprises, by Way of example, an intensity discontinuity D
`betWeen the sample It_3C and the sample IFZC, said discon
`tinuity being due to a small local movement. This small local
`movement translates as ascending edge of the temporal
`signal I('c) betWeen the instant t—3 and the instant t—2, a high
`signal “plateau” betWeen the instants t—2 and t+1, and a
`“depression” at the instants preceding the instant t—3.
`This temporal signal also exhibits small saWteeth betWeen
`the instants t—7 and t—3 because, due to the noise, the
`intensity level of a pixel having the same coordinates (x, y)
`in the images of the sequence varies continuously, thus
`causing a snoWfall aspect. The ?ltering enables this aspect to
`be suppressed by smoothing the temporal signal I('c) to a
`constant mean value in relation to a given pixel.
`The temporal ?ltering in accordance With the invention
`can adapt itself to an intensity discontinuity such as D. Thus,
`FIGS. 4A, 4C and 5A shoW the temporal signal P("c)
`smoothed by various versions of the method in accordance
`With the invention.
`FIG. 3B shoWs a noisy temporal signal I('c) having, by
`Way of example, a noise peak D‘ Which appears betWeen the
`instants t—3 and t—1, that is to say at the instant t—2. Such a
`noise peak D‘ is distinct from an intensity discontinuity such
`as D of FIG. 3A in that it does not correspond to a real spatial
`movement phenomenon. Such a noise peak can be ?ltered
`by means of the method proposed by the invention.
`The ?ltered sample PtA, corresponding to the present
`noisy sample If, is expressed as:
`
`(1)
`
`The ?ltered sample PIA, or temporal signal ?ltered at the
`instant t, is a linear combination of noisy samples observed
`at the causal instants from j=t—kO until j=t—1, of the noisy
`sample observed at the present instant t, and the noisy
`sample observed at the anti-causal or future instant t+1.
`In the formula (1) of the ?ltered signal PIA, a coef?cient
`or Weight bJ-C, and bpflA is applied to the causal samples
`I,_1C, IFZC etc. and anti-causal samples IIHA, respectively,
`the Weight attached to the present sample If being ?xed to
`the value 1.
`The denominator of the formula (1) is a normaliZation
`factor Whose presence is based on the fact that the sum of the
`Weights applied to the various samples must be equal to 1 in
`order to ensure that the mean value of the ?ltered signal PtA
`equals the mean value of the noisy signal If.
`
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`Where otj-c is the probability of intensity continuity betWeen
`the successive causal samples after It_kC until ItP. According
`to the relation (2) a Weight bt_kC relating to a causal sample
`It_kC equals the product of all probabilities of intensity
`continuity otj-c after the causal instant j =t—k+1 until the
`present instant t.
`Thus, the determination of the Weight attached to a given
`causal sample according to the formule (1) is based on the
`hypothesis that said given sample is retained and taken into
`account in the linear combination, be it exclusively in as far
`as it does not deviate too much from the present sample,
`Which signi?es that it relates to the same object.
`According to this hypothesis, for example the formulation
`of the causal Weight bt_1C, relating to the causal sample
`It_1C, is realiZed by Writing that bt_1C is a function of the
`absolute value of the difference |IEP—I,_1C|, or, in other
`Words, that b,_1C is a function of the difference:
`
`Where Pt_1C is the sample already ?ltered at the preceding
`instant t—1, so that presumably it is less noisy than It_1C as
`a result of ?ltering. If the difference betWeen the samples If)
`and PHLC is small, a “high” value near 1 is assigned to the
`corresponding Weight bt_1C. If this difference is large, a
`value close to Zero is assigned to the Weight bt_1C. In that
`case the sample It_1C is practically not taken into account.
`Subsequently, in this example the formulation of the
`second causal Weight bt_2C, relating to the causal sample
`IFZC, is realiZed by Writing that the Weight b,_2C is a function
`of not only the difference betWeen the sample at the instant
`t and the sample at the instant t—1, but also a function of the
`difference betWeen the sample at the instant t—1 and the
`sample at the instant t—2. Thus, the Weight b,_2C has a high
`value near 1 if exclusively samples are taken into account
`Which have not been subjected to modi?cations, other than
`those due to the noise, With respect to the present sample If,
`ie if the condition is imposed that the differences betWeen
`the samples taken into account must be small. This leads to
`the formulation of causal Weights bJ-C as products of func
`tions of intensity differences appearing in the temporal
`signal, ie to the formulation of these Weights as products of
`the probability of intensity continuity betWeen the samples
`preceding the present sample to be ?ltered.
`Thus, the formulation of the causal Weight relating to the
`sample It_1C is:
`
`(23)
`Where the function of is a probability of intensity continuity
`betWeen the present sample If and the ?ltered sample Pt_1C.
`The intensity continuity betWeen the sample at the instant t
`and the sample at the preceding instant t—1 translates as a
`
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`8
`noisy sample If observed at the present instant t. This
`innovation term is multiplied by a term Which is called a
`causal gain term Ktc Which varies betWeen 0 and 1. This ?rst
`causal equation (3C) produces a ?ltered value PIC Which is
`referred to as a causal ?ltered value Which Will be modi?ed
`and enhanced by the ?ltering due to the calculation of the
`second equation (3A).
`The second non-recursive equation (3A), referred to as
`anti-causal, takes into account the causal ?ltered sample PIC
`resulting from the ?rst recursive causal equation (3C), modi
`?ed by an anti-causal innovation term Which is noW formed
`by the difference betWeen the ?ltered sample Pic, resulting
`from the recursive equation (3C), and the noisy sample 1H1A
`observed at the future instant t+1. This anti-causal innova
`tion term is multiplied by a term Which is called the
`anti-causal gain term K? which also varies betWeen 0 and
`1.
`
`The gains KEG et K? are not Kalman gains, but completely
`different gain factors Which are given by the folloWing
`recursive and non-recursive equations (4C) and (4A) respec
`tively:
`
`7
`high continuity probability of. An intensity discontinuity
`between the samples translates as ottC close to 0.
`The formulation of the causal Weight relating to the
`sample IFZC is:
`
`c_ c
`c
`bri2 _ar xaril
`
`(2b)
`
`Where (Xt_1C is the probability of intensity continuity
`betWeen the sample at the instant t—1 and the sample at the
`instant t—2.
`The intensity continuity of the samples betWeen the
`instant t and the instant t—2 implies intensity continuity
`betWeen the instants t and t—1 and intensity continuity
`betWeen the instants t—1 and t—2, Which translates as the
`product of the probabilities (2b) in Which the probability of
`intensity continuity betWeen the instants t and t—1 is high
`and the probability of intensity continuity betWeen the
`instants t—1 and t—2 is also high.
`The formulation of the causal Weight relating to the
`sample It_3C is then:
`
`Thus, in the eXample of FIG. 3A the probability of
`continuity 094C is very close to Zero because of the intensity
`discontinuity D betWeen the samples It_3C and I,_2C. The
`Weight bt_3C obtained by the product of the probabilities,
`including said probability ot,_2C, also tends toWards Zero.
`The Weight bt_4C also tends toWards Zero, because its
`formulation is a product Which also contains said probability
`close to Zero. As soon as a probability close to Zero is present
`in the product of the probabilities supplying a Weight bJ-C, the
`presence of this probability cancels this product and all
`products corresponding to the instants preceding the instant
`at Which said probability close to Zero appeared. As a result,
`the Weights bJ-C of the causal samples corresponding to said
`preceding instants are also close to Zero. In the speci?c
`example of FIG. 3A the causal Weights are:
`
`10
`
`15
`
`25
`
`For the transition betWeen the equations (1), (2) and (2‘)
`and the equations (46), (5C) and (4A), (5A), it is to be noted
`that the causal gain (Ktc) equals the inverse of the sum of the
`causal Weights:
`
`35
`
`jail mp
`
`The formulation of the anti-causal Weight bt+1A, relating
`to the future sample, is a function of the absolute value of an
`intensity difference AA and is given by:
`
`and that, as has already been stated, the causal coef?cient of
`continuity probability of is the Weight of the causal sample
`It_1C preceding the present sample If:
`
`br+1A=arA
`
`(2)
`
`45
`
`C
`C
`at =bril
`
`where of‘ has a speci?c value Which Will be described in
`detail hereinafter.
`As a result of the speci?c formulations (2) and (2‘) of the
`causal and anti-causal Weights bJ-C et b, HA, respectively, the
`equation (1) Whose calculation Was very complex is trans
`formed into tWo groups of three simple relations each, a ?rst
`group performing a recursive ?ltering operation Which is
`referred to as causal ?ltering Whereas a second group
`performs a ?ltering operation Which is referred to as anti
`causal ?ltering.
`The linear equation (1), in Which the Weights have the
`speci?c form (2) and (2‘), leads to a formulation of a
`recursive equation (3C and a non-recursive equation (3A)
`Which are formulated as folloWs:
`
`The transition betWeen the equations (1)+(2)+(2‘) and the
`equations (3C)+(3A) is based on the ?rst system of recursive
`causal equations:
`
`P rC=P I’1C+KIC><(I rP_P r*1C)>
`
`being the causal integration equation (3C)
`
`55
`
`C Kril
`
`being the causal gain factor (4C) and
`
`The ?rst recursive equation (36), referred to as causal
`integration, takes into account the ?ltered sample P,_1C at
`the instant t—1, modi?ed by What is called hereinafter a
`causal innovation term Which is formed by the difference
`betWeen the ?ltered sample P,_1C at the instant t—1 and the
`
`65
`
`causal continuity coef?cient (5C),
`and on the second system of non-recursive anti-causal
`equations:
`
`SAMSUNG EXHIBIT 1020
`Page 12 of 20
`
`

`

`being the anti-causal integration equation (3A)
`
`A
`
`6
`K1
`
`being the anti-causal gain factor (4A),
`
`A
`
`(IA-FA A—
`I-
`SIC’
`
`being the anti-causal continuity coef?cient (5A) in Which AA
`represents the absolute value of an intensity difference
`betWeen the future instant t+1 and a second instant to be
`considered, so either the present instant or a past instant,
`Where Sf is a normaliZation factor, and Where FC and FA are
`functions to be de?ned hereinafter.
`The iteration of the equation (3C), commencing at t-ko in
`time, and the subsequent calculation of the equation (3A)
`enable the equation (1) to be retrieved exactly as formulated
`above.
`The formulation (3C)+(3A) of the integration equations
`thus corresponds to this linear combination of samples,
`associated With speci?c Weights Which depend on the con
`tinuity of intensities betWeen the samples.
`Generally speaking, the evaluation of the present ?ltered
`causal sample PIC is performed by the iteration of the causal
`integration equation (3C) Which corresponds to a linear
`combination of the noisy causal and present samples,
`Whereto Weights have been applied, a causal Weight being
`evaluated as a probability of continuity betWeen the causal
`sample Whereto said Weight is attached and the present
`sample, the Weight of the noisy present sample being ?xed
`at 1. In order to evaluate a causal Weight, the probability of
`continuity betWeen a given causal sample and the present
`sample can be evaluated as the product of the probabilities
`of continuity betWeen the successive samples after the given
`sample until the present sample.
`Furthermore, the evaluation of the ?ltered present anti
`causal sample PtA is performed by calculation of the anti
`causal integration equation (3A), corresponding to the evalu
`ation of a linear combination betWeen the ?ltered present
`causal sample PIC and the anti-causal noisy sample 1H1A
`Whereto a speci?c Weights has been attached.
`Actually, the equation (1) given above can also be Written
`as:
`
`It has already been stated that:
`
`5,911,012
`
`Assuming that:
`
`10
`
`hi1
`
`:
`
`‘11A
`(1 / KIC) + 11f‘
`
`_ KIA
`
`the linear expression of the ?ltered anti-causal sample is
`then obtained:
`
`[+1
`
`its Weights being(1—KtA) for PIC et K? for IH1A. The
`anti-causal gain K? in this case depends exclusively on the
`anti-causal continuity coef?cient of‘ and on the sum of the
`Weights attached to the samples involved in the causal linear
`?ltering operation.
`In a general version of the invention, the ?ltered sample
`PIC can be supplied by any linear ?ltering process (be it
`recursive or not) Whose inverse of the sum of the coef?cients
`provides a factor utiliZed as the causal gain Ktc.
`Formulation of the causal continuity coefficient of‘
`The causal continuity coef?cient ottc is de?ned as a
`function of the difference AC=|ItP—Pt_1C|
`It is advantageous to determine Whether the difference AC
`relates to a discontinuity such as D in FIG. 3A or Whether it
`relates exclusively to noise.
`In order to determine the noise participation, the differ
`ence AC is normaliZed by a factor SIC Which takes into
`account the variance of the noise relative to each sample of
`this discontinuity AC. The standard deviation of the noise,
`referred to as OB and measured in intensity levels for each
`sample I], can be estimated by means of any knoWn method
`or should be estimated a priori.
`According to a method proposed, by Way of example, for
`the determination of the standard deviation GB of the noise
`in relation to FIG. 3C, a noisy signal is represented by the
`curve 1(1); the mean amplitude of this signal approximately
`equals its arithmetical mean mB evaluated over a number of
`N samples, ie between the present instant j=t, for example,
`and the past instant j=t-N+1, taking into account the noisy
`samples II_N+1C .
`.
`. If);
`therefore, this offers the mean value searched
`
`(8C)
`
`The standard noise deviation GB is the mean deviation
`betWeen a noisy signal and its mean value mg. The variance
`of the noise, referred to as 052 and being the mean value of
`the square of the intensity deviations due to the noise of the
`N samples considered With respect to the mean value mB,
`can be calculated by means of the relation:
`
`(9%
`
`The standard noise deviation GB is then obtained by calcu
`lat