Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai, China, September 1-4, 2005
Exponential Entropy Driven HUM on Knee MR Images Edoardo Ardizzone, Roberto Pirrone and Orazio Gambino
Abstract— A very important artifact corrupting Magnetic Resonance Images is the RF inhomogeneity. This kind of artifact generates variations of illumination which trouble both direct examination by the doctor and segmentation algorithms. Even if homomorphic filtering approaches have been presented in literature, none of them has developed a measure to determine the cut-off frequency. In this work we present a measure based on information theory with a large experimental setup aimed to demonstrate the validity of our approach.
I. INTRODUCTION In the last years a lot of studies have been presented on a particular artifact that afflict medical images and in particular Magnetic Resonance Images (MRI): the RF inhomogeneity. In a few words, the machine magnetic field has intensity variations depending on the scan sequence, patient tissues and kind of coil used. In particular, surface coil image is more sensible to this artifact but noise is not strong; on the contrary, body coil image has the opposite behavior. The effect of this artifact on the image is a non uniform illumination which can be seen either as a white stripe, or a more or less extended peak of luminance or an image shade variation along a direction, especially if images are produced by open coil devices like the one used for lower and upper limbs imaging. The approaches used to suppress this artifact can be subdivided in two classes: discovering the degradation model or modifying classic segmentation algorithms to classify the tissues taking into account the degradation model. In the first class of methods, a phantom,usually a container which holds a known substance, may be introduced in the machine, in order to obtain an image afflicted by magnetic field intensity variations without any interaction with tissues and discover the degradation model. With this special experimental setup a surface can be discovered which approximates the illumination variations[7][10] and it is used to correct the image. If body-coil image and surface-coil image of the same slice are available, it can be used a preconditioned gradient algorithm to discover the surface[2]. In [1] the bias artifact is modelled by Legendre polynomials and the algorithm estimates each uncorrupted class computing the image by a least-square technique, and assuming that the image is made by piecewise zones with constant intensity level. The above methods can be considered evolutions of homomorphic filtering [16] approaches moving from[8] where the HUM (Homomorphic Unsharp Masking) has been developed. This approach either uses directly the phantom pixel to perform the correction or performs a log subtraction The Authors are with the Universita’ di Palermo - Dipartimento di Ingegneria Informatica - viale delle Scienze - edificio 6-C.A.P. 90128-PALERMO (ITALY)email: ardizzon,
[email protected] [email protected]
0-7803-8740-6/05/$20.00 ©2005 IEEE.
of the image with a blurred version of the same one using low pass filters[11]. The trend of last years has been to modify classical segmentation algorithms in order to take into account, in their iterative steps, the artifact effect; this kind of approach starts with EM algorithm[12][6][4] while in the recent years some modified fuzzy c-means (fcm) algorithms[3][14][15][9] have been used due to their higher speed with respect of the first one. An unusual solution to the problem is proposed in [13] which uses an approach exclusively based on information theory, minimizing the entropy of the inverse degradation model with Brent-Powell algorithm. The question is open because the NEUROVIA PROJECT [18] shows that this artifact isn’t uncorrelated with tissues. This project was born as a comparative study of six different algorithms with the aim to determine the best approach on encephalic MR, thanks to a large and accurate experimental setup in various environment conditions: the result is that each algorithm shows a different bias estimation, due to its theoretical approach, and by the interaction of the tissues with the magnetic field. We perform our experimental setup on images of knee which are not shown in any study presented above and are corrupted by heavy bias artifact. The rest of the paper is arranged as follow. Section2 and 3 describe the proposed method for bias removal. In Section 4 the experimental setup is detailed. In Section 5 an evaluation of the results is performed while in Section 6 and 7 some conclusions and future works are presented. II. T HE E XPONENTIAL E NTROPY D RIVEN HUM Often in medical imaging the image is composed by a background and a foreground so if homomorphic filtering is performed a streak artifact is produced on the boundary. Guillemaud [5] proposed a homomorphic filter where the Region of Interest (ROI) is considered to avoid the artifact cited above. The algorithm is explained in detail as follows: 1) define a binary image (background=0;foreground=1) in order to define the region of interest named ROI 2) perform a log-transform to the original image with background suppressed: Ilog = log(I(ROI)) 3) apply a low-pass butterworth filter of 2nd order in frequency domain obtained from the previous step: If ilt = F F T −1 (BT Worder=2 (F F T (Ilog ))) 4) perform the same butterworth filtering to the binary image: ROIf ilt = F F T −1 (BT Worder=2 (F F T (ROI))) 5) due to anti-transform operation, the images will be complex data, so perform magnitude of each pixel value: Imag = ABS(If ilt ) ROImag = ABS(ROIf ilt )
1769
11.5
11
10.5
10
local entropy
9.5
9
8.5
8
7.5
7 corrected bias 6.5
Fig. 1. Cut-off frequency 0.99 - From Left to Right: original, restored and bias images
0.002
0.004
0.006
0.008
0.01 0.012 frequency
0.014
0.016
0.018
0.002
0.004
0.006
0.008
0.01 0.012 frequency
0.014
0.016
0.018
0.002
0.004
0.006
0.008
0.01 0.012 frequency
0.014
0.016
0.018
0.02
11.5
11
10.5
10
local entropy
9.5
6) divide pixel-by-pixel the images obtained at the step 5: Log(Bias) = Imag /ROImag 7) subtract the image at step 2 with that one obtained at the previous step and perform an exponential transform: Irest = exp(Ilog − Log(Bias)) 8) perform a contrast stretching of the image at step 7 in order to obtain the initial dynamic: Irest − min(Irest ) Icorr = · max(I) (1) max(Irest ) − min(Irest )
In fig.2 this information transfer is shown visually thanks to the pictures and mathematically by the bias local entropy diagram (BLED) and corrected local entropy diagram (CLED). The entropies are calculated only on ROI, because the background do not contain any information about the artifact. III. BFCF D ETERMINATION A first frequency candidate for bfcf is the intersection between the BLED and CLED, because they have the same quantity of information so it is sure that a part of information is migrated from the image to the bias. There isn’t any prove that the point found in this way is the best solution, but the
8
7.5
7 corrected bias 6.5
0.02
11.5
11
10.5
10
local entropy
9.5
9
8.5
8
7.5
7 corrected bias 6.5
0.02
11.5
11
10.5
10
9.5 local entropy
The Bias image can be obtained in this way: IBias = exp(Log(Bias)) (2) Here the same stretching as in (1) is added. An unresolved problem of this method is that the butterworth filter cutoff frequency (bfcf) is not specified. This is an important parameter because the entity of homomorphic process depends on it. If it is too low, no effect will be visible on Icorr ; on the contrary, with very high values of bfcf Icorr tends to exhibit uniform gray levels and, consequently, the tissues contrast is lost. For values of bfcf close to 1, a total inversion between Icorr and IBias is produced: IBias will appear identical to the initial image I while Icorr is made by a quasi-uniform gray levels in the ROI, as shown in fig.1. This fact is a classic image processing finding because step 7 corresponds to a high-pass filtering so the lower frequencies which are subtracted from I are given to IBias . From the information theory point of view, I has more information than necessary: it contains both not corrupted information and bias information, so the filtering can be considered like a sort of information transfer. An ideal information measure applied both to Icorr and IBias as a function of the cut-off frequency should increase in the case of IBias , and decrease in the case case of Icorr . The Shannon Entropy can be reported as an information measure: H=− p(x) · log2 [p(x)] (3)
9
8.5
9
8.5
8
7.5
7 corrected bias 6.5
0
0.002
0.004
0.006
0.008
0.01 frequency
0.012
0.014
0.016
0.018
0.02
Fig. 2. From Up-to-down: sequences at growing cut-off frequency. Leftto-Right: corrected, diagram frequency-entropy,bias estimated
experimental setup shows good results over 90% of cases, as it can be noticed in the third row of fig.2 and the first rows of fig.3 and 4. During this experimentation we noticed a streak artifact introduced by the filter which was generated by the presence of dark zones whose intensity is similar to the background, as shown in fig.3. A solution to this problem is to identify these zones with a fuzzy c-means algorithm [19] applied to the corrected image using three cluster (dark, medium and bright pixels). In this way it can be created a mask, not a ROI, where this dark zones are absent in such a way that the streak artifact is not present when the correction is applied again. Even if the bfcf is generally lower than the one found with the ROI, sometimes the bfcf is too high, as shown in fig.4, where the contrast is visually lost. The BLED curve has a profile like a capacitance charge function; this behavior is exhibited by all the BLED curves, some of them are shown in fig.5b. Thanks to the experimental setup, it can be noticed that goods values of bf cf are located in a zone of the curve close to the end of the transient phase, as it can be seen in figs 2,3 and 4. Therefore, a BLED can be modelled by the charge capacitance function: x
y(x) = k1 + k2 e− τ
(4)
where x and τ , instead of time variable and time constant in the original formula, are respectively the frequency variable and a ”frequency constant”. A sure value of bf cf is given by 5τ , which is the end of the transitory. The coefficients k1 , k2 , τ can be found minimizing the quadratic
1770
11.5
error:
12
k1
τ
n 2 x(i) 1 k1 + k2 · e− τ − H [x(i)] E= 2 i=1
11
local entropy
local entropy
10
where x(i) are the BLED discrete values of the frequency and H [x(i)] are the entropy values of the BLED. The minimization is performed using the Nelder-Mead [17] algorithm, because it exhibits a better stability than gradient descendant method, using as start values the average of the last ten values of H in the BLED for k1; k2 is constrained to be negative due to the BLED profile so it can be set initially to -1 and τ was set to 0.001.
9
8
7
6
6.5
0
0.05
frequency
a)
5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
frequency
b)
Fig. 5. UP: the charge of a capacitor model on a BLED and a family of BLED curves. 5
6 4 2
12
11
10
3
local entropy
9
8
1 7
6
5 restored bias 4
0
0.005
0.01
0.015
0.02
0.025 frequency
0.03
0.035
0.04
0.045
0
0.005
0.01
0.015
0.02
0.025 frequency
0.03
0.035
0.04
0.045
0.05
11
10
local entropy
9
8
7
6
corrected bias 5
Fig. 6. UP: original image, exponential ed-hum corrected, spm2 restored, labels. Down: Coefficient of variations and Coefficient of contrast tables.
0.05
Fig. 3. Comparison of the result applying two different masking on the slice of fig.1. From Left-to-Right: restored image, masking used, local entropy diagram, bias estimated.
12
11
10
local entropy
9
8
7
6
5 corrected bias 4
0
0.005
0.01
0.015
0.02
0.025 frequency
0.03
0.035
0.04
0.045
0.05
11
10
of the original ones where the illumination is focalized on a limited portion of the anatomical structure. All the corrected images doesn’t present false negatives or positives due to a grey levels distortion and the boundaries between the tissues are preserved. After the visual inspection, an handmade segmentation of a slice has been performed to measure the coefficient of variation cv inside of the located zones and the coefficient of contrast cc between only the adjacent zones:
local entropy
9
cv(zone) =
8
7
6
restored bias 5
0
0.005
0.01
0.015
0.02
0.025 frequency
0.03
0.035
0.04
0.045
0.05
Fig. 4. The intersection of the local entropy doesn’t offer sometimes a good bfcf.
IV. E XPERIMENTAL S ETUP The method has been performed on images decoded from DICOM file format, that is without any optical scanner acquisition which may introduce other artifacts. The device is an ESAOTE ARTOSCAN C with a magnetic field intensity of 0.18 Tesla. The dataset which we present in this paper is a complete study of the knee on sagittal plane which consists of 19 T1-weighted images on 5 subjects acquired with the following parameters: Spin Echo sequence, Repetition time: 980 ms, Echo time: 26 ms, Slice thickness: 4mm, Flip Angle= 90. The useful resolution of FOV is 170x170 pixels with 12 bit of pixel depth. V. T HE MEDICAL EVALUATION AND MEASURES All the restored images show an enhanced contrast among the tissues by the visual impression of the doctor. The bias suppression offers a panoramic vision of the image, instead
σ(zone) µ(zone)
cc(zone1 , zone2 ) =
µ(zone1 ) µ(zone2 )
where σ and µ are, respectively, standard deviation and the mean value of a segmented zone. As it can be seen in fig.6, the cv of our method is better than SPM2 algorithm[20] in some zones while the cc is generally better. VI. C ONCLUSIONS The price to pay for the bias suppression is a reduction of contrast, as shown in fig.6; this parameter often isn’t taken into consideration by other studies and our model satisfies both image features. It is obvious that E 2 D −HU M can obtain the same performance of SPM2 just increasing the bf cf ; in fact, the method can be implemented in semi-automatic way to allow the doctor to select manually the cut-off frequency, starting from the one suggested by the algorithm. This is not a marginal fact: the doctors use visual inspection of the image with the aim of to determine the disease so an interactive allowing trial and error analysis procedures. Even if most of the algorithms presented above use principally a certain objective function to optimize, they don’t care the doctor’s image perception. The method presented in this paper exhibit several good properties: no hypothesis are made on tissue classes, it can be applied on great variety
1771
11
of medical images while often the application presented in other papers is oriented principally on encephalic MRI; the implementation is quite simple, it is an improvement of classical filtering approach in image analysis; there is no needs of preliminary experimental setup to ”tune” the filter; the method is fast and it can be performed on the fly.
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.035
0.04
0.045
0.05
0.035
0.04
0.045
0.05
11
VII. F UTURE W ORKS
10
9
Despite other modified segmentation algorithms which include in their objective function the bias effect, it is possible to try a classical scheme preprocessing+segmentation where the correctness of the frequency could be found via comparison between results and a pre-segmented model. This fact is particularly interesting on encephalic MR images both normal and in presence of lesions. This is particularly true for images obtained by open coil devices which are used when the patient has a claustrophobic behavior on a classical device, or during surgery operations.
8
7
6
5
0.005
0.01
0.015
0.02
0.025
0.03
10
9.5
9
8.5
8
7.5
7
6.5
6
0
0.005
0.01
0.015
0.02
0.025
0.03
11
VIII. ACKNOWLEGEMENT This work has been partially supported by Istituto Radiologico PIETRO CIGNOLINI - Policlinico dell’Universit´a di Palermo. Particulars thanks to Eng. Daniele Peri for his technical support, Dr. Gian Piero De Luca and Dr. Claudio Cusumano for their medical support and Dr. Prof. Giuseppe De Maria for his availability. R EFERENCES [1] Styner, M.; Brechbuhler, C.; Szckely, G.; Gerig, G.: Parametric estimate of intensity inhomogeneities applied to MRI Medical Imaging. IEEE Transactions on Medical Imaging 22 (2000)153–165 [2] Shang-Hong Laia; Ming Fangb: A dual image approach for bias field correction in magnetic resonance imaging. Magnetic Resonance Imaging 21 (2003)121–125 [3] Mohamed N. Ahmed; Sameh M. Yamany; Nevin Mohamed: A Modified Fuzzy C-Means Algorithm for Bias Field Estimation and Segmentation of MRI Data. IEEE Transactions on Medical Imaging 21 (2002) 193–199 [4] Van Leemput K.;Maes F.; Vandermeulen D. and Suetens P.: Automated Model-Based Bias Field Correction of MR Images of the Brain. IEEE Transactions on Medical Imaging 18 (1999) 885–896 [5] Guillemaud, R.: Uniformity Correction with Homomorphic filtering on Region of Interest. IEEE International Conference on Image Processing 2 (1998) 872–875 [6] Guillemaud, R.; Brady, M. : Estimating the bias field of MR images. IEEE Transactions on Medical Imaging 16 (1997) 238–251 [7] Dawant B.M.; Zijdenbos A.P.; Margolin R.A.: Correction of Intensity Variations in MR Images for Computer-Aided Tissue Classification. IEEE Transactions on Medical Imaging 12 (1993) 770–781 [8] Axel L.; Costantini J.; Listerud J.: Intensity Correction in Surface Coil MR Imaging. American Journal on Roentgenology 148 (1987) 418– 420 [9] Lei Jiang,Wenhui Yang:A Modified Fuzzy C-Means Algorithm for Segmentation of Magnetic Resonance Images. Proc. VIIth Digital Image Computing: Techniques and Applications.Sun C., Talbot H., Ourselin S. and Adriaansen T. Editions. (2003) 225–231 [10] Tincher M.; Meyer C.R.; Gupta R.; Williams D.M.: Polynomial Modelling and Reduction of RF Body Coil Spatial Inhomogeneity in MRI. IEEE Transactions on Medical Imaging 12 (1993) 361–365 [11] Brinkmann B. H. , Manduca A. and Robb R. A.: Optimized Homomorphic Unsharp Masking for MR Grayscale Inhomogeneity Correction. IEEE Transactions on Medical Imaging. 17 (1998) 161–171 [12] Wells W.M.; Grimson W.E.L.; Kikins R.; Jolez F.A.: Adaptive Segmentation of MRI Data. IEEE Transactions on Medical Imaging. 15 429–442 (1996) [13] Likar B.; Viergever M.A.; Pernus F.: Retrospective Correction of MR Intensity Inhomogeneity by Information Minimization. IEEE Transactions on Medical Imaging 20 (2001) 1398–1410
0
11
10.5
10
9
8
7
6
5
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
11
10
9
8
7
6
5 11
10
9
8
7
6
5
Fig. 7. Some results. Each row:original and restored images, local entropy diagram
[14] Pham D.L.; Prince J.L.: Adaptive Fuzzy Segmentation of Magnetic Resonance Images. IEEE Transactions on Medical Imaging. 18(9), (1999) 737-752 [15] Pham D.L.; Prince J.L.: An Adaptive Fuzzy C-Means Algorithm for Image Segmentation in the Presence of Intensity Inhomogeneities. Pattern Recognition Letters. 20(1), (1999) 57-68 [16] Gonzalez R.C.; Woods R.E.: Digital Image Processing. Prentice Hall Ed. [17] Nelder, J. A. and Mead, R. A Simplex Method for Function Minimization. Comput. J. 7, 308-313, 1965. [18] Arnold JB; Liow J-S; Schaper KS; Stern JJ; Sled JG; Shattuck DW; Worth AJ; Cohen MS; Leahy RM; Mazziotta JC; Rottenberg DA. Quantitative and Qualitive Evaluation of Six Algorithms for Correcting Intensity Non-Uniformity Effects. Neuroimage (2001) 13(5) 931–943. [19] Bezdek J.C. :Pattern Recognition with Fuzzy Objective Function. Plenum Press 1981. [20] www.fil.ion.ucl.ac.uk/spm/software/spm2/
1772
