DETECTION OF THE MIDSAGITTAL PLANE IN MR IMAGES USING A SHEETNESS MEASURE FROM EIGENANALYSIS OF LOCAL 3D PHASE CONGRUENCY RESPONSES Ricardo J. Ferrari*, Carlos H. Villa Pinto, Camilo A. Ferri Moreira BipGroup - Department of Computer Science Federal University of S˜ao Carlos S˜ao Carlos, SP, Brazil *
[email protected] ABSTRACT The midsagittal plane (MSP) separates the cerebrum into left and right hemispheres and its detection has a number of useful applications in brain image processing. We propose an automatic technique for the detection of the MSP in magnetic resonance (MR) images that uses a sheetness measure obtained from eigenanalysis of local matrix of second-order moments of 3D phase congruency responses to determine those voxels most likely to belong to the MSP. A weighted least-squares fitting algorithm is used in a coarse-to-fine iterative manner to find the best fitting plane (the MSP) through the selected voxels. Unlike most of the proposed approaches, which are mainly based on symmetric measures of the brain, our technique uses a direct measure to find the MSP. Our technique was applied to 202 MR images (40 clinical and 162 synthetic) and it has shown to be very effective for both symmetrical and asymmetrical brain images. Quantitative assessment, using the angle between unit normals of the detected and reference MSPs, yielded to a mean absolute angle value bellow 0.5◦ . Index Terms— Midsaggital plane, 3D Phase Congruency, Principal Moments of Inertia, Sheetness Measure 1. INTRODUCTION Based on the assumption that the human brain exhibits a high level of bi-fold symmetry, one common approach used to detect brain pathologies and structural changes in Magnetic Resonance (MR) images is to compare the left and right brain hemispheres in the search for violations of a devised symmetric measure. In fact, symmetry is routinely employed by neuroradiologists to assist their assessment of brain images and therefore there is no surprise why researchers aim to mimic such analysis. However, in order to assess cerebral hemispheric symmetry in brain images, detection of the midsagittal plane (MSP) is required. The MSP is the plane formed from the medial longitudinal fissure (LF) that separates the cerebrum into left and right hemispheres. Reliable automatic detection of this plane has a number of useful applications in brain image processing such as image registration, asymmetry analysis applied to detection of brain tumors and quantification of rates of atrophy in Alzheimer’s disease. This study proposes an automatic technique for the detection of the MSP in MR images. Unlike other approaches that rely mainly on brain hemisphere symmetry, our method uses a voxel-based sheetness measure, devised from eigenanalysis of a 3 × 3 × 3 matrix of 3D phase congruency (PC) responses, to determine those voxels most likely to belong to the MSP.
2. RELATED WORK Due to the importance and use of brain asymmetry analysis [1], a number of automatic methods have been proposed in the last decade to detect the MSP in 3D medical images. Volkau et al. [2] proposed to extract the MSP from 3D neuroimages using a two stages method that is based on the Kullback and Leibler’s divergence (KLD) to measure brain symmetry. First, slices along the sagittal direction are analyzed with respect to a reference slice to determine the coarse MSP. Then, the MSP is calculated by using a local search algorithm. Their algorithm was quantitatively assessed on multiple modality data sets (MRI and MR angiography (MRA)) using the angular and distance errors between the detected MSP and the ground-truth plane. Results have shown an average distance of 1.25 pixels and angular deviation of 0.63°. Puspitasari and Volkau [3] extended an earlier algorithm [2] for the MSP identification from MRI for CT images by estimating patient’s head orientation using model fitting, image processing, and atlas-based techniques. Their algorithm was validated on 208 clinical scans acquired with slice thickness ranging from 1.5 to 6 mm and severe head tilt. According to the authors, the proposed algorithm is robust to head rotation, and correctly identifies the MSP for standard clinical CT scans. In [4] a bilateral symmetry measure (SM) computed from edge features was proposed to extract the MSP in brain images. The SM is essentially a score that indicates how similar the left and right sides of an edge image are with respect to a candidate cutting plane. The image edges were obtained by using a 3D Sobel operator, followed by application of a binary threshold technique. A total of 164 clinical images were used in their experiments, including CT and MR (T1-w and T2-w) images. Results were assessed using the average distance along the z axis for all points (x, y) inside the image between the ground-truth plane and the detected plane. To detect the MSP, Jayasuriya and Liew [5] used the fact that the LF is the only major plane that appears darkest in Tl-w MR images. First, image centroid was computed on each individual 2D slice. Next, a straight line that passes through the centroid was rotated from 0◦ to 180◦ in 1◦ intervals and an intensity score (summation of intensity values along the line) was computed. Then, after the full line rotation, the rotation angle given the minimum score was used to identify the LF on that slice. Finally, the best plane was defined by taking the mode of the angles obtained from all slices. The algorithm was tested on 12 T1-w axial MR scans. Kuijf et al. [6] used the method proposed in [2] to initially detect a MSP candidate that was further deformed to represent the midsagittal surface. The midsagittal surface was represented as a
bicubic spline with control points equally placed in a regular grid on the MSP candidate. The control points were allowed to move in the left-right direction in order to optimize (using the KLD as a cost function) the surface configuration. MR images of fifty subjects with brain torque and fifty random subjects from patients with symptomatic atherosclerotic disease were used to assess the method. The extracted midsagittal planes and surfaces were compared to manual delineations by assessing the absolute volume of misclassified cerebrum tissue. Schwing and Zheng [7] proposed to extract the MSP in 3D brain MR images using an hierarchical image landmark detection technique. Their method starts by first detecting a global object using a marginal space learning [8] technique, which permits to infer initial estimates of five important brain landmarks. After that, the MSP is obtained by fitting a plane to the detected landmarks using a least squares minimization technique. Experiments to validate the method were conducted on 509 volumes, including male and female patients of all ages with various diseases. Quantitative assessment was performed using the plane normal error and the in-plane error on 40 expert-annotated data sets. Huisi Wu et al. [9] proposed a method to extract the MSP from brain images that uses the scale-invariant feature transform (3D SIFT). Different from other methods, which rely on gray level similarity, 3D edge registration and parameterized surface matching to determine the MSP, their method works by matching distinctive 3D SIFT feature points, following by an iterative least-median of squares plane regression. The assessment was conducted by using synthetic images simulating brain tumors and 136 clinical images of patients with stroke, Alzheimer and tumors. As indicated by the authors, their method cannot perform well for data with high levels of noise due to the difficulty in extracting consistent and stable 3D SIFT feature points.
with scale s and orientation Θ. S is the number of scales and T is an estimated threshold value [13] controlling the noise level of the amplitude responses. The small constant ε is used to avoid division by zero. The term W (u, Θ) is a weighting function that penalizes frequency distributions that are particularly narrow and is defined as
3. METHODOLOGY
where ωi is the central radial frequency of the filter i and σβ is the standard deviation controlling the filter bandwidth [15]. ω is a point in the frequency space expressed in Cartesian coordinates. To obtain filters that are all geometric scaling of a reference filter, the term σ ηβ = ωβi must be held constant for varying ωi [12]. The central radial frequency of each scale is defined as ωi = ωmax/mS−1 , where m is a multiplicative factor that scales between center frequencies of successive filters, ωmax is the maximum radial frequency center of the filters. To extend the filters to 3D, a Gaussian on the angular distance is used to control the spread of the angular frequency component as
In this study, we propose an algorithm for detection of the MSP in brain MR images that uses a voxel-based sheetness measure, devised from eigenanalysis of a 3 × 3 matrix of 3D PC responses, to determine those voxels in the image most likely to belong to the MSP. First, the sheetness measure is computed for all image voxels, resulting in a map of intensities ranging from 0 to 1 that clearly enhances the LF voxels. Next, a weighted least squares algorithm is used (in a coarse-to-fine manner) to find the best fit plane using the positions of the enhanced voxels from the sheetness intensity map. A detailed description of the proposed method is given as follows. 3.1. 3D Phase Congruency The PC model provides a measure of feature significance that is invariant to intensity and contrast variations of an image [10, 11]. A bank of quadrature pairs of 3D log-Gabor filters, as proposed by Dosil et al. [12], was used in this work to implement the 3D PC measure as PC(u, Θ) =
∑Ss=1 W (u, Θ) max [0, As (u, Θ)∆Φs (u, Θ) − T ] , ∑Ss=1 As (u, Θ) + ε
(1)
where u = (x, y, z) is a spatial location and Θ = (θ , φ ) represents the filter orientation on a sphere of unit radius, with θ ∈ (0, π/2] and φ corresponding, respectively, to the elevation and azimuth angles. As (u, Θ) is the amplitude response of the filter at location u, computed by convolution of the MR image with a 3D log-Gabor filter
W (u, Θ) =
1
h i, n An (u,Θ) γ c− N1 A∑max (u,Θ)+ε )
(2)
1+e
where γ = 15 and c = 0.45 are, respectively, a gain factor and a cutoff value. To improve localization on blurred features, a new phase deviation expression was proposed by Kovesi [14] as ∆Φs (u, Θ)
=
cos(φ s (u, Θ) − φ (u, Θ))− sin(φs (u, Θ) − φ (u, Θ) ,
(3)
where φs (u, Θ) and φ (u, Θ) are, respectively, the phase angle and the mean phase angle, computed over all scales. This formulation takes into account the fact that, at a point of PC, the cosine of the phase deviation should be large and the absolute value of the sine of the phase deviation should be small. 3.2. Design of the 3D log-Gabor filters The design of the bank of 3D log-Gabor filters [12] as well as the image filtering were conducted in the Fourier domain. In 1D, a logGabor transfer function has the form ln2 ωωi , ω )|ωi ,σβ = exp − R(ω (4) σ 2 ln2 ωβi
ω , Θ)2 α(ω ω , Θ)|ηα = exp − G(ω 2 , π/N 2 ηαa,0
(5)
ω ·n ω , Θ) = arccos( kω where α(ω ω k ), with n = (cos θ · cos φ , cos θ · sin φ , sin θ ). Parameter ηα controls the angular standard deviation of the filter, once the number of azimuth angles in the equator of the unit radius sphere, Na,0 , is fixed. Finally, the 3D log-Gabor transfer function is obtained by multiplying Equations 4 and 5 as ωk 2 kω 2 ln ω , Θ) ωi α(ω − ω , Θ)|ωi ,ηβ ,ηα = exp − H(ω 2 . (6) σβ π/N 2 2 ln ωi 2 ηαa,0
In our implementation of the bank of 3D log-Gabor filters, eleva-
tion is uniformly sampled while the number of azimuth angles (Na ) decreases with elevation in order to keep constant the density of filters [12]. Therefore, the number of filters varies with the elevations as ( 1 ; if Ne > 1 and i = Ne − 1 Na = , (7) Na,0 · cos(i · 2(Nπ−1) ) ; otherwise e
where Ne is the number of elevations. 3.2.1. Adjusting the filter bank parameters Filter parameters were experimentally adjusted to provide minimal overlap between adjacent isotropic filter responses with an uniform middle-spectral coverage. Parameter values that best approximate these requirements were: S = 5, Na,0 = 6, Ne = 4, ωmax = 0.2 cycles/pixels and ηα = 1.2 (providing filters with an angular standard deviation equal to 25◦ ). By setting ηβ = 0.745 and m = 1.6, filters with bandwidth of approximately 1 octave are obtained [14].
structures (sheet, tube and blob) mapped by the second-order tensor ellipsoid of the local structure. Here, eigenvalues are sorted in order of increasing value (λ1 ≤ λ2 ≤ λ3 ). The idea behind eigenanalysis of the MPC matrix (similar to the work of Descoteaux et al. [16, 17] using Hessian matrix) is to extract the principal moments of inertia in which the local structure of an image can be decomposed. Therefore, the following combination of the components to define the voxel-based sheetness measure is used, ! " !# −R2sheet (u) −R2blob (u) ϒ(u) = exp · 1 − exp · (11) 2α 2 2β 2 " !# " !# 2 (u) −Rtube −R2noise (u) 1 − exp · 1 − exp , 2γ 2 2δ 2 where α, β , γ and δ correspond to parameters controlling the sensitivity of each corresponding sheetness measure component. In this work, their values were set to α = δ = 0.15 and β = γ = 0.25. Experimental results have shown that the algorithm is robust with respect to slight changes of these parameters.
3.3. Sheetness measure To enhance voxels that are most likely to belong to the MSP, we consider a local matrix of second-order moments of 3D PC responses centered around each spatial location u in the image and define a sheetness measure based on the eigenvalues of this matrix. The local moment of order p + q + r is computed as p q (8) M pqr (u) = ∑ xΘ yΘ zrΘ , Θ
where xΘ yΘ zΘ
= = =
PC(u, Θ) cos θ cos φ PC(u, Θ) cos θ sin φ PC(u, Θ) sin θ ,
and, consequently, a 3 × 3 matrix written as M200 (u) MPC (u) = M110 (u) M101 (u)
(9)
of second-order moments can be M101 (u) M011 (u) . M002 (u)
M110 (u) M020 (u) M011 (u)
(10)
In order to reduce the effect of PC responses perpendicular to 2θ the sagittal plane, Equation 8 is multiplied by the factor 1 − π . This multiplicative factor reaches its maximum value when θ (the elevation angle) is zero and goes to zero when θ approaches 90◦ , i.e., on the equator of the sphere of unit radius. The relations among the eigenvalues of the MPC matrix, summarized in Table 1, were devised to distinguish between all three
Eigenvalue ratios
Sheet
Tube
Blob
Noise
Rsheet (u) = λ2 /λ3 Rblob (u) = (2λ3 − λ2 − λ1 )/λ3 q Rnoise (u) = λ12 + λ22 + λ32
0 2
1 0 √ λ3 3
// //
λ3
1 1 √ λ3 2
Rtube (u) = (λ3 − λ2 + λ1 )/λ3
1
0
1
//
0
Table 1: Relations of eigenvalues of the MPC (u) with local geometric structures.
3.4. Best fit plane technique In order to detect the MSP, a weighted least-squares (WLS) fitting algorithm [18] is applied, in a coarse-to-fine iterative manner, to the voxels enhanced by the sheetness measure in Equation (11). First, an approximated fitting plane is obtained by using all voxels with intensity values greater than 0.01. No weight is given to the voxels in this case. Then, the following two steps are performed iteratively: (i) define a searching region for the refined plane as a small margin of the current detected plane; (ii) fit a new plane using only the voxels inside the defined searching region. In this case, the sheetness measure values are used as weights by the WLS fitting algorithm. The iterative procedure, working with margin widths in decreasing steps of 2 voxels, stops when the searching margin, initially set to 21 voxels, is reduced to 3 voxels.
4. RESULTS AND DISCUSSIONS Results of the proposed technique for the detection of MSP on different brain MR images are presented in the next subsections.
4.1. Synthetic data (BrainWeb data set) A total of 162 synthetic normal brain MR images (T1-w, T2-w and PD acquisitions) from the publicly available McGill University BrainWeb MRI simulator [19] were processed by our proposed technique. The images were simulated with different slice thickness (1, 3 and 5 mm), noise level (0, 1, 3, 5, 7 and 9%) and intensity inhomogeneity (0%, 20% and 40%), resulting in 54 images for each MR image acquisition. Other three brain asymmetric images of Multiple Sclerosis were also used. For all processed images, our technique successfully extracted the MSP. For visual analysis, Figure 1(a)-(i) presents the worst case scenario, i.e., images corrupted by 40% of bias field and 9% of noise level, for the three image acquisitions. Even in this unfavorable situation, the MSP can be still detected in all three image acquisitions, including the PD image, which provides the lowest brain tissue contrast.
4.1.1. Quantitative assessment
(a)
(b)
(c)
(d)
(e)
(f)
Since all simulated image volumes from the BrainWeb simulator are in stereotaxic space, the midsagittal plane is perfectly parallel to the yz plane. Based on this fact, we compare the automatically detected plane and real MSP (ground-truth) using the angle between their unit normals. All 162 synthetic images were used in this experiment. Summary statistics calculated using all computed angles resulted in a mean absolute value (mav) of 0.4932◦ with standard deviation (std) of ±0.2548◦ and standard error (se) of ±0.0347◦ ; the median absolute value (medav) was 0.4086◦ . These statistics confirm the stability of our technique in the presence of noise and image inhomogeneity (bias field). Individual analyses were also conducted for T1-w, T2-w and PD images and the results were: mav (±std) = 0.3517 ± 0.0164◦ for T1-w, 0.4308 ± 0.0296◦ for T2-w and 0.6972 ± 0.3632◦ for PD; se = 0.0038◦ for T1-w, 0.0069◦ for T2-w and 0.0856◦ for PD; and medav = 0.3573◦ for T1-w, 0.4272◦ for T2-w and 0.6905◦ for PD. Results of the proposed technique have shown lower values for all statistics metrics of T1-w images. This can be explained by the fact that T1-w images most clearly differentiate brain gray matter, white matter and cerebrospinal fluid. 4.2. Simulating asymmetries and rotation
(g)
(h)
(i)
(j)
(k)
(l)
To test the proposed technique against asymmetric and rotated brain images, we randomly positioned in clinical images three ellipsoids of different sizes inside the brain region and also rotated the raw image data by different angles in x, y and z-axes. After that, the MSP detector was applied to the images and the results were visually assessed. Again, the MSP detector correctly detected the MSP in all images. An example is presented in Figure 1(j)-(l) for the purpose of visual analysis. 4.3. Clinical and pathological data Evaluation of the proposed technique on real clinical data was carried out by using 30 MRI cases (T1-, T2-weighted, and PD images) from the IXI database (available at: http://www.braindevelopment.org) and 10 cases of MS patients (FLAIR images) from the UNIFESP hospital (http://www.hospitalsaopaulo.org.br/quemsomos). The proposed technique maintained high robustness in detecting the MSP, even in images of pathological cases with a reasonable degree of brain asymmetry.
(m)
(n)
(o)
5. CONCLUSIONS
(p)
(q)
(r)
Fig. 1: Results of the proposed technique applied to different MR image contrasts. 1st, 2nd and 3rd columns correspond, respectively, to the axial, coronal and sagittal views. (a)-(i) synthetic BrainWeb images, corrupted by 9% of noise and 40% of bias field; (j)-(l) images with synthetic ellipsoids and rotation applied; (m)-(o) image of a brain tumor patient; (p)-(r) image of a MS patient.
We proposed a new MSP detection technique that is based on a voxel-based sheetness measure computed from the eigenvalues of the local matrix of second-order moments of 3D PC responses. Experiments with synthetic and real MR brain images (with and without asymmetries) from different acquisition (T1-w, T2-w, PD and FLAIR) were performed to prove the effectiveness of our technique. Although a more thorough investigation is required, quantitative results using synthetic data with precise indication of the MSP position have shown that our technique can produce accurate results comparable to the state-of-the-art methods for extraction of the MSP in MR images. As an improvement, research has been conducted to extend our technique to deal with brain images of subjects with a considerable amount of naturally occurring brain torque, in which the midsagittal surface deviates to a large extent from a plane.
6. ACKNOWLEDGMENTS
[13] P. Kovesi, “Phase congruency: A low-level image invariant,” Psychological Research, vol. 64, pp. 136–148, 2000.
The authors are thankful to FAPESP (grants nº 2014/11988-0, 2015/08288-9 and 2015/02232-1) and CAPES for the financial support given to this research.
[14] P. Kovesi, “Image features from phase congruency,” Videre: Journal of Computer Vision Research, vol. 1, no. 3, pp. 2–26, 1999.
7. REFERENCES
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