Understanding Phase Maps in MRI: A New Cutline Phase Unwrapping Method

Transcription

1 966 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 Understanding Phase Maps in MRI: A New Cutline Phase Unwrapping Method Sofia Chavez, Qing-San Xiang*, and Li An Abstract This paper describes phase maps. A review of the phase unwrapping problem is given. Different structures, in particular fringelines, cutlines, and poles, contained within a phase map are described and their origin and behavior investigated. The problem of phase unwrapping can then be addressed with a better understanding of the source of poles or inconsistencies. This understanding, along with some assumptions about what is being encoded in the phase of a magnetic resonance image, are used to derive a new method phase unwrapping which relies only on the phase map. The method detects cutlines and distinguishes between noise-induced poles and signal undersampling poles based on the length of the fringelines. The method was shown to be robust to noise and successful in unwrapping challenging clinical cases. Index Terms Magnetic resonance imaging, phase maps, phase unwrapping, signal processing. I. INTRODUCTION PHASE maps can contain very important inmation in many fields of study. Since the magnetic resonance imaging (MRI) signal is composed of real and imaginary components [1], a phase map can, in theory, always be derived. Inmation, such as field inhomogeneity or the velocity of blood flow, can be encoded in the phase of the MRI signal [2]. However, due to the ambiguity in interpreting the inmation conveyed by the phase of the signal, the magnitude of the complex MRI signal is the inmation that is primarily used clinical diagnostics. This will inevitably result in the loss of important inmation which is and can only be encoded in the phase of the signal. The phase of any complex signal represents a rotation, with direction and amplitude. However, given any complex data, the phase can only be derived as modulo 2 which gives a scalar value (known as the principal value) contained in a given range, usually and the true phase rotation cannot be known unambiguously from a single scalar value. For example, rotations of and are indistinguishable since falls outside the unambiguous range and is hence wrapped around to give a principal value of. This wrapping procedure consists of adding or subtracting a multiple of 2 such Manuscript received June 28, 1999; revised March 6, This work was supported in part by a grant from the Whitaker Foundation. The Associate Editor responsible coordinating the review of this paper and recommending its publication was Z. P. Liang. Asterisk indicates corresponding author. S. Chavez is with the Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada. *Q.-S. Xiang is with the Department of Physics and Astronomy, and the Department of Radiology, University of British Columbia, Vancouver, BC V6T 1Z1, Canada ( xiang@physics.ubc.ca). L. An is with Schlumberger Technology Corporation, Sugar Land, TX USA. Digital Object Identifier /TMI that the result is contained within. If phase maps are to convey any useful inmation, the exact direction and amplitude of rotation of the signal in one pixel with respect to that in another must be known. Any error made in assigning a phase value to a pixel will result in an erroneous quantification of the underlying physics. For example, flow may be misinterpreted as going in the wrong direction and with the wrong magnitude (off by a factor of three) if a rotation of is not distinguished from a rotation of. When phase maps contain no wrappings, each principal value indicates the exact amplitude and direction of rotation. These types of maps will not be further discussed since they already contain readily accessible inmation, just like magnitude maps. This paper begins in Section II by describing the characteristics and properties of wrapped phase maps in general and then, in Section III, MRI phase maps are discussed. Such maps must be well understood in order to appreciate the problem of phase unwrapping. In Section IV, the motivation that led to the new unwrapping method is given along with a description of the steps involved. The ideas this new method were first presented at the ISMRM meeting in 1997 [3]. Results are presented in Section V followed by a discussion in Section VI and conclusion in Section VII. II. PHASE LOOPS: POLES Wrapped phase maps may contain two types of borderlines called fringelines and cutlines [4]. It is important to distinguish between these structures when trying to understand the inmation contained in a phase map. Fringelines are borderlines between two adjacent pixels where phase wrapping appears to occur since it denotes a boundary. However, the principal phase values on either side of a fringeline may in fact be the true phase values and hence represent a physical reality. In such a case, this fringeline will not have resulted from a wrapping procedure and will also be a cutline. Cutlines are borderlines between two adjacent pixels where the signal has undergone a relative rotation of more than (in either direction). Cutlines are often more difficult to discern than fringelines since they do not necessarily follow an obviously contrasting line. In most cases, fringelines and cutlines will overlap in some areas but not in others. When no cutline is present, it is straightward to determine the exact value of the phase at any point by adding or subtracting a multiple of 2 each time a fringeline is crossed. This technique is known as fringe counting or fringe scanning [5]. This is due to the fact that a fringeline in such a case has been caused by the wrapping procedure so the true phase can be determined from /02$ IEEE

2 CHAVEZ et al.: UNDERSTANDING PHASE MAPS IN MRI: A NEW CUTLINE PHASE UNWRAPPING METHOD 967 Fig. 1. (a) Phase map simulation with closed fringeline and no poles. (b) The profile of (a) along the dotted line. The wrapped signal can be unwrapped in this case by adding 2 when crossing the fringeline from bright to dark. The true (unwrapped) phase difference between points A and B is less than. the principal phase value by an analogous unwrapping procedure. This is only possible when no cutlines are present so the true phase varies by less than between adjacent pixels. Such cases occur when a fringeline is closed with no open ends as in Fig. 1. However, the difficulty in phase map interpretation arises when a fringeline has two open ends due to the presence of a cutline as in Fig. 2. In these cases, the four pixels ming each of the ends result in phase loops known as inconsistencies or residues. The term pole will be used here to emphasize that the values and associated with such four-pixel structures can be of either sign. In fact, one end of an open fringeline is a positive pole and the other is a negative pole. Such a pair will be referred to as a dipole structure (see Fig. 2). If the boundary conditions are such that the phase map is connected at both ends, the image has the topology of a torus as shown in Fig. 3. This is the realistic case MRI since the complex image is the result of a Fourier Transm. In such a case, there are always an equal number of positive and negative poles since all open fringelines have both ends defined within the phase map. For all further discussions, all poles considered belong to a dipole (no monopoles [6] are present). An interesting feature of the dipole structures is that the pairing up of positive and negative poles can be affected by a global phase offset. Although a global phase offset will not affect the pole positions, it will affect their connections as the fringelines are shifted. In other words, a positive pole may no longer connect to the same negative pole a given phase offset. This is demonstrated in Fig. 4. Given a two-dimensional (2-D) phase map, a pole,, can be defined mathematically as a 2 2 pixel loop which the curl of the 2-D wrapped phase gradient is nonzero Fig. 2. (a) Phase map simulation with open fringeline and a dipole structure. (b) The profile of (a) along the dotted line. The wrapped signal cannot be easily unwrapped because errors will be made at the poles. The cutline causing the poles is the borderline between grey and dark region, assumed to terminate at the poles on either end. This cutline will cause the unwrapping to be path dependent. The true phase between points A and B is greater than. Fig. 3. The boundary conditions in MRI are such that the phase map is connected at both ends resulting in a torus. Fig. 4. Phase maps derived from a single random complex field with various phase offsets: (a) 0, (b) 45, and (c) 90. The opposing poles of a given dipole,, change connection from (a) to (b) whereas those of another dipole,, change connection from (b) to (c). and is the wrapping operator which gives a result contained within the range by adding or subtracting a multiple of 2 to whatever is in the square brackets. More specifically where (1) (2) (3) (4) To evaluate this, the following relations of the wrapping operator can be used: (5) (6) (7)

3 968 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 where (8) and where the notation: indicates. The expression, thus, yields (9) where again. Equations (9) and (1) give the result:, corresponding to positive and negative poles. Hence, any 2 2 pixel loop will be a pole if 1 and it will not be a pole only the case 0 which according to (8) results only if Fig. 5. Zoom of a random phase map showing several poles (circled) and their fringeline connections (dashed borderlines). (10) Equation (10) can, theree, be referred to as the no pole condition. This mulation is analogous to that of conservative (irrotational) and nonconservative (rotational) fields in mechanics or electromagnetic theory and, thus, also leads to considerations of line integral path independence or dependence, respectively [7]. In particular, a closed-path line integration of a phase map will equal the sum of the enclosed poles. Theree, closed-path line integration of the phase gradient map is path independent (consistent) as long as no poles or an equal number of positive and negative poles are enclosed by the path. The cutline phase unwrapping methods ensure this by placing cuts between the dipoles (see Section IV). III. POLE SOURCES IN MRI As defined in the previous section, poles are present at the ends of the open-ended fringelines. Since poles result from a true phase difference of more than between the signal in two adjacent pixels, the borderline between these two pixels is a cutline. In fact, the shortest possible cutline is a single pixel edge in length with positive and negative poles at either end sharing two common pixels, one on either side of the edge. Theree, open-ended fringelines indicate the presence of a cutline which can be assumed to begin and end at the poles but the rest of it is not easily detectable [4]. Open-ended fringelines or cutlines may be caused by at least three different factors in MRI and hence there are at least three different sources of poles. One such source is noise. Because the noise in MRI is a small random vector added to the complex signal, the noise will determine the phase in areas of low relevant signal. Since this type of noise is random, the phase in these regions will randomly fluc- tuate and an open-ended boundary may result. Fig. 4 shows a phase map corresponding to a smoothed noise field. Several poles are indicated by the circles and squares. In addition, Fig. 5 shows a zoomed section of a purely random phase map where several poles are circled and their corresponding fringelines are indicated by the grey dashes. In fact, it can be shown statistically that in a purely random noise phase field of pixels, it is expected that there will be approximately poles, with half positive and half negative (see the Appendix). In such a case, it does not make sense to assume this is due to wrapping since in the noise, the principal phase value can be considered the true phase value. The result is that in noisy regions, all fringelines are cutlines and no unwrapping needs to be done. Another source of poles is spatial undersampling of the phase. This will result when the phase sensitivity and spatial resolution of the imaging sequence are such that two adjacent pixels have a true signal rotation of more than. In theory, most physical fields mapped by the phase of the MRI signal are continuous. This means that, in theory, a fine enough spatial sampling will always eliminate such large phase differences between adjacent pixels. However, in practice this is not the case since many other factors will limit the spatial resolution of the MR image. Fig. 6 demonstrates how poles appear when the sampling resolution of a continuous function is made very coarse. There are at least two different cases in MRI, when these types of undersampling poles can be seen. The first case is that of velocity-encoded imaging, where the flow is encoded in the phase of the signal. Such a case is shown in Fig. 7. Such poles are usually avoided by reducing the flow sensitivity of the imaging sequence. However, the reduced flow sensitivity can cause the loss of other important inmation by reducing the range of blood flow velocities that can be distinguished.

4 CHAVEZ et al.: UNDERSTANDING PHASE MAPS IN MRI: A NEW CUTLINE PHASE UNWRAPPING METHOD 969 Fig. 6. Four phase maps are shown with corresponding central horizontal profiles under them. A continuous function in (a) is sampled with different resolutions: (b) 16 pixels, (c) 32 pixels, (d) 42 pixels. Poles only appear in (d) as indicated by the circles and the larger than phase jumps (j1j >) between adjacent pixels. The other case of undersampling poles is due to main field inhomogeneity. This inhomogeneity may result from magnetic susceptibility differences such as those that occur at tissue air interfaces. An imaging sequence that uses a gradient echo is sensitive to this inhomogeneity and, thus, may result in a discontinuous phase map, where the phase of the signal in adjacent pixels varies by more than. Very nice gradient echo images demonstrating such open-ended fringelines with undersampling poles was obtained by Reichenbach et al. [8, figure 5]. The main distinction between poles arising from undersampling and those arising from noise is in the length of the fringelines connecting the dipoles. In regions of low signal-tonoise ratio (SNR), dominated by noise, the phase coherence is also low since the values of the pixels are random. This means that a boundary appearing somewhere does not affect the probability that adjacent pixels will also have a boundary. The result is that, in such regions, a fringeline will not tend to be very long. In contrast to this, in regions of high SNR, the phase coherence is high since the phase is dominated by the relevant signal which is in general, a slowly varying continuous field. In such a case, if there is a boundary at one location, there is a higher chance that the regions around it will also contain a boundary. The result is that in regions of high SNR, the fringelines tend to be longer than those in the noisier regions. Fig. 7. Velocity-encoded phase map of blood circulating in the heart. The fringeline is open-ended. One pole of either sign is present at the fringeline ends. IV. UNWRAPPING METHODS A. Review of Phase Unwrapping In order to access the inmation contained in a wrapped, principal value phase map, it is often necessary to first unwrap it successfully. The problem of phase unwrapping can be stated mathematically as finding given and the fact that where is the wrapping operator, is an integer and. There-

5 970 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 path-following method phase unwrapping. Since it belongs to the first category of phase unwrapping methods, the second category will not be further discussed. The goal is not to compare its permance to that of other methods. Comparisons of the permance of different methods can be found elsewhere [7], [10]. Part B is not intended to be an exhaustive review of path-following methods but rather a brief discussion in order to motivate the development of this new method while setting its place amongst the long history of phase unwrapping work. B. Path-Following Methods Fig. 8. (a) Phase map representing flow in a longitudinal direction. (b) Phase map with streaks resulting from conventional phase unwrapping algorithm. e, the problem is basically one of finding the correct value at every pixel. If the true phase difference between adjacent pixels is contained within the unambiguous range, this problem becomes quite trivial: a line integration of the gradient (phase difference between adjacent pixels) along any path, will result in an unwrapped phase map. Furthermore, at every pixel can be determined from the value of the resulting unwrapped phase map if is known the starting point of the path of integration. For example, if the starting point the gradient integration is taken to be a pixel which 0 and, no correction will be necessary and the resulting unwrapped phase map from the integration represents the true phase map. If 0, all the pixels in the resulting unwrapped map will be shifted by 2 where is the number of cycles by which the initial pixel value is wrapped. This is the most basic and straightward method to phase unwrap and is what is commonly used to solve this problem. However, if the phase of adjacent pixels does not obey the condition, such a straight ward solution will fail. This is when the problem of phase unwrapping becomes challenging and the focus of many publications. The reason the condition must be met is due to the fact that this is implicitly assumed when integrating the gradient since the gradient can only be known modulo 2. When this condition is not met, the result of the integration becomes path dependent. In such cases, integrating across neighboring pixels which is outside the range will cause errors to propagate across the rest of the unwrapped image which will then be off by some unknown multiple of 2. This is what sometimes appears as streaks if the integration is permed along rows or columns. Fig. 8 shows a typical phase map image that results when the straightward unwrapping approach fails. Most of the work in phase unwrapping has been done in other fields such as optical engineering. In particular, most new phase unwrapping developments are synthetic aperture radar (SAR) interferometry. However, the applications can easily be extended to MRI. Good reviews of phase unwrapping methods are given by Judge and Bryanston-Cross [9] and Ghiglia and Pritt [7]. The present methods of phase unwrapping can be grouped into two major categories: i) path-following methods and ii) minimum normalization methods [7]. This paper presents a new Path-following methods deal with the idea of guiding the integration of the gradient map so as to avoid any error propagation. More specifically, these methods rely on integrating the gradient map along a particular path which avoids using any true phase differences which are outside the unambiguous range to unwrap. In this category are the disk growing method [11], cut-line methods [12] [14] and region grow methods such as minimum and maximum spanning tree methods [15] [17]. The temporal phase unwrapping proposed by Xiang [18] can also be included in this category since it proposes to integrate the phase differences along another dimension in order to avoid integrating across a true spatial phase jump outside the range. This method will only work if the familiar condition: is obeyed pixels in temporally adjacent frames. This condition along with motion between temporal frames will further restrict and complicate the application of this procedure [19]. Path-following phase unwrapping methods often result in streak-free phase maps. This is because their main goal is to block error propagation although the result may not always be correct. As long as the poles are connected or blocked somehow from entering the unwrapping process, any integration path, starting at the same initial pixel, will give the same result (be consistent, see Section II). This result will, however, be dependent on the exact placement of cuts or criterion used to grow. As appreciated by Ghiglia and Pritt [7], the choice of cuts is not obvious, some criterion must be used. Given the description of fringelines and cutlines in Section II, it is clear that in order to obtain the true phase map, unwrapping should be blocked along the cutlines. This is because it is along these borderlines that true phase differences are outside the unambiguous range. This reveals the choice of terminology since cuts should be used to block the unwrapping along the cutlines. However, most cut-line phase unwrapping methods connect the poles and block the unwrapping but they do not explicitly look cutlines. Instead, many use arbitrary criteria like minimizing the cut lengths [7], [9] to determine the placement of the cuts. The reason so many of these methods are often successful is because they deal with noise induced poles. As discussed previously, noise poles will have short fringelines/cutlines associated with them so the minimization of length criterion is justified. As long as all the poles in a phase map are the result of noise, cuts connecting these along shortest paths result in good representations of the true phase map. This is due to the fact that even if the cuts do not follow the cutlines closely, only small areas of the phase map are affected. However, most of these methods will fail

6 CHAVEZ et al.: UNDERSTANDING PHASE MAPS IN MRI: A NEW CUTLINE PHASE UNWRAPPING METHOD 971 to result in true phase maps if there are long range dipoles due to undersampling by either phase-encoded velocity or magnetic field inhomogeneity. It is in these cases, with longer cutlines, that any error in the placement of the cuts will affect a larger region of the map. Phase unwrapping methods which explicitly place cuts do not deal with such dipoles very successfully. Region grow methods, which guide the unwrapping without explicitly determining the placement of the cuts, have been shown to successfully unwrap phase maps with large order open-ended fringelines [14]. This is to be expected if the criterion used to guide the growing, such as minimum phase gradient, results in circumvention of the cutlines. In these cases, there will be effective cuts implicitly placed along the cutlines since there is a direct correspondence between the region grow path and the cuts [16]. The limitation of these methods is the speed [9]. The correspondence by An et al. [20] introduces a new implementation of a minimum spanning tree phase unwrapping method where the speed has been greatly improved, reducing it from order to. However, these methods are also difficult to track since the effective cuts are not as easy to see. This may make the result more difficult to understand. Flynn s phase unwrapping guided by a quality map [21] is very similar to the phase unwrapping presented here. It combines the ideas of cut-line methods with region grow methods. Region grow methods use some other inmation about the data contained in the phase map (quality map) to help guide the growing [7]. Cut-line methods place explicit cuts between the dipoles to avoid path-dependent results. The idea behind Flynn s quality guided method is to choose the cut placements according to a quality map so that not only the poles are blocked from the unwrapping path, but additional inmation, is also incorporated. The main difference between this method and the method presented in this paper, is that the quality map used by Flynn is magnitude-based whereas the score map proposed here is only phase dependent. The pole-guided-cutline method can be seen as a generalization of Flynn s method to be used whenever a magnitude-based, quality map is not readily available. Ghiglia and Pritt [7] also present a hybrid phase unwrapping algorithm that uses a quality map to guide the placement of cuts. It is called the mask cut algorithm. In their text, several phase-based quality maps are suggested. However, the score map proposed in this paper differs from those quality maps in that it differentially treats the regions of the image. In doing so, the score map provides unique inmation: it highlights the relative phase gradients only in regions that are relevant the placement of cuts. In other words, although a region of the image may contain large phase gradients, it will not be highlighted in the score map if no long-range dipoles need to be connected in that area of the image. No other quality map seems to provide this type of differential phase gradient inmation. It is believed that the robustness of the score map to the effects of noise (see Section VI) can be attributed to this differential treatment of the regions. This is in contrast to many of the quality maps suggested by Ghiglia and Pritt [7]. C. New Cutline Phase Unwrapping The understanding of the source of the poles led to a new idea finding the cutlines explicitly. The idea relies on a score map generated by tracking the fringelines which connect the dipoles. This score map is then eroded (thinned) so as to obtain the cuts. Kramer and Loffeld [4] suggested a similar (perhaps more complicated) fringline tracking procedure cutline detection. However, in contrast to the method presented in this paper where the fringeline tracking is used to produce a score map, Kramer and Loffeld used fringeline tracking to directly extract portions of cutlines. As pointed out in their discussion, this led to several limiting factors their method. Also, their paper concentrates on cutline detection without attempting to phase unwrap which can be a problem if the entire cutline is not found. A new method of finding the location the cuts can be derived from the understanding of the phase map as described in previous sections. This led to the new phase unwrapping method which will be referred to as a pole-guided-cutline phase unwrapping method. As described above, poles are point-like (2 2) inconsistencies present in phase maps. Nonetheless, it is well appreciated that removal of the poles (i.e., replacement of the pole pixels by a fixed value) alone is not sufficient to guarantee a streak-free phase unwrapping result [22]. In fact, removal consecutively creates adjacent poles along a path connecting positive and negative poles until they superimpose and cancel in a process called annihilation. This phenomenon is the basis the cutline detection that is proposed in this paper. The first step is to distinguish between noise dipoles and those which are longer-range and may be due to undersampling. This is accomplished by tracking the fringelines and then classifying the poles according to their relative fringeline lengths. Fringeline tracking is done by consecutively chasing the poles. At each step a pole is removed (all four pixels are replaced by zero phase value), thus creating a new one in one of the eight possible adjacent 2 2 loops containing at least one of the original pole pixels. This process continues until no new poles are found at which point dipole annihilation has occurred. The result of the tracking will be a two-pixel-wide line of phase equal to zero which delineates the fringeline. A zoomed version of the phase map can be used these steps since otherwise, removal of a pole displaces all surrounding poles. If all fringelines in the image are tracked simultaneously, the result will be an initial reduction in the total number of poles as the noise dipoles are annihilating. When the total number of poles no longer decreases with pole removal, the leftover poles can be classified as longer-range and their corresponding fringelines treated accordingly. This will allow simple noise extraction without any thresholding. Furthermore, if this method results in any noise poles being mistaken long-range poles and treated accordingly this will not affect the result. It will only increase the amount of unnessary calculations. The cuts connecting noise poles can be determined from a single fringeline tracking procedure since the fringelines are cutlines in the noise. However, determination of the cuts connecting the other dipoles will require a lot more work. The longer cutlines the other long-range dipoles are determined by repeating the fringeline tracking several global phase offsets and superimposing the resulting lines. The idea, as also stated by Kramer and Loffeld [4], is that those parts of the fringeline connecting a given dipole that do not vary much with

7 972 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 Fig. 9. Four velocity-encoded images I, II, III and IV. The original phase maps are shown in (a). Unwrapped results from the pole-guided-cutline method are shown in (b). phase offset are part of the cutline. This is due to the fact that the smaller the phase gradient in a region, the more the phase offset will cause the fringeline to vary in that region. The converse is true as well. The sought cutline will then be visible in the superimposed map, which can now serve as a score map, as the most highlighted connection of the dipoles. This line can be extracted by first region growing from one pole along the regions of higher signal (more overlap) until a pole of opposite polarity is reached and an island is med. Then, an erosion procedure described by Xiang et al. [23] is used to convert the island into a thin cut. Alternatively, the mask-thinning algorithm described by Ghiglia and Pritt [7] or Flynn [21] their mask cut phase unwrapping algorithms could be used. A more efficient implementation of the described fringeline tracking and phase offsetting is to change the phase of the value used pole removal with no need to do a global phase offset. In other words, instead of setting the pixels to zero phase, they are consecutively set to another value where and the number of overlapping fringelines. Once the cuts are all determined, unwrapping can proceed in any fashion as long as it is blocked at the cuts. The implementation used in this paper is to unwrap into but not past the cuts. If some pixels are surrounded by cuts or if the cuts are thick (greater than three pixels wide) in some areas, there will be some left-over, not unwrapped pixels at the end of all the unwrapping. This will occur mostly in noisy regions. These pixels can be unwrapped at the very end, layer by layer. The result will be some unwrapping in the noise but it will not corrupt the result. as in II where all the high signal fringelines are closed. Such a case is very simple to unwrap since no score map and long-range cutline determination is necessary. The fringelines in the noise can simply be used as cuts the unwrapping. Fig. 10 shows the different stages of the cutline method one particular example (Fig. 9 I). The original velocity-encoded phase map and the resulting unwrapped map are shown in (a) and (b) of Fig. 9 I, respectively. Once the classification of poles has been done, the location of long-range dipoles can be used to zoom in on the relevant part of the original phase image. This relevant region is shown in all images of Fig. 10. In (a), arrows point to the long-range dipoles with white indicating positive and black negative. Many of these dipoles are clearly undersampling dipoles since they occur at the ends of high signal (flow) fringelines. However, a fringeline in the noisy region (right) was seen to have a similar length and hence the associated dipoles are considered in the long-range cutline treatment. As previously mentioned, this will not affect the result but only extend the processing time. Noise dipole fringelines (cuts) resulting from fringeline tracking are shown in (c). The interesting score map that is obtained by overlaying the dipole fringelines corresponding to various phase offsets is shown in (d). This map is used to score the pixels when perming the region grow and score-guided-erosion (SGE) described by Xiang et al. [23] since the brightest pixels should be used to find the cuts. Image (e) gives the resulting cuts which can be seen to follow the brighter parts of (d) while connecting the dipoles pointed out in (a). The cuts shown in (c) and (e) are used to obtain the unwrapped result in (b). V. RESULTS In this section, some results from the previously described phase unwrapping method are presented. Fig. 9 shows four original phase maps (I IV) with the results of the pole-guided-cutline phase unwrapping method. It is interesting to note that the cutline method works even when there are no long-range dipoles VI. DISCUSSION The better understanding of phase maps presented in this paper is based on a better understanding of the pole structures. Although the aim here was to exploit this knowledge in order to successfully phase unwrap, such an understanding of phase maps could be extended to many other applications. Since poles

8 CHAVEZ et al.: UNDERSTANDING PHASE MAPS IN MRI: A NEW CUTLINE PHASE UNWRAPPING METHOD 973 Fig. 10. (a) Velocity-encoded phase map with long-range positive (white) and negative (black) dipole locations indicated by arrows. (b) Unwrapped result using the pole-guided-cutline method on (a). (c) Noise dipole cuts. (d) The score map obtained by overlaying long-range dipole fringelines various phase offsets. (e) Cuts resulting from the score-guided-erosion (SGE) permed on (d) given the dipole locations indicated in (a). are always present in the noise (in fact, there are one third of the number of noise pixels) their localization could be used to distinguish between noise and signal in other applications such as tissue segmentation or statistical estimates of SNRs etc. This pole-guided-cutline phase unwrapping method relies on the score map to obtain the cuts but even if there is a lot of noise corrupting the image, the score map produced should still be useful since all the dipoles are connected at every phase offset step. The noise is not expected to disrupt the final connectivity between dipoles as was the case [4]. This is also in contrast to the mask cut method [7], [21] which relies on a good quality map and fails when the quality map is too corrupted. To test the robustness of the pole-guided-cutline phase unwrapping method presented in this paper, the effects of noise were studied. A simulated phase map was produced with varying levels of Gaussian noise added to the corresponding real and imaginary maps. The SNR is calculated, in a flat region of the magnitude image, as the average signal divided by the standard deviation. The unwrapped results are shown in Figs. 11 and 12. For the first simulation shown in Fig. 11, there is no cutline so all poles are due to noise. The pole-guided-cutline method result was not affected by the noise. In Fig. 12, there is a long-range cutline present along the bottom of the circle and again, the poleguided-cutline method result was not affected by the noise. In conclusion, the detection of the long-range cutline connecting undersampling poles is immune to the effects of noise, making the pole-guided-cutline method very robust. An important limitation of any 2-D phase unwrapping method is the inability to unwrap a fully wrapped region, with no detectable fringelines. If the three-dimensional (3-D) inmation is available, either CINE velocity-encoded images with inmation or field inhomogeneity mapping with dimensions, unwrapping in three dimensions is expected to improve the results. One example of such a fully wrapped region in the plane is that of a fully wrapped vessel in CINE velocity-encoded MRI as seen in the lower part of Fig. 9 III. For such a case, no poles will be detected in the plane so there will be no fringeline tracking in the signal. This is the same as what happens in the closed fringeline case (such as in Fig. 1) which can be successfully unwrapped by any straightward method if the noise is treated somehow. However, the fully wrapped vessel there is no good (unwrapped) signal to provide the inmation needed to successfully unwrap. If the fully wrapped vessel is treated in three dimensions, the problem can be solved. The unwrapped signal from temporal frames adjacent to a fully wrapped vessel frame can provide the necessary inmation temporal phase unwrapping [18]. This direction the unwrapping will be favored when avoiding large phase jumps in any of the three dimensions. This is because although no poles are found in the plane, a ring of poles and hence large phase jumps in the and dimensions can be detected all along the vessel edge as shown in Fig. 13. This ring of poles will lie parallel to the plane, at the border between and as well as between and. These poles will block any spatial unwrapping from the noise region into the vessel while allowing temporal unwrapping within the vessel. The pole-guided-cutline unwrapping method avoids areas of large phase jumps when perming the phase unwrapping integration. Extended to three dimensions, it should favor the time dimension as a path to enter the fully wrapped vessel since it is spatially surrounded by large phase jumps. The mulation of Section II can easily be extended to three dimensions since the

9 974 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 Fig. 11. Simulations of a wrapped phase map with no cutlines is shown in row (a). From left to right, the SNR deteriorates as indicated in the top row. Row (b) shows the result from applying the pole-guided-cutline phase unwrapping method to the corresponding image in (a). Fig. 12. Simulations of a wrapped phase map with a cutline along the bottom is shown in row (a). From left to right, the SNR deteriorates as indicated in the top row. Row (b) shows the result from applying the pole-guided-cutline phase unwrapping method to the corresponding image in (a). curl in (1) can be applied to a 3-D field. The result will be a pole vector with discretized components (either 0, 2 or 2 ) in each of the three directions. Three-dimensional poles can hence be searched in a 3-D phase map where fringelines are now fringesurfaces and cutlines are cutsurfaces. In theory, the same steps as those used the 2-D cutline phase unwrapping method can be used to find the surfaces which should block the unwrapping. The third dimension also has the potential of providing additional inmation about the dipoles. In two dimensions, it was shown that fringelines will change dipole connections depending on the phase offset (see Fig. 4). For the pole-guided-cut- line method, fringelines in the noise are treated as cutlines hence the noise dipoles are connected according to the fringeline connections with no phase offset. For the longer range, high signal dipoles, correct connections between the dipoles and placement of the cuts relies on the presence of a larger phase gradient that is outlined by the score map. In both cases, assumptions are made about the phase map in order to determine the dipole connections. However, if a series of 2-D phase maps is available, revealing the history of the appearance and annihilation of dipoles, the correct dipole connections can be directly observed without relying on any assumptions. In such a case, the third dimension will provide that extra inmation.

10 CHAVEZ et al.: UNDERSTANDING PHASE MAPS IN MRI: A NEW CUTLINE PHASE UNWRAPPING METHOD 975 Fig. 13. (a) Sketch representing the phase map of a wrapped vessel at three different moments in time: t ; t, and t with 3-D poles shown by the dashed rings. The vessel is fully wrapped at t. (b) The profiles shown in the first column represent the phase as a function of y. The second column shows there will be no poles detected in any of the three (x; y) phase maps. The third column shows that there will be poles present in the (y; t) map at the edge of the vessel. These poles will arise due to the large phase jumps between the different time steps the same point in space (as shown point A). If all possible poles are searched in three dimensions, the result will be a ring of poles lying along the edge of the vessel, parallel to the (x; y) plane, at the border between t and t as well as between t and t as shown by the dashed lines in (a). Fig. 14. Resulting pole rings in three dimensions showing the dipole connections. To study this, a 3-D random phase map was produced. This phase map consists of a series of 2-D phase maps varying gradually such that the first and last maps have no poles. The third dimension is considered to represent time. For such a 3-D map, 3-D poles were detected and the presence of a pole (in any direction) was flagged. The result was a series of rings floating in three dimensions, clearly connecting the dipoles as shown in Fig. 14. These connected poles in three dimensions would be separate in any 2-D slice. VII. CONCLUSION A better understanding of phase maps has revealed several interesting properties of such maps and the structures contained within them. The problem of phase unwrapping can indeed become quite challenging if poles are present in the phase map. Regions of pure noise will always result in poles so it is essential to have a good way to deal with poles when phase unwrapping. Although magnitude thresholding is commonly used to deal with the noise, this paper suggests that a better understanding of the phase map can lead to phase unwrapping that is independent of any such unreliable thresholding. The better characterization of phase maps has led to a new phase unwrapping cutline method that exploits many of the properties of such maps. This method was shown to be immune to noise effects and quite robust. It was successful in unwrapping challenging clinical cases on which a standard unwrapping methods would have failed. Future work should involve extension into a 3-D phase unwrapping method since the third dimension could provide the necessary inmation to overcome some of the limitations of any 2-D phase unwrapping method. APPENDIX For a purely random phase map with torus topology and pixels, there will be approximately poles in all. Furthermore, since such boundary conditions will lead to an equal number of positive and negative poles, there will be approximately positive poles and negative poles. To prove these statements, let be the

11 976 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 21, NO. 8, AUGUST 2002 probability distribution the wrapped phase difference of any two 4-connected pixels. Since the difference is wrapped, any value will be as likely to occur so will be a step function which, when normalized, can be written as &. (11) Using the no pole condition expressed in (10), let, and be the three wrapped differences involved in (10). These differences can be any three of the possible four associated with a 2 2 pixel loop. The probability of being a pole, either positive or negative,, will, thus, be equal to the probability of the sum of these three phase differences being equal to a value outside the range. Here, the unambiguous range is considered to exclude the end-point simplicity. In other words, given the normalized probability distribution the sum of three wrapped phase differences, written as simplicity, can be calculated as follows: (12) where indicates integration over the variable which is simply the sum:. Since the wrapping operator makes it such that and are all independent of each other, can be easily calculated given by simple convolution ( ) (13) where is given in (11). The result of the convolution, after normalization, gives (14) where. Evaluating (12) using (14) gives the result 1/3. It is clear from the symmetry of the graph shown in Fig. 15 that if and are the probabilities of being a positive pole and a negative pole, respectively, 1/6. If a random phase map, with torus topology, is made up of pixels, there will be a total of potential poles hence predicts that approximately poles will be present if is large. This result has been verified random phase maps. Fig. 15. Normalized probability distribution the sum of three wrapped phase differences: P where W = W + W + W. The shaded region represents the fractional area which will be equal to the probability of being a pole: P = 1/3. REFERENCES [1] M. Wood, Fourier imaging, in Magnetic Resonance Imaging, 2nd ed, D. D. Stark and W. G. Bradley, Eds. St. Louis, MO: Mosby Year Book, [2] I. R. Young and G. M. Bydder, Phase imaging, in Magnetic Resonance Imaging, 2nd ed, D. D. Stark and W. G. Bradley, Eds. St. Louis, MO: Mosby Year Book, [3] S. Chavez and Q. S. Xiang, 2D phase unwrapping based on dipole connections, in Proc. 5th Annu. Meet. ISMRM, Vancouver, BC, Canada, Apr. 1997, Paper [4] R. Kramer and O. Loffeld, A novel procedure cutline detection, Int. J. Electron. Commun., vol. 50, no. 2, pp , March [5] S. Nakadate and H. Saito, Fringe scanning speckle-pattern interferometry, Appl. Opt., vol. 24, no. 14, pp , [6] J. R. Buckland, J. M. Huntley, and S. R. E. Turner, Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm, Appl. Opt., vol. 34, no. 28, pp , Aug [7] D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping. New York: Wiley, [8] J. R. Reichenbach, R. Venkatesan, D. A. Yablonskiy, M. R. Thompson, S. Lai, and E. M. Haacke, Theory and application of static field inhomogeneity effects in gradient echo imaging, J. Mang. Reson. Imag., vol. 7, no. 2, pp , Mar./Apr [9] T. R. Judge and P. J. Bryanston-Cross, A review of phase unwrapping techniques in fringe analysis, Opt. Lasers Eng., vol. 21, pp , [10] J. Strand and T. Taxt, Permance evaluation of two-dimensional phase unwrapping algorithms, Appl. Opt., vol. 38, no. 20, pp , July [11] C. DeVeuster, P. Slangen, Y. Renotte, L. Berwart, and Y. Lion, Diskgrowing algorithm phase-map unwrapping: Application to speckle interferograms, Appl. Opt., vol. 35, no. 2, pp , Jan [12] R. M. Goldstein, H. A. Zebker, and C. L. Werner, Satellite radar interferometry: Two-dimensional phase unwrapping, Radio Sci., vol. 23, no. 4, pp , July/Aug [13] J. M. Huntley, Noise-immune phase unwrapping algorithm, Appl. Opt. Letters to the Editor, vol. 28, no. 15, Aug [14] D. J. Bone, Fourier fringe analysis: The two-dimensional phase unwrapping problem, Appl. Opt., vol. 30, no. 25, pp , Sept [15] D. P. Towers, T. R. Judge, and P. J. Bryanston-Cross, Automatic interferogram analysis techniques applied to quasiheterodyne holography and ESPI, Opt. Lasers Eng., vol. 14, pp , [16] N. H. Ching, D. Rosenfeld, and M. Braun, Two-dimensional phase unwrapping using a minimum spanning tree algorithm, IEEE Trans. Image Processing, vol. 1, pp , July [17] M. Takeda and T. Abe, Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: A comparative study, IEEE Trans. Image Processing, vol. 1, pp , July 1992.

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