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A Correlation-Based Approach to Calculate Rotation and Translation of Moving Cells Cyrus A. Wilson and Julie A. Theriot
Abstract—We present a noniterative image cross-correlation approach to track translation and rotation of crawling cells in time-lapse video microscopy sequences. The method does not rely on extracting features or moments, and therefore does not impose specific requirements on the type of microscopy used for imaging. Here we use phase-contrast images. We calculate cell rotation and translation from one image to the next in two stages. First, rotation is calculated by cross correlating the images’ polar-transformed magnitude spectra (Fourier magnitudes). Rotation of the cell about any center in the original images results in translation in this representation. Then, we rotate the first image such that the cell has the same orientation in both images, and cross correlate this image with the second image to calculate translation. By calculating the rotation and translation over each interval in the movie, and thereby tracking the cell’s position and orientation in each image, we can then map from the stationary reference frame in which the cell was observed to the cell’s moving coordinate system. We describe our modifications enabling application to nonidentical images from video sequences of moving cells, and compare this method’s performance with that of a feature extraction method and an iterative optimization method. Index Terms—Biological cells, biomedical image processing, image motion analysis, image registration, microscopy, motion estimation.
I. INTRODUCTION
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UANTITATIVE motion analysis of crawling eukaryotic cells is a challenging problem. Attempts to make sense of the information-rich image sequences by reducing them to simple parameters often fail to capture the biologically relevant phenomena under investigation. A specific experimental condition might affect aspects of a cell’s movement without altering the trajectory of the cell centroid, or cause aberrant shape while preserving the overall aspect ratio. More sophisticated measurements may be specific to a position within the cell, or best expressed as a spatially varying distribution, or perhaps a higher dimensional function of location. For example, the persistent moving of crawling cells depends on the proper spatial coordination of actin polymerization, actin depolymerization and myosin contraction, with actin polymerization biased toward the cell’s leading edge and myosin contraction strongest at the rear [1]. For quantitative observations of such processes to be interpretable in a biological context, to be followed as a function
Manuscript received November 29, 2004; revised May 2, 2005. C. A. Wilson was supported by the National Institute of General Medical Sciences under a graduate training grant for the Stanford University Program in Cellular and Molecular Biology. J. A. Theriot was supported by the American Heart Association under Grant 0240038N. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Erik H. W. Meijering. The authors are with the Department of Biochemistry, Stanford University, Stanford, CA 94305 USA (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TIP.2006.873434
of time in a single cell, to be comparable across different cells as a function of experimental conditions, and to be statistically analyzable, it is important that they be expressed in the coordinate system of the cell. In the case of a motile cell, the entire cell coordinate system moves relative to the frame of reference in which the cell was observed. It, therefore, becomes necessary to calculate a mapping between the stationary lab reference frame and the moving cell reference frame, which in turn demands an accurate and objective method for determining the position and orientation of the cell in every frame of a video sequence. Tracking the position and orientation of a moving cell can be accomplished by calculating a centroid of image intensity, and second-order moments of the intensity distribution, respectively [2], [3]. However, this requires that the cell have a simple enough shape that these parameters are sufficient to reliably follow its motion. It also requires that the distribution of pixel values in the image be related to the distribution of cell mass, or otherwise strongly representative of cell shape. This can impose constraints on the choice of cell type and on the way in which it is imaged, such as necessitating the use of a volume marker. In certain cases these constraints cannot be met due to the nature of the system or the limitations of an experiment. It may be possible to track the cells based on other features, but again if the features discard too much information they might be inadequate for complex motions, and if they are difficult to extract reliably they might fail for complex appearances. A more desirable strategy, then, is one that uses information derived from the entire appearance of the cell: one which does not involve feature extraction or assumptions about shape, and one which does not depend on the mode of imaging, as long as the cell’s appearance—as depicted by image pixel intensity distribution—is roughly similar from image to image in a sequence. A correlation-based approach meets these qualifications. Given two images—a template image and test image—which are identical but for a translation of the objects in those images, the coordinates of the maximum value of the cross correlation between those images recovers precisely the relative translation between the images [4]. Cross correlation between approximately similar images retrieves the approximate relative translation between those images. Since such a method is based directly on appearance in terms of pixel values, even phase-contrast images—easy to acquire but often complicated and difficult to interpret quantitatively—can be used for determining the cell’s trajectory, as long as the appearance does not change too dramatically between consecutive images in the sequence. If images have been obtained in multiple channels (e.g., phase-contrast and multiple fluorescence wavelengths) this method can be applied to all channels in order
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to determine the degree to which the movement of different subcellular structures is correlated. Alternatively, information from multiple channels can be combined for computing the cell’s moving frame of reference, with appropriate weighting given to each channel as determined by the investigator. , The cross correlation of two images and , denoted can be calculated efficiently from the product of their Fourier transforms and [4], [5] (1) where indicates the complex conjugate of , and is the Fourier transform operator. Unfortunately, cross correlation can only recover translations. If an object rotates enough, its relative translation cannot be found by two-dimensional (2-D) cross correlation. Image noise and structures in the image other than the object of interest further confound 2-D cross correlation. Cross correlation is only one of several approaches to image registration [6], [7]. A general strategy in the field of medical image registration for dealing with transformations beyond simple translation is an iterative local search to optimize a similarity measure over the parameter space of possible transformations [8]. This approach is well suited to nonrigid registration of common features extracted from different types of images, such as MRIs versus PET scans. However, applying this method to intra-modal image pairs using an image intensity based similarity measure, as would be desirable for application to a series of images in a video microscopy sequence, involves substantial image resampling to calculate the similarity measure and its partial derivatives at each iteration of the search [9]. This can be computationally intensive, and the transformation estimated for best fit is only a local optimum. In this paper, we describe the implementation of a two-stage cross correlation based method to estimate a cell’s rotation and translation between images of a video sequence, in such a manner that iterative search through rotation-translation space is not required. Rotation is isolated using the translation-invariant magnitudes of the 2-D Fourier transforms of the two images. Applying a polar transform to each magnitude spectrum converts rotation to be represented as translation; we can then calculate the angle of rotation by cross correlation. This enables us to estimate the change in orientation of the cell in a single calculation. The idea underlying this approach is not new [10], [11]. However, we combine this with nonlinear intensity scaling, dynamic masking, and spatial frequency bandpass filtering to make possible the successful application of this strategy to objects with nonrigid and highly dynamic appearances such as biological cells. Having calculated the rotation angle, we rotate the original template image by this angle around the cell center, eliminating rotation. This, combined with intensity scaling, masking, and filtering as in the first pass, isolates translation, which we then calculate by cross correlation. We have found that this two-step method works robustly for tracking the cell’s moving frame of reference for crawling cells in phase-contrast video microscopy sequences—a challenging application—and its performance is superior to iterative optimization and feature extraction methods with respect to both accuracy and computational speed.
II. CONCEPTUAL APPROACH: REPRESENTING ROTATION AS TRANSLATION Given two images of an object, if the object is translated in one image relative to the other, the displacement can be found from the cross correlation of the images. If images can be transformed in such a way that rotation of an object in the original images is converted to translation in the new representation, then that translation (and thus the original rotation) can be easily retrieved with cross correlation. For example, rotation around the central origin can be transmuted into translation via a polar transform [10]: The rectangular axes of the polar-transformed image are the radius and angle in the original image (in the convention, we follow here, angle in the original image maps to the horizontal axis of the polar transform image, and radius maps to the vertical axis of the polar transform). Rotation about the origin in the original image becomes translation along the angle axis in the polar transform. However, this representation is not invariant to translation—that is, translations in the original image drastically alter the polar representation. Furthermore, rotation around a center other than the origin (the location designated as a radius of zero for the polar transform; often the center of the image) results in a more complicated transformation in the image’s polar transform. Therefore, if we do not know the center of rotation and the translation between images, we cannot use the polar transform directly. Note that simple cross correlation cannot find the translation without already knowing the rotation, and that neither method yields a center of rotation. However, the 2-D Fourier transform has two properties which make it extremely well-suited to the problem of finding an image representation in which rotation about any center is represented as a translation, and in which translation of the original image is disregarded entirely. First, rotation of an image about any center results in a rotation of the 2-D Fourier transform magnitude about its origin [4], [11]. This is illustrated by the rotation of an object, off-center in the field of view, around its own center in Fig. 1; the Fourier transform (real and imaginary components) rotates around its origin between (a) and (b). Second, since position information is contained in the phase of the Fourier transform, the magnitude of the Fourier transform (magnitude spectrum) is invariant to translation, as shown in Fig. 1; the magnitude spectrum does not change between (b) and (c). If we then calculate the polar transform of the magnitude spectrum, we obtain a representation in which rotation in the original image, about any center, converts to centered rotation of the Fourier spectrum, and translation is ignored. The polar transform of the magnitude spectrum then converts centered rotation into translation (Fig. 1). With this transform, then, rotation of an object can usually be determined with simple cross correlation, without any information regarding the center of that rotation or the translation of that object. Given a test image which contains a rotated, translated template, we can perform polar transforms on the magnitude spectra of the test and template images, and then calculate the cross correlation of the images in this representation to elegantly and efficiently find the angle of rotation in most cases. With this information, we can then rotate the template image so it lines up with the test image, and use cross correlation
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Fig. 1. Representing rotation as translation. Rotation [between (a) and (b)] of the glyph around its own center, though it is not at the origin of the image, results in rotation of the 2-D Fourier transform magnitude around the origin. This converts to a horizontal translation of the polar transform of the magnitude spectrum (Fourier magnitude). Translation [between (b) and (c)] of the rotated changes the phase but not the magnitude of the Fourier transform; therefore the polar transform of the magnitude spectrum is unaffected. See the supplementary material at http://ieeexplore.ieee.org for an animated version of this figure.
as before to find the relative translation between the two. This approach is not entirely foolproof: If the magnitude spectra of the images are radially symmetric, for example, then the angle of rotation cannot be recovered in the manner just described. Such a scenario is unlikely in practice, however.
This strategy has been described previously as an approach to image registration for images that have been translated, rotated, and/or scaled relative to each other [10]–[15] (if, instead of a polar transform, a log-polar transform is applied to the magnitude spectrum, rotation results in translation in one dimension
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Fig. 2. Ideal image registration compared to that of similar but nonidentical images. Example cross correlations for the calculation of rotation are shown for a pair of ideal images and a pair of real images. The ideal images are identical but for rotation and translation; image one is a rotated and translated version of image zero. The real images are two images, taken 60 s apart, from a movie of a crawling cell: a fish epidermal keratocyte. The cell’s appearance is similar in the two images but not identical, as the cell is not a rigid object. The polar transforms of the magnitude spectra (second row) of the images are used to calculate cross correlations and phase correlations (third row). Polar transforms and correlations have been cropped for space. Correlation values along the centers of the correlation surfaces are plotted in the fourth row. Images and plots of the ideal and real cross correlations have been scaled identically; similarly the phase correlations have been scaled identically. Phase correlation is especially sensitive to differences in the images other than rotation and translation (note the absence of a peak for the real images). Simple cross correlation is more forgiving; however the calculation as previously described is not sufficiently accurate for this application (Fig. 3).
as before, and scaling results in translation in the other dimension [11]. This representation is also known as a Fourier–Mellin Transform [12], [13]). These implementations have been shown to accurately calculate, in the presence of additive noise, imposed transformations that change an image’s position, orientation, and/or scale. In [13] and [14], the method was applied to separately acquired images of the same object or scene, but the main changes in appearance were geometric transformations (movement of the image acquisition apparatus between images). However, the extent to which this method can find the most likely angle of rotation for images of an object which has changed by more than rotation, translation, and scaling has not yet been determined. We find that differences in the appearance of a cell in two images 60 s apart have a significant effect on the correlation of the polar transforms of the magnitude spectra (Fig. 2). The phase correlation, preferred because it gives a narrower peak than cross correlation [11], [13], [14], [16] yields a sharp peak for two simulated “ideal” images, but no peak for the real images. Cross correlation gives a peak for both the ideal and real cases. For the real images, however, the calculated rotation
angle, as well as the subsequently calculated translation, do not approximate the cell’s movement as well as the results obtained by applying the improved method we present here (Fig. 3). Given that a cell’s appearance may change substantially over time, we generally cannot track the cell in each image in a sequence by registration with the sequence’s initial image, but rather with a more recent image, adding up the calculated transformations along the way. This means that a constant error in each registration does not merely produce a constant error for each point in the track, but instead results in a cumulative error. Consequently, the two-stage cross-correlation strategy as described in previous implementations is unsuitable for application to tracking cell motion. We introduce our adaptations for this application below. III. APPLICATION TO DYNAMIC CELLS In the analysis of images of moving cells from video microscopy sequences, we are not actually looking at two identical images where one has been translated and/or rotated with
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Fig. 3. Application of the modified approach to track rotation and translation of moving cells. The two-stage cross-correlation approach, as previously described [10], [11], is suboptimal for tracking the movement of nonrigid objects. This is illustrated by the registration (a) of the two real images from Fig. 2 using rotation and translation calculated as in prior work. An improved registration (b) is obtained with rotation and translation computed using the modified approach (c) presented here. Nonlinear intensity scaling and bandpass filtering select the intensity range and spatial scale, respectively, of the features whose motion we are approximating as rigid.
respect to the other. We are looking at two different images; cells are nonrigid, dynamic objects. While we might characterize some overall motion of a cell as a combination of translation and rotation, that is a simplification. When the cell as a whole “translates” and “rotates,” not every point on the cell translates with the cell center or rotates about the cell center. In fact, this description of motion does not apply directly to the individual physical components of the cell but rather to a larger scale arrangement: a configuration whose motion emerges from the dynamic remodeling of its component structures. For example, much of the actin cytoskeleton remains stationary or moves slowly backward relative to the substrate during keratocyte movement while the cell as a whole moves forward [17]. It is thus only at a specific spatial scale that there appears to be an overall arrangement which is rotating and translating. For example, forward motion at the cell’s leading and trailing edges is not concerted movement of the components of the membrane (a fluid structure) but a movement of the enclosed cellular space
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that is surrounded by membrane. In contrast, the nucleus moves forward approximately as a unit. Yet, when observing the entire cell, it appears that the edges and the nucleus, with their contrasting modes of motion, move roughly in concert. Bulk movement of our system can be summarized as rotation and translation at a certain scale; therefore our adaptation to extend this technique to dynamic cells involves applying a spatial frequency bandpass filter at each of the correlation steps to select only spatial information at that scale. Our procedure for tracking the rotation and translation of a cell from each image to the next in a video sequence, diagrammed in Fig. 3, is as follows. For the current image (designated image zero), a mask is applied to select the region of interest (the cell) to form the template image. The remainder of the image may contain structures which do not move or rotate with the cell, and thus should be ignored. This prevents other objects in the video frame besides the moving cell from contributing to the spatial frequency map and therefore obscuring the result. The polar transform of the magnitude spectrum is then calculated for the template image (generated from image zero) and for image one (the next image in the sequence). After bandpass filtering to focus attention on only the spatial scale over which meaningful rotation is occurring, the cross correlation is calculated; the location of the maximum corresponds to the angle the cell rotated between image zero and image one. The masked image zero is then rotated around the center of the mask region by the angle just calculated to form a template image in which the cell has the same orientation as it does in image one. After additional bandpass filtering, the cross correlation of these two images is calculated; the location of the maximum is the relative translation of the cell between image zero and image one. Now that the rotation and translation relative to image zero are known, the mask can be positioned at the location of the cell in image one—which will become image zero in the next iteration, in the simplest implementation relevant for application to a video sequence—to form a template image. Another option is to use the initial image in the sequence as image zero at each iteration, and assign each subsequent image to be image one, calculating the rotation and translation in each image relative to the orientation and position in the starting image rather than in the previous image. However, for later images, the appearance of the cell may have evolved sufficiently such that it cannot be reliably correlated with the starting image (note that, even when correlating with the previous image, it may be necessary to occasionally update the mask shape if the cell shape is rapidly changing). In the following two sections, we present for the two calculation stages (rotation and translation) the implementation details that enable successful application of this approach to dynamic cells. IV. CALCULATION OF ROTATION Two images, 60 s apart, from a video of a crawling keratocyte (fish epidermal cell) are shown in Fig. 2 (“real”). The cell experiences rotation as well as translation over the interval between the two images: image zero and image one. As shown in Fig. 4, the magnitude spectrum of image one is rotated relative to that of image zero, by the same angle of the cell’s rotation.
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Fig. 4. Representations of a cell’s rotation. The cell from Fig. 2 is shown at 0 s (image 0) and 60 s (image 1). The magnitude spectrum (center region enlarged, inset) of image 1 is rotated by the same angle relative to the magnitude spectrum of image 0 as is the cell. Correspondingly, the polar transform of the image 1 magnitude spectrum is translated relative to that of image 0 (the black vertical arrows point to approximately the same feature in the two polar transforms; the gray arrow in the second polar transform indicates the position of the black arrow in the first, for reference). See the supplementary material at http://ieeexplore.ieee.org for an animated version of this figure.
This rotation results in a horizontal translation—the horizontal axis is the angle axis—of the polar transform of the magnitude spectrum. A. Nonlinear Intensity Scaling For various types of images, including phase-contrast and epifluorescence, the image features corresponding to the cell fall in a specific range of pixel intensities, and confounding objects in the image field often tend to fall outside of this range. To improve the specificity with which the cross-correlation calculations find our object of interest, we can select a range of pixel values somewhat analogously to our selection of a spatial region with a mask and a spatial scale with a bandpass filter (below). Our first step, before masking, transforming, or filtering the images is to apply a sigmoidal scaling to the pixel values. This stretches the contrast in the range of interest, while flattening out intensities above and below. We determine our pixel value range of choice by drawing a line from inside the edge of the cell to outside the edge; the minimum and maximum values along that line define the range to be stretched. We observed that flattening out features outside the intensity range of interest can help to decrease the influence of cell and tissue debris (floating past or being carried by the cell) on the cross correlations. The cell shown in Figs. 2–9 is carrying a round remnant of debris with features at intensities above our selected range. Nonlinear scaling, applied to the cell image in Figs. 4–9, attenuates the variations, making the debris appear
Fig. 5. Selecting the region of interest. We use a mask derived from the outline of the cell in the initial image of the sequence; the mask is rotated and translated to the cell’s current calculated orientation and position. The mask need not be shaped like the cell, as long as the masked region does not include features other than the cell. We apply a Gaussian lowpass filter to the mask so that the frequency content it contributes is below our spatial frequency passband (hence the lack of dependence on mask shape). We subtract from image zero the average value of the pixels outside the mask, and then multiply this image with the mask. The neutral gray in the masked image (right) corresponds to a pixel value of 0, due to scaling of the image for the figure.
uniform as it does in the figures, and reduces its frequency content such that it is nearly eliminated by the bandpass filter, as seen in Fig. 7. Without this initial sigmoidal intensity scaling step, the debris had a pronounced texture, and the error it contributed to the rotation and translation calculations could not be reduced by bandpass filtering alone. This was the case for phase-contrast video sequences of other keratocytes as well. B. Region of Interest In order to specify the template (the cell), we are looking for in image one, we select the region in which the cell is located in image zero by applying a mask. This step is critical for video
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Dictyostelium discoideum, for example, change shape rapidly, but tend to be somewhat spherical overall; for such cells a mask might be made from a circle of some radius around the center (as specified manually or from a volume marker centroid). Alternatively, one might use an outline but automatically update it periodically to keep up with shape changes. We apply a lowpass filter to the mask by convolving the binarized mask with a Gaussian. We want to avoid introducing new frequency components into the template image; the lowpass filtering ensures that any spatial frequency content contributed by the mask is below the lower limit of the bandpass filter we will be applying later. The mask, which now smoothly drops off to zero, is ready to be applied to the image. We subtract from image zero a constant background intensity—the average intensity of pixels outside the mask in image zero—then we multiply by the mask. Fig. 5 shows application of the mask to image zero to generate the template image. C. Calculation of Magnitude Spectrum and Polar Transform
Fig. 6. Calculation of rotation. We calculate the polar transform of the magnitude spectrum of the masked image zero (shown) and of the windowed image one. We then apply a spatial frequency bandpass filter to each. In this representation, spatial frequency is an axis; therefore, filtering can be accomplished simply by taking a slice across the image. We only take angles 0 to , as the region from to 2 is a repeat of 0 to due to the conjugate symmetry of the Fourier transform. The location of the maximum of the cross correlation of these two sections corresponds to the angle of the cell’s rotation between image zero and image one. We calculate the centroid of the peak; the angle coordinate of the centroid (or the angle plus ) is the angle of rotation.
sequences of moving cells, which typically include several cells in a single image field. Our mask is derived from the outline of the cell in the initial image of the movie, rotated and translated as necessary according to the calculated motion of the cell up to the current image zero. For our purposes, we compute the outline using an “active contours,” or “snakes,” approach based on [18]. Briefly, a rough outline is drawn around the cell by the user, and then iteratively deformed toward the cell edge. Once the outline has been computed, it is filled in and then dilated such that it spans the whole cell even after lowpass filtering. For best results, the mask should include the entire cell but not include other features in the image; however it does not need to be based on a cell outline. An outline might be a poor choice for some modes of imaging, or for cells which change shape substantially over the course of the video sequence. Neutrophils and
Since the Fourier transform treats a signal as periodic, the discontinuities at the boundaries of an image contribute artifacts to its Fourier transform. It is important that the magnitude spectrum we calculate not include such artifacts, as they may obfuscate the important features in the representation in which we will be calculating the angle of rotation, decreasing our ability to reliably find that angle. We therefore want both our template image and our test image to be continuous at the boundaries, meaning that the pixel intensities should gently drop off to a constant value near the boundaries. The boundaries of the template image (the masked image zero) are already continuous, due to the application of the mask (however, if the mask had nonzero values at a boundary, an additional window may need to be applied). To ensure continuities at the boundaries of the test image (image one) we can optionally apply a Hann window [19] in the spatial domain. The window is applied to image one the same way the mask was applied to image zero above; we subtract from image one its average intensity, then multiply by the window, which smoothly falls to zero near the boundaries. Though we include this windowing in the implementation as described here, we find that in practice it is generally unnecessary, and, in some cases, the uneven weighting of image one due to windowing can interfere the accuracy of the calculations. We did not apply this window when calculating rotations and translations in the comparison section. When we calculate the polar transform of the magnitude spectrum, we use bicubic interpolation to reduce the appearance of edges between neighboring pixels in the polar transform representation. Previous work uses a log-polar transform at this step, allowing simultaneous calculation of rotation and scale [11]. Since we will be applying a spatial frequency bandpass filter to look at motion at a specific scale, we cannot simultaneously calculate scale at this step. Therefore, we calculate a polar transform, not a log-polar transform, of each magnitude spectrum. D. Bandpass Filter As mentioned earlier, the “rotation” and “translation” that we are characterizing are not direct properties of individual compo-
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Fig. 7. Calculation of translation. We rotate the masked image zero (shown before and after rotation) by our calculated rotation angle of the cell, such that the cell has the same orientation in each image (note that image 1 is shown after windowing). We apply a spatial frequency bandpass filter to the rotated, masked image zero and to the windowed image one; then we calculate the cross correlation. The centroid of this peak corresponds to the translation of the cell between the images (note that, for simplicity, the bandpass filtering here is equivalent to that used in the calculation of rotation; the infinitely steep frequency cutoffs of that filter are responsible for the ringing artifacts in the filtered images).
nents of the cell but of an emergent configuration at a particular spatial scale. Therefore, to calculate the angle of what we are defining as “rotation” of the cell, we apply a bandpass filter to select the spatial frequency range corresponding to the scale at which rotation occurs. Conveniently, one of the axes—the vertical axis here—of the polar transform of the magnitude spectrum corresponds to spatial frequency. We can perform the filtering simply by taking a horizontal slice across this representation of the image, as depicted in Fig. 6. Furthermore, as we will only be using a slice of the polar transform, that is the only portion of the polar transform we need to calculate. This speeds up the method significantly, as the polar transform is the most computationally expensive step in our implementation. We only compute the left half of this slice (angles 0 through ) is a repeat of 0 through , due to because the area through the conjugate symmetry of the Fourier transform of a real image [11]. This means that after calculating the angle of rotation , the calculation of translation must be computed for the template (the cross correlation with the larger rotated by both and maximum value corresponds to the actual rotation angle). If, on the other hand, it can be assumed that the rotation angle lies within certain limits, it may not be necessary to test both and . For our cells we can safely assume that . We do not apply an additional window to the bandpass filtered polar transform of the magnitude spectrum before calculating the cross correlation. This signal is periodic in the angle dimension, so there is no discontinuity across the vertical (left and right) edges. The discontinuity across the horizontal edges has the effect of constraining the maximum of the cross correlation to lie along the line corresponding to no spatial frequency displacement. This is appropriate for our purposes, as we only want the displacement in angle (rotation) at this step.
E. Cross Correlation In previous applications of this approach to image registration, phase correlation has been found to give a narrower peak than correlation as computed here [11], [13], [14], [16]. The phase correlation technique assumes that images f and g differ only by translation. Under that assumption, their phase correlation (the inverse Fourier transform of the cross-power spectrum) will have a peak at the displacement corresponding to the relative translation between the images, and will be near zero everywhere else [14]. However, our filtered polar-transformed magnitude spectra (Fig. 6) differ by significantly more than translation. We found that the phase correlation of these representations failed to reliably produce a peak corresponding to the relative displacement between them (Fig. 2). For our similar, but not identical, images, standard cross correlation is required. The cross correlation is performed using (1) the complex conjugate of the Fourier transform of the filtered, polar-transformed magnitude spectrum of the template image is multiplied by the Fourier transform of the filtered, polar-transformed magnitude spectrum of image one. The inverse Fourier transform of this product gives the correlation. The resulting cross correlation is shown in Fig. 6. For a precise calculation of rotation angle, we desire a subpixel location of the cross-correlation peak. We obtain this by computing a centroid. Each above-threshold pixel in a neighborhood centered at the maximum pixel is weighted in the centroid calculation by the difference between the pixel value and the threshold. In practice this procedure gives better localization than the nearest integer value. However, it should be noted that as this is a discrete case, the Fourier shift theorem does not hold exactly for subpixel shifts.
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V. CALCULATION OF TRANSLATION A. Rotation of Template Around Region Center Having determined the angle the cell rotated between image zero and image one, image zero can be rotated by that angle to generate a new template image in which the cell has the same orientation as it does in image one. Cross correlation can then be used to compute the translation of the cell. However, the displacement that will be calculated depends on the center around which image zero was rotated. In our current implementation, we define the cell center as the centroid of the vertices of the cell outline (originally represented as a polygon) in the initial image, displaced according to the cell’s translations calculated for the subsequent frames up to the current image zero. The cell center does not need to be defined based on this criterion; other possibilities include using the location of the cell nucleus, or, if the cell has been imaged with a volume marker, the center of volume. The center need not be computed automatically either, as it only needs to be defined in the first image of a sequence; it is subsequently updated with the recovered translation information. The spatial transformation to rotate the masked image zero around the specified center is then the composition of three transformations translation of an object located at the region center to the origin, rotation by the angle calculated in the previous section, and translation from the origin back to the region center [20]. The transformation is then applied to the image via an inverse mapping [2] with bicubic interpolation. By composing a single transformation we avoid resampling the image more than once. Multiple rounds of resampling would be undesirable because they would introduce artifacts in the image, which hampers accurate cross correlation of the transformed image with image one. The rotated, masked image zero is shown in Fig. 7. B. Bandpass Filtering We apply a spatial frequency bandpass filter when calculating translation for the same reasons as when we calculate rotation. When we calculated rotation we were working with a representation in which spatial frequency was an axis, and bandpass filtering could be accomplished by slicing. This time we are working in the spatial domain of the images, so filtering can be performed by convolution in the spatial domain or multiplication in the frequency domain. We do the latter, simultaneously with the cross-correlation calculation. There would be no additional computational cost to using a filter kernel without infinitely steep frequency cutoffs; however, for consistency in this demonstration, we use a filter kernel which is spectrally equivalent to the earlier bandpass filtering operation. The infinitely steep cutoffs produce the ringing artifacts that can be seen in Fig. 7. These have no significant effect on the calculation of translation. C. Cross Correlation The cross correlation, Fig. 7, is calculated from the product of the Fourier transforms (1). We calculate the centroid of the peak as before to get a translation displacement with subpixel precision.
Fig. 8. Testing the result. Registering the first image to the second using the calculated angle and displacement allows us to visually gauge the accuracy of the computed values. Image zero is rotated and translated (a) such that the cell has the same orientation and position as it does in image one (b). The average (c) of these two images shows good overlap, indicating that the calculated rotation and translation match the actual motion of the cell. The difference (d) reveals that discrepancies (light or dark areas) are in regions of the cell that changed shape or appearance.
At this point, we can assess how accurately the calculated angle and displacement match the rotation and translation of the cell from image zero to image one by registering image zero such that the cell has the same orientation and position as it does in image one. The registration of image zero is the composition of three spatial transformations: translation of the cell from its center (as located in image zero) to the origin, rotation, and translation from the origin to the location of the cell center in image one. The registered image zero is shown together with the original image one in Fig. 8. The average and difference images in Fig. 8 reveal that the only differences are in areas in which the cell shape changed slightly; the calculated rotation and translation closely match the cell’s overall motion. VI. CELL FRAME OF REFERENCE As mentioned in the Introduction, one of the motivations for tracking a cell’s motion is to establish a mapping between the stationary frame of reference of the lab and the moving frame of reference of the cell. This allows us to transform measurements made in the lab reference frame to cell coordinates, or to transform entire image sequences to the cell reference frame for visualization purposes, such that the cell stays still and the world travels by. This can aid in observing, for example, how local interactions of one part of the cell with its environment can then affect another part of the cell due to the cell’s motion: In the cell reference frame, cell-substratum adhesions formed at the front of the cell “travel” to the back of the cell where they must be dismantled if the cell is to continue forward.
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In order to generate a spatial transformation from an image in the sequence to the cell frame of reference, it is necessary to know the position and orientation of the cell in the image. This is simply the orientation and position of the cell at the beginning of the sequence, rotated and translated according to the cell’s motion calculated above. Therefore, we need to define the cell’s position and orientation in the initial image. We have already defined the position—the cell “center” we used as a center of rotation; but we still need to know which direction is front or back: We need to define an orientation, by manual or automatic means. In this implementation, we define the cell’s orientation in the initial image based on the same outline we used to define a center. We compute a major and minor axis by a principle components analysis of the vertices of the outline (represented as a polygon). The orientation of the minor axis (negated if necessary such that it is not against the direction of cell movement) is taken to be the orientation of the cell. Having updated the position and orientation of the cell for the current frame, given its calculated translation and rotation, we generate the spatial transformation to the cell reference frame as the composition of two transformations: translation of the cell from its center in the lab reference frame to the origin, and rotation of the cell from its orientation in the lab frame of reference to an orientation of zero. Optionally, the rotation can be to a difso ferent orientation; for visualization we typically rotate to the cell is pointed up. The final effect is a movie of cell motility as it would be imaged by a camera poised above the center of the cell and moving with it. Coordinates of a location in the lab frame can be converted to the cell frame coordinates via a forward mapping, or the entire image can be transformed into the cell frame using an inverse mapping. Fig. 9 shows the result of transforming an image sequence such that the cell is centered and pointed up in each image. With this method, there are two possible approaches to calculating the rotation and translation a cell has undergone to reach its orientation and position in a given frame. The strategy we have described thus far involves calculating at each image the rotation and translation relative to the previous image, and summing those rotations and translations to get the total change in orientation and position relative to the initial image. The cumulative nature of this approach has a disadvantage in that small errors in individual calculations can add up to give an overall drift over long sequences, and a large error for a given image will throw off all subsequent images. Another approach is to directly calculate the rotation and translation relative to the initial image: using the initial image as image zero for each calculation. However, since the calculation is based on correlation and the cell’s appearance changes over time, this alternate strategy is unreliable for long intervals. A strategy that incorporates calculations over multiple interval lengths might combine the strengths of both above approaches, reducing drift while still handling changes in appearance. Periodically updating the mask may also be appropriate if over time the cell’s shape becomes drastically different of that from the mask. VII. COMPARISON Are the position and orientation measures obtained by this method well suited to defining a cell frame of reference? How
Fig. 9. Cell frame of reference. We used the method presented here to track the cell’s rotation and translation between images 10 s apart. Having defined the position and orientation in the initial image, we determine the orientation and position in each image by adding the cell’s rotation and translation to the orientation and position in the previous image. We then rotate and translate each image such that the cell is centered in the image and oriented upward. See the supplementary material at http://ieeexplore.ieee.org for an animated version of this figure.
well does our correlation-based approach perform for this application next to other methods that might be used to track
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cell translation and rotation? We compared our procedure with two other methods that might be used to compute a cell’s position and orientation in phase-contrast microscopy image sequences. We did not include in our comparison methods based on thresholded regions or on moments of image intensity distributions, even though they are most commonly used, because such methods are attached to a specific mode of imaging, therefore imposing constraints such as the use of a fluorescent volume marker. The methods selected below can be applied to a fluorescent volume or membrane marker movie, or to a phase-contrast movie, or can be adapted to other methods that do not lend themselves to thresholding such as Nomarski or birefringence imaging. We implemented each approach in MATLAB; computation times are reported for execution in MATLAB 7.0.1 on a 2-GHz G5 processor. A. Methods The first method is based on feature extraction; it obtains the cell position and orientation in each image from a polygonal cell outline. The outline is generated by the same active contours approach as described above, manually initialized around the cell in the initial image. In subsequent images, the snake is initialized with the final snake configuration from the previous image. We define the cell’s position in each image as the centroid of the outline vertices. For orientation, we employ the ellipticity of the characteristic keratocyte shape: The cell has a major axis perpendicular to the direction of motion, and a minor axis parallel to the motion. We therefore calculate a major and minor axis of the outline by performing a principle components analysis on the coordinates of the outline vertices. The minor axis (negated if necessary such that it is not opposite the direction of motion) is then taken to be the cell’s orientation (note that this is how we earlier defined the cell orientation in the initial image of the sequence). The second method is image registration by local iterative optimization of an image similarity measure. We apply intensity scaling, masking, bandpass filtering as we do for our twostage cross-correlation strategy. Then, we minimize the sum of squared pixel intensity differences between registered images as a function of rotation angle, horizontal displacment, and vertical displacement. The optimization algorithm we use is a large-scale trust-region method using preconditioned conjugate gradients [21], [22]; it is provided by the “lsqnonlin” function in MATLAB’s Optimization Toolbox. We found that on our keratocyte images this algorithm minimizes the squared differences more successfully and with fewer iterations than the Optimization Toolbox’s available alternatives. However, for images taken 10 s apart, as we used in this comparison, the cell moves too far between images to be found by local iterative optimization without an initial estimate of rotation and translation. Therefore, we perform the optimization first at half-resolution—at which rotation and translation can be found successfully without an initial guess—and then refine the half-resolution estimate by optimization at full resolution. For intervals of 20 s, not used in this comparison, we found it necessary to start at quarter-resolution. In addition, we present the results of using the transformation returned by our cross-correlation method as an initial estimate for further refinement by iterative optimization.
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B. Dataset We apply each method to calculate the rotation and translation of keratocytes over 10-s intervals in 87 movies. All cells were imaged at a single magnification and under the same conditions. Images are 512 512 pixels. If the microscope stage was moved during a movie (to bring a cell that is moving out of the field of view back into full view), then we only use images from before the first repositioning of the stage. While our cross-correlation method is global and can follow the cell as long as enough of it is in the field before and after moving the stage, the local iterative optimization cannot recover large movements of the cell; nor can the feature extraction, as it uses the cell outline from the previous image in finding the outline in the current image. With this restriction, this gave 731 intervals from the 87 image sequences. Each method was applied to calculate cell rotation and translation over each of these intervals. C. Results We compared the accuracy and computation time of each method on each interval (Fig. 10). Here, accuracy is evaluated by how well the rotation and translation returned by each method minimizes the sum of squared pixel value differences between the images flanking the interval. This is the same measure that the iterative optimization method seeks to minimize; it is calculated after intensity scaling, masking, and bandpass filtering. We show the distributions of sums of squared differences for each method; we include the data prior to registration for reference. Unsurprisingly, the feature extraction approach performs relatively poorly. The appearance of cells in phase-contrast images is complex—modern transmitted light microscopy methods are designed to reveal the slightest variations across otherwise transparent cells—making it difficult to extract an outline with consistency between images. Furthermore, the outline considers only the cell margin, not the entire appearance of the cell. Motile cells typically exhibit dynamics at the cell edge which are not strongly coupled to the motion of the whole cell. The accuracies of the iterative optimization approach and our two-stage cross-correlation strategy are similar. Though the difference is small, our method achieves better results than the iterative optimization, as assessed by a paired Wilcoxon signed rank test: . Most likely, this is because the iterative optimization can get trapped in a local minimum. If we use our cross-correlation approach to calculate an initial transformation estimate, then iteratively optimize it, we get the best by paired score overall [Fig. 10(a), far right]; signed rank test. Though consistent, it is still only a minor improvement over the cross-correlation approach alone. The two-stage cross-correlation method we present in this paper is by far the fastest. We show in Fig. 10(b) the amount of time each method takes to calculate rotation and translation of a cell between a pair of images, assuming the mask for image zero has already been provided. Our method takes a median 1.67 s, whereas the iterative optimization takes a median 52.57 s with variation by the number of iterations. The median number of iterations is ten at half-resolution followed by eight at full resolution. The large computational expense of the iterative optimiza-
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without extracting features, and despite the complexities of a cell’s appearance in a phase-contrast image. The method does not require that the images show clearly defined cell edges that can be extracted computationally or that the distributions of image intensity have a specific relationship to the shape of the object; therefore, it can be applied to a variety of modes of imaging. The requirement is simply that the motion of interest is manifested in the movement of the intensity distribution. A. Generality
Fig. 10. Comparison of methods for calculating cell rotation and translation. We compared the performance of our approach (labeled "cross-correlation") with a feature extraction method and an iterative optimization method for calculating rotation and translation over 731 intervals between images 10 s apart taken from movies of 87 keratocytes. We also show the results of using the transformation calculated by our two-stage cross-correlation method as an initial guess for refinement by iterative optimization. For each pair of images, the masked image zero is registered to image one (both after intensity scaling and bandpass filtering) using the calculated rotation and translation; we then sum the squared differences in pixel intensity values between the images. A smaller difference between the registered images suggests a more accurate calculation of rotation and translation. The distributions of sums of squared differences before and after registration with transformations calculated by each method are shown as box-and-whisker plots (a). We also report the time it takes to compute rotation and translation between each pair of images (b). Boxes indicate the spread from the bottom quartile (25th percentile) to the top quartile (75th percentile). Medians are indicated by the filled circles and adjacent text annotations. Whiskers extend to the first and 99th percentiles; data points in the first and 99th percentiles are displayed individually.
tion method comes from resampling the image four times per iteration: to calculate the sum of squared pixel differences for the current transformation and the partial derivatives with respect to rotation, horizontal displacement, and vertical displacement. VIII. CONCLUSION AND DISCUSSION Our application of this modified correlation-based approach successfully tracks the rotation and translation of moving keratocytes based on image intensity, without an iterative search,
Being able to track a cell’s translation and rotation without requiring that it be imaged with a fluorescent volume marker opens significant new experimental opportunities. Fluorescence microscopy setups for live-cell imaging are limited in how many channels can be acquired in a single experiment. If an experiment calls for simultaneous or near-simultaneous observation of different labeled molecules or structures, the limit might be reached before a volume marker is added to the list. This can be due to limitations of the equipment, the need for sufficient separation between emission spectra of the fluorophores being used, sensitivity of the cells to certain wavelengths of light, or photo-reactivity of drugs with which the cells are being treated. In other situations, the experimental requirements accommodate imaging a volume marker, but the cells are being observed long enough—to capture a rare event for example—that photobleaching of the volume marker is a substantial problem. Transmitted-light imaging (such as phase-contrast or Nomarski microscopy) is already frequently combined with fluorescence imaging of specific molecular markers in live-cell imaging experiments, and is used by investigators to evaluate the overall behavior of the cell during the time of observation. In comparison with the use of a fluorescent volume marker, transmitted light imaging is less likely to conflict with other experimental requirements—especially if the illumination is not intense—and it is not subject to bleaching. It is also less likely to be prohibited by equipment limitations, assuming the microscope has a transmitted light path. B. Rigid and Non-Rigid Transformations In order to extend the correlation-based strategy, previously applied to images which differed by a rigid transformation, to our nonideal application—cells with similar appearances from image to image, whose movement resembles rigid motion—it was necessary to specify the scale, by spatial frequency bandpass filtering, for which we wanted to approximate motion as rotation and translation. In contrast to the straightforward selection of the spatial region and pixel intensity range containing the cell, the choice of appropriate frequency range for the bandpass filter is not obvious. Empirically, we found a passband that was suitable to a collection of cells imaged under the same conditions; this spatial frequency range corresponds to variations with periods between 2 and 6 m. However, a change in mode of imaging or cell type will likely require a corresponding change in frequency range. In the future we hope to automate the determination of optimal spatial frequency range. This discussion of approximating nonrigid transformations as rigid raises the question of whether it is even appropriate to summarize cell dynamics as rigid motion. As we pointed out early
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on, excessive simplification of complex phenomena can sacrifice biologically relevant details. On the other hand, some extent of reduction is prerequisite to interpretation. In modern biological imaging, the central challenge of quantitative analysis is not the conversion of information to a quantitative form—the image data are already quantitative—but of making sense of the overwhelmingly rich information. We view this characterization of movement as rotation and translation as a first step, and as a base for further analysis of cell motion. The observed changes include nonrigid transformations as well as changes in appearance not due to spatial transformations (or at least not due to any spatial transformations on a scale large enough to be captured by the imaging method). Having separated out the overall movement of the cell, we can further deconstruct the remaining dynamics. We can also interpret observations and measurements in the context of locations in or paths traversed through the cell. In the case of the actin cytoskeleton in keratocytes, different remodeling processes are localized to specific regions of the moving cell, but the filament meshwork near the front of the cell is nearly stationary relative to the substratum—which means it is moving in the frame of reference of the cell [17]. The architecture at a specific point in the meshwork depends on where in the cell that portion of the meshwork has been. How that architecture will affect the cell’s movement depends on where that portion of the meshwork is going. The spatiotemporal mapping between the frame of reference of the actin meshwork and that of the cell is intimately related to the organization of motility processes and the feedback between them. Comparing or connecting the mappings (rigid or nonrigid) between frames of reference derived from simultaneously acquired channels has the potential to help illuminate the coordination between the underlying biological processes. ACKNOWLEDGMENT The authors would like to thank Z. Pincus and K. Keren for critical reading of the manuscript. This paper has supplementary downloadable material available at http://ieeexplore.ieee.org, provided by the authors. This includes three QuickTime movies, which show animated versions of Figs. 1, 4, and 9. This material is 49 MB in size.
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[8] B. F. Hutton and M. Braun, “Software for image registration: Algorithms, accuracy, efficacy,” Sem. Nucl. Med., vol. 3, pp. 180–192, Jul. 2003. [9] P. Thevénaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process., vol. 7, no. 1, pp. 27–41, Jan. 1998. [10] D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt., vol. 15, no. 7, pp. 1795–1799, Jul. 1976. [11] B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process., vol. 5, no. 8, pp. 1266–1271, Aug. 1996. [12] Y. Sheng and H. H. Arsenault, “Experiments on pattern recognition using invariant Fourier-Mellin descriptors,” J. Opt. Soc. Amer. A, vol. 3, no. 6, pp. 771–776, Jun. 1986. [13] Q. Chen, M. Defrise, and F. Deconinck, “Symmetric phase-only matched filtering of Fourier-Mellin transforms for image registration and recognition,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 16, no. 12, pp. 1156–1168, Dec. 1994. [14] S. Ertürk, “Translation, rotation and scale stabilization of image sequences,” Electron. Lett., vol. 39, no. 17, pp. 1245–1246, Aug. 2003. [15] E. DeCastro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 9, no. 5, pp. 700–703, Sep. 1987. [16] J. L. Horner and P. D. Gianino, “Phase-only matched filtering,” Appl. Opt., vol. 23, no. 6, pp. 812–816, Mar. 1984. [17] J. Lee, A. Ishihara, J. A. Theriot, and K. Jacobson, “Principles of locomotion for simple-shaped cells,” Nature, vol. 362, pp. 167–171, Mar. 1993. [18] C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process., vol. 7, no. 3, pp. 359–369, Mar. 1998. [19] S. L. Marple, Digital Spectral Analysis: With Applications. Englewood Cliffs, NJ: Prentice-Hall, 1987, pp. 136–144. [20] J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice. Boston, MA: Addison-Wesley, 1996, pp. 208–210. [21] T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM J. Optim., vol. 6, no. 2, pp. 418–445, May 1996. [22] , “On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds,” Math. Progr., vol. 67, no. 2, pp. 189–224, 1994.
Cyrus A. Wilson received the B.S. degree in molecular biophysics and biochemistry from Yale University, New Haven, CT, in 2000. He is currently pursuing the Ph.D. degree in biochemistry at Stanford University, Stanford, CA. His research interests include quantitative cell biophysics, image analysis, modularity and abstraction in complex biological systems, and biological representations of information. Mr. Wilson is a member of the Biophysical Society and the American Society for Cell Biology.
REFERENCES [1] S. M. Rafelski and J. A. Theriot, “Crawling toward a unified model of cell motility: Spatial and temporal regulation of actin dynamics,” Annu. Rev. Biochem., pp. 209–239, 2004. [2] B. Jähne, Practical Handbook on Image Processing for Scientific and Technical Applications, 2nd ed. Boca Raton, FL: CRC, 2004, pp. 271–272. [3] F. S. Soo and J. A. Theriot, “Large-scale quantitative analysis of sources of variation in the actin polymerization-based movement of Listeria monocytogenes,” Biophys. J., vol. 89, pp. 703–723, 2005. [4] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2002, pp. 194–211. [5] R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. New York: McGraw-Hill, 1986. [6] B. Zitová and J. Flusser, “Image registration methods: A survey,” Image Vis. Comput., vol. 21, pp. 977–1000, 2003. [7] F. Maes, D. Vandermeulen, and P. Suetens, “Medical image registration using mutual information,” Proc. IEEE, vol. 91, no. 10, pp. 1699–1721, Oct. 2003.
Julie A. Theriot received concurrent B.S. degrees in physics and biology from the Massachusetts Institute of Technology, Cambridge, in 1988, and the Ph.D. in cell biology from the University of California, San Francisco, in 1993. She has been on the faculty of the Stanford University School of Medicine, Stanford, CA, since 1997, with joint appointments in the Department of Biochemistry and the Department of Microbiology and Immunology. Previously she was a Whitehead Fellow at the Whitehead Institute for Biomedical Research. Her research interests include cell motility, protein polymerization and large-scale self-organization in the cytoskeleton, cell shape determination, and the cell biology of bacterial infection. Dr. Theriot is a member of the American Society for Cell Biology, the American Society for Microbiology, the Biophysical Society, and the American Physical Society.
