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WO2006050364A2 - Debruitage d'image avec filtrage adaptatif a base d'informations theoriques - Google Patents

Debruitage d'image avec filtrage adaptatif a base d'informations theoriques Download PDF

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WO2006050364A2
WO2006050364A2 PCT/US2005/039491 US2005039491W WO2006050364A2 WO 2006050364 A2 WO2006050364 A2 WO 2006050364A2 US 2005039491 W US2005039491 W US 2005039491W WO 2006050364 A2 WO2006050364 A2 WO 2006050364A2
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image
pdf
intensities
pixel
entropy
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WO2006050364A3 (fr
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Suyash P. Awate
Ross T. Whitaker
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University Of Utah Research Foundation
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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T5/00—Image enhancement or restoration
    • G06T5/70—Denoising; Smoothing
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
    • G06T2207/20—Special algorithmic details
    • G06T2207/20004—Adaptive image processing
    • G06T2207/20012—Locally adaptive

Definitions

  • Nonlinear filtering approaches are typically based on either varia ⁇ tional methods, which result in algorithms based on partial differential equations (PDEs); or statistical methods, which result in nonlinear estimation problems.
  • PDEs partial differential equations
  • Nonlinear filters can overcome some of the limitations of linear filters, but they introduce some problems such as higher computational costs and the additional tuning of extra free parameters. Furthermore, most linear and nonlinear approaches enforce specific geometric or statistical assumptions on the image.
  • Vese has modeled textured images using functional minimization and partial differential equations by decomposing images as a sum of two functions — a cartoon-like image (bounded variation) and a texture image.
  • Weickert has proposed a coherence enhancing flow (not derived as a variation of an image energy), which preserves and enhances textures that exhibit a homogeneous structure tensor.
  • Several authors have proposed higher-order flows that correspond to piecewise-linear image models. These nonlinear PDE models have proven to be very effective, but only for particular applications where the input data is well suited to the model's underlying geometric assumptions.
  • the parameter tuning is a challenge because it entails fuzzy thresholds that determine which image features are preserved (or enhanced) and which are smoothed away.
  • the statistical approaches to nonlinear filtering fall into several classes.
  • One class is the methods that use robust statistics, the most prevalent being the median filter.
  • the median filter enforces a constant image model with an allowance for outliers; iterative applications of the median filter to ID signals result in piecewise-flat solutions.
  • Miller has proposed robust statistics for fitting higher-order models.
  • Bilateral filtering is a robust, nonlinear filtering algorithm that replaces each pixel by the weighted average over a neighborhood with a fuzzy mechanism for excluding outliers.
  • these statistical methods are essentially mechanisms for fitting simple geometric models to local image neighborhoods in a robust way.
  • MRFs Markov random fields
  • the Markov property for images is based on the assumption of spatial dependency or predictability of the image intensities — it implies that the probability of a pixel having a particular intensity depends only on the intensities of its spatial neighbors.
  • they describe an algo ⁇ rithm that relies on Gibbs sampling to modify pixel intensities.
  • Gibbs sampling assuming the knowledge of the conditional distributions of the pixel intensities given the intensities of their neighbors, generates a Markov chain of pixel intensities which converges point wise to the desired denoised image.
  • conditional probabilities for image neighborhood configu ⁇ rations play a similar role to the image energy in the variational approaches.
  • MRF image models often include extra parameters (hidden parameters) that explicitly model intensity edges, allowing these models to achieve piecewise-constant solu ⁇ tions.
  • these conditional probabilities encode a set of probabilistic assumptions about the geometric properties of the signal (noiseless image) .
  • Figure 1 shows the results of filtering on the Lena image using some of the prevalent nonlinear techniques, demonstrating their typical characteristics.
  • Perona and Malik's diffu ⁇ sion Figure Ic) eliminates the noise on the cheeks but introduces spurious edges near the nose and the lips.
  • Bilateral filtering ( Figure l(d)), which is essentially an integral form of anisotropic diffusion, tends to smooth away fine textures resulting in their elimination, e.g. on the lips. Both of these algorithms entail two free parameters (scale and contrast), and require significant tuning.
  • the coherence enhancing diffusion (Figure l(e)) forces specific elongated shapes in images, as seen in the enlarged nostril and the lips' curves.
  • Huang analyzes the statistical properties of the intensity and range values of natural images. These include single pixel statistics, two-point statistics and derivative statistics. They found that the mutual information between the intensities of two adjacent pixels in natural images is rather large and attributed this to the presence of spatial correlation in the images.
  • Lee and Silva analyze the statistics of 3 x 3 high-contrast patches in optical images, in the corresponding high-dimensional spaces, and find the the data to be concentrated in clusters and low-dimensional manifolds exhibiting a nontrivial topology. The work in this prior paper relies on the hypothesis that natural images exhibit some regularity in neighborhood structure in a parametric manner.
  • the literature shows several statistically-based image processing algorithms that do rely on information theory, such as the mean-shift method, which moves the samples uphill on a PDF associated with the data. This process produces a steady state in which all of the data have values corresponding to the nearest local maximum of the PDF (assuming appropriate windowing strategies).
  • the mean-shift procedure thus, can be said to be a mode s eeking process.
  • the mean-shift algorithm operates only on image intensities (be they scalar or vector valued) and does not account for neighborhood structure in images.
  • mean shift resembles a kind of image-driven thresholding process (particularly in the algorithm proposed by Comaniciu, in which the density estimate is static as the algorithm iterates).
  • DUDE addresses the problem of denoising data sequences generated by a discrete source and received over a discrete, memoryless channel.
  • DUDE assumes that the source aaid the received data sequences take values from a finite population of symbols and that the -transi ⁇ tion probabilities over the channel are known.
  • DUDE assumes no knowledge of the statistics of the source and yet performs (asymptotically) as well as any denoiser (e.g., one that knows the source distribution), thereby making DUDE more widely applicable.
  • DUDE assigns image values based on the similarity of neighborhoods gathered from image statistics, which resembles the construction of conditional probabilities in the proposed method.
  • the DUDE approach is limited to discrete-valued signals whereas the proposed method addresses continuous-valued signals, such as those associated with grayscale images. While the DUDE algorithm is demonstrably effective for removing em replacement noise, it is less effective in case of additive noise.
  • the present invention provides an unsupervised, information-theoretic, adaptive filter that improves the predictability of pixel intensities from their neighborhoods by decreasing the joint entropy between them. Accordingly, the present invention can programatically discover the statistical properties of the signal and can thereby reduce noise in a wide spectrum of images and applications.
  • PDFs probability density functions
  • PDF probability density function
  • a sample space is the set of all possible outcomes of a random experiment (A random experiment is an experiment whose outcome is not certain).
  • a random variable (RV) is a function that maps samples in the sample space to values.
  • the samples comprise pixel/neighborhood locations, and the values comprise the intensities.
  • a probability density function (PDF) is a function on the output of a RV that gives the probability of the value of the RV obtained on the performance of a random experiment.
  • One embodiment of the invention treats the collection of pixels in a neighborhood as a random variable which has an associated sample space and PDF.
  • the shape/structure of the PDF is used to modify pixel intensities and thereby filter or process the image.
  • the processing of images in a manner that uses multi-dimensional PDFs has not been done in the past.
  • the structure of the PDF is used to modify the intensities of center pixels based on the gradients of the PDF.
  • the PDF generated from samples in the entire image, provides the statistical distribution or likelihoods of intensities in neighborhoods.
  • only the center pixel for a neighborhood is modified by examining the gradient along one dimension (corresponding to that of the center pixel) to infer an intensity update for that pixel.
  • more dimensions may be used or changes can be made to two or more pixels in the neighborhood.
  • the pixel intensities are modified in order to optimize an information-theoretic measure, such as entropy, associated with PDFs of image neighborhoods, in order to filter or denoise images.
  • Entropy is a metric related to the shape of a PDF and pixel intensities can be modified so as to reduce entropy (resulting in a corresponding change in the PDF) using gradient descent methods.
  • the present invention also provides automatic scale selection for the nonparametric mul ⁇ tivariate density estimation of PDFs associated with image neighborhoods by optimizing the particular information-theoretic measure (such as entropy) of the PDF with respect to the scale or width of the Parzen window.
  • a nonparametric density estimation technique such as Parzen windowing
  • Parzen win ⁇ dow technique constructs an estimate by superimposing kernels placed at the locations of each data sample in the multi-dimensional space. Parzen windowing entails the setting of the scale parameter for the kernels used in the estimation process. Determining the optimal value of this parameter is crucial for the success of the filtering method.
  • the present invention also uses an anisotropic distance metric in the sample space in order to construct rotationally invariant neighborhood masks. Square neighborhoods gen ⁇ erate results with artifacts exhibiting preferences for grid-aligned features. A solution is to weight the intensities, making neighborhoods more isotropic. Such fuzzy weights can be incorporated by using an anisotropic distance metric in the multi-dimensional sample space. The weights correspond to a relatively isotropic kernel in the image space.
  • Another embodiment of the invention can use subspaces of image-neighborhood PDFs in order process pixels at image boundaries. Images have finite extent and there exists some neighborhoods near the image boundaries that lie partly outside the image where there is no data. Typical image boundary conditions, e.g.
  • replicating pixels or toroidal topologies can produce neighborhood intensities distorting the feature-space statistics.
  • the boundary neighborhoods can be handled by collapsing the feature space along the dimensions corresponding to the neighbors falling outside the image. Accordingly, the square regions crossing image boundaries can be cropped and processed by analyzing the PDFs associated using the lower-dimensional subspace.
  • the present invention provides an unsuper ⁇ ised information-theoretic adaptive filter for image denoising and denoises pixels by comparing pixel values with other pixels in the image that have similar neighborhoods.
  • the underlying formulation relies on an information- theoretic measure of goodness combined with a nonparametric model of image statistics.
  • the information-theoretic optimization measure relies on the entropy of the patterns of intensities in image regions. Entropy is a nonquadratic function of the image intensities, and therefore the filtering operation is nonlinear.
  • the present system and method operates without a priori knowledge of the geometric or statistical structure of the signal or noise, but relies instead on some very general observations about entropy of natural images. It does not rely on labeled examples to shape its output, and is therefore unsuper ⁇ ised.
  • a statistical representation of the image is automatically derived from the input data and constructs a filtering strategy based on that model, it is adaptive. Moreover, the free parameters are automatically adjusted using a data-driven approach and information-theoretic metrics. Because the present invention is nonlinear, nonparametric, adaptive, and unsupervised, it can automatically reduce image noise in a wide spectrum of images and applications.
  • the present invention can be applied to images which tend to be two-dimensional in nature.
  • image is generally defined as a digital signal, consisting of one or more floating point or integer values, organized on a regular grid over any number of dimensions. The values are sometimes called intensities. Individual grid points are sometimes called pixels.
  • the invention can be applied to one dimensional type data, such as audio data, volumetric three-dimensional data, or data obtained from multiple modalities (where the appropriate computing power is available) .
  • the dimensions of the data should not be considered to limit the applications of the present invention because the filtering of the present invention may be used in many dimensions if desired.
  • An uppercase letter, e.g. X denotes a random variable (RV), which may be scalar /vector- valued, a,s necessary.
  • a lowercase letter, e.g. x denotes the value of a particular sample from the sample space of X.
  • p(x) denotes the probability density function (PDF) for X. Applying a function /(•) on X yields a new RV, /(X). When the function is the PDF p(x) itself, we refer to the new RV as p(X).
  • the Shannon entropy of a RV measures the information, or uncertainty, associated with the RV .
  • the differential entropy h(X) is
  • E p denotes the expectation of the RV with X drawn from the PDF p(x).
  • the conditional entropy between two continuous RVs X and Y measures the uncertainty remaining in X after observing Y. It is defined as the weighted average of the entropies associated with the conditional PDFs.
  • h(X, Y) is the joint entropy of the RVs X and Y .
  • a neighborhood or image neighborhood as used in the present discussion can be generally defined as the set of points on the grid in proximity to a specified point. In other words, the set of grid points which is within a. certain, usually small, distance from the specified point.
  • S (1, . . . , n) X (1, . . . , m).
  • a region r (s) G S centered at a pixel location s is the ordered set (i.e. vector) of pixels ⁇ t: ⁇ t — ⁇ d ⁇ , where the set ordering is based on the values of the two spatial coordinates of pixels t, and d denotes the region size.
  • the present invention filters images by increasing the predictability of pixel intensities from the neighborhood intensities.
  • This uses an entropy measure for intensities in image regions.
  • RV X S ⁇ — >• M. that maps each pixel in the image to its intensity.
  • the statistics of X are the grayscale statistics of the image.
  • x(s) — I(s) I(s)
  • every intensity in the image is seen as a realization of the RV X.
  • the random vector Y (X(s + ⁇ ), . . . , X(s + o c ⁇ ), X(s + o c+1 ), . . . , X(s + O n )) which captures the statistics of intensities in pixel neighborhoods.
  • the present method relies on the statistical relationship between the intensity of each pixel and the intensities in a set of nearby pixels defined by its neighborhood.
  • the present system and method employs a gradient descent to minimize entropies of the conditional PDFs.
  • the gradients of h(X ⁇ Y) have components corresponding to both the center pixel, x(s), and the neighborhood, y(s), and thus the entire neighborhood, (x(s), y(s)), would be updated for a gradient descent scheme.
  • the proposed entropy measure, Ji(X ⁇ Y — y(s)) makes sense for several reasons.
  • maximizing mutual information (I (X, Y)) and multiinformation (M(Zi, . . . , Z n )) penalizes joint entropy, it rewards higher entropy among the individual RVs.
  • the high-level structure of the method of the present invention is as follows.
  • the noisy input image namely /, consists of a set of intensities x(s). These values form the initial values of a sequence of images /°, J 1 , / 2 , . . .. 2.
  • Step 5 Based on a suitable stopping criterion, terminate, or go to Step 2. (Appendix B discusses more about stopping criteria.)
  • This method includes two important parameters.
  • the first is the size of the image neigh ⁇ borhoods (the parameter d in the previous discussion). Typically, values of 1 or 2 suffice (giving regions of sizes 3 x 3 or 5 x 5 and feature spaces of dimensions 9 or 25, respectively). However, as we shall see in later sections, more complex or noisy images may perform better with larger neighborhoods for reliable denoising.
  • the second free parameter is in the stop ⁇ ping criterion. For most natural images we would not expect the steady states of this filter to be an acceptable result for a denoising task — that is, we expect some degree of variation in neighborhoods.
  • the method is consistent with several conventional techniques for enforcing fidelity to the input data; such mechanisms inevitably introduce an additional parameter.
  • each step of the method operates on a single pixel (Step 4 above) is merely a gradient descent on the center pixel, the interactions from one iteration to the next are quite complicated.
  • the updates on the center-pixel intensities in Step 4 affect, in the next iteration, not only the center pixels but also the neighborhoods. This is because the image-regions, r(s)Vs, overlap and the set of pixels that form the centers of regions is the same as that which form the neighborhoods.
  • the present filtering consists of two kinds of processes. One is the first-order optimization process, which computes updates for pixels based on their neighborhoods.
  • the other second-order process causes updates of the neighborhoods based on the role of those pixels as centers in the previous iteration.
  • the result of the filtering can be seen as the quest for the steady state of the combination of these two processes.
  • the present invention exploits the Markov property of the images, but in a different context. Rather than imposing a particular model on the image, the present invention estimates the relevant conditional probability density functions (PDFs) from the input data and updates pixel intensities to decrease the randomness of these conditional PDFs.
  • PDFs conditional probability density functions
  • some of past filtering work has relied on the hypothesis that natural images exhibit some regularity in neighborhood structure, but the present method discovers this regularity for each image individually in a nonparametric manner.
  • the present invention also determines the mathematical relationship between a mean- shift procedure and entropy reduction and is thereby a generalization of the mean-shift method, which incorporates image neighborhoods to reduce the entropy of the associated conditional PDFs.
  • the random vector Z ⁇ (X, Y) is sphered, by definition.
  • a sphered random vector is one for which the marginal PDFs of each individual RV have the same mean and variance.
  • each marginal PDF is simply the grayscale intensity PDF, p(x), of the image.
  • This invention relies on the neighborhoods in natural images having a lower-dimensional topology in the multi ⁇ dimensional feature space. This is also a general property for multivariate data. Therefore, locally (in the feature space) the PDFs of images are lower dimensional objects that lend themselves to better density estimation.
  • the literature shows Parzen windows as an effective nonparametric density estimation technique.
  • the Parzen- window density estimate p(z), in an n-dimensional space, is
  • ⁇ A ⁇ denotes the cardinality of the set A
  • Zj is a shorthand for Z(SJ).
  • the set A is a randomly selected subset of the sample space.
  • One embodiment of the invention chooses to use G(z, ⁇ ) as the n-dimensional Gaussian
  • ⁇ is the n x n covariance matrix.
  • Equation 1 gives entropy as an expectation of a RV.
  • the approximation for entropy follows from the result that the sample mean converges, almost surely, to the expectation as the number of samples tends to infinity.
  • Equations 1, 3, and 5 give
  • the set A which generates the density estimate p(zj), should not contain the point s » itself — because this biases the entropy estimates.
  • the samples in set A are, typically, a small fraction of those in B, chosen at random.
  • the relatively small cardinality of A has two important implications. First, it significantly reduces the computational cost for the entropy estimation, from O(
  • a gradient descent entropy optimization technique results in a stochastic-gradient method.
  • the stochastic-gradient effectively overcomes the effects of spurious local maxima introduced in the Parzen-window density estimate using finitely many samples.
  • the proposed entropy-estimation scheme is important not only for computational efficiency but also for effective entropy minimization. 5 Conditional Entropy Minimization
  • Entropy minimization in the present method relies on the derivative of the entropy with respect to the center-pixel value of the samples in B.
  • Each pixel intensity in the image undergoes a gradient descent, based on the entropy of the conditional PDF estimated from A.
  • the gradient descent for X 1 ⁇ x(s t ) for each S 1 € B is
  • the Parzen- window density estimate shows a great deal of sensitivity for different values of ⁇ .
  • the particular choice of the standard deviation ⁇ , and thereby ⁇ , for the Gaussian in Equation 3 is a crucial factor that determines the behavior of the entire process of entropy optimization.
  • this choice is related to the sample size ⁇ A ⁇ in the stochastic approximation.
  • ⁇ A ⁇ we propose to use the ⁇ that minimizes the joint entropy, which we will call the optimal scale for a data set. This can be determined automatically at each iteration in the processing. Our experiments show that for sufficiently large ⁇ A ⁇ the entropy estimates and optimal scale are virtually constant, and thus
  • the square neighborhoods de ⁇ scribed in Section 3.2 show anisotropic artifacts, and favor features that are aligned with the cardinal directions.
  • isotropic filtering results we use a metric in the feature space that controls the influence of each neighborhood pixel so that the resulting mask is more ro- tationally symmetric. In this way directions in the feature space corresponding to corners of neighborhood collapse so that they do not influence the filtering.
  • a similar strategy enables us to handle image boundaries without distorting the statistics of the image. That is, pixels at image boundaries rely on the statistics in lower-dimensional subspaces corresponding to the set of neighborhood pixels lying within the input image.
  • the mean-shift procedure moves each sample in a feature space to a weighted average of other samples using a weighting scheme that is similar to Parzen windowing. This can also be viewed as moving samples uphill on a PDF.
  • Comaniciu and Meer propose an iterative mean-shift method for image intensities (where the PDF does not change with iterations) that provides a mechanism for image segmentation.
  • Each grayscale (or vector) pixel intensity is drawn toward a local maximum in the grayscale (or vector-valued) histogram.
  • This is exactly the mean-shift update proposed by Fukunaga. Note that here the PDFs on which the samples climb get updated after every iteration.
  • the mean-shift method is a gradient descent on the entropy associated with the grayscale intensities of an image. We observe that samples x(s) are being attracted to every other sample, with a weighting term that diminishes with the distance between the two samples.
  • the updates in this invention have the same form, except that the weights are influenced not only by the distances/similarities between intensities x(s) but also by the distances/similarities between the neighborhoods y(s). That is, pixels in the image with similar neighborhoods have a relatively larger impact on the weighted means that drive the updates of the center pixels.
  • Figure 3 shows the result of 11 iterations of the present invention on the Lena image with spatially local sampling (explained in Appendix E).
  • the present method preserves and enhances fine structures, such as strands of hair or feathers in the hat, while removing random, noise.
  • the results are noticeably better than any of those obtained using other methods shown in Figure 1.
  • a relatively small number of iterations produce subjectively good results for this image — further processing oversimplifies the image and removes significant details.
  • the fingerprint image in Figure 4 shows another example of the structure-enhancing tendencies of the present system and method, which enhances the contrast of the light and dark lines without significant shrinkage.
  • a kind of multidimensional classification of image neighborhoods can be performed — therefore features in the the top-left are lost because they resemble background more than ridges.
  • Figure 5 presents the results of other denoising strategies for visual comparison with the present method.
  • We see that the piece-wise smooth image models associated with anisotropic smoothing, bilateral filtering, and curvature flow (Figure 5(a)-(c)) are clearly inappropriate for the this image.
  • a mean-shift procedure (Figure 5(d)) on image intensities (with the PDF not changing with iterations) yields a thresholded image retaining most of the noise.
  • Figure 6 shows the results of processing an MRI image of a human head for 8 iterations.
  • This example employs the local sampling strategy (explained in Appendix E), and shows the ability of the present invention to adapt to a variety of grayscale features. It enhances structure while removing noise, without imposing a piecewise constant intensity profile. As with the Lena example, more iterations tend to slowly erode important features.
  • Figure 7 gives a denoising example involving large amounts of noise.
  • the checks are 4x4 pixels in size and the defined neighborhoods are 5x5 pixels. All of the edges can be restored and the corners and the image boundaries show no signs of artifacts.
  • Figure 10(a) shows that the RMS error (root of the mean squared difference between pixel intensities in the filtered image and the noiseless image) decreases by 90 percent.
  • Figure 12(c) shows that the joint entropy and the Parzen window size, ⁇ , decrease monotonically as the filtering progresses. For this example, a multi-threaded implementation takes roughly 1 hour for 100 iterations with a pair of Intel®XeonTM2.66 GHz Pentium 4 processors (shared memory).
  • Figure 8 shows the results of applying this invention to another corrupted binary im- age.
  • the smoothness of the resulting circle boundary demonstrates the effectiveness of this method in preserving rotational invariance, as explained in Appendix C.
  • the edges of the square are also well restored, but, unlike the checkerboard example, the corners are rounded.
  • the presence of many similar corners in the checkerboard image form a well defined pat ⁇ tern in feature space, whereas the corners of the square are unique features (in that image) and hence, treats them more like noise — those points in the feature space are attracted to more prevalent (less curved) features.
  • Figure 9 shows the application of 15 iterations of this method to an image of hand-drawn curves (with noise).
  • the present invention learns the pattern of black-on- white curves and forces the image to adhere to this pattern. However, mistakes may be made by the present method when curves become too close, exhibit a very sharp bend, or when the noise introduces ambiguous gaps.
  • Figure 10(b) shows comparisons of the performance with three different region sizes, i.e. 3 x 3, 5 x 5 and 7 x 7, that reflect the advantage of larger neighborhoods. For higher levels of noise, we find that larger neighborhoods are able to better discern patterns in image regions and yield superior denoised images.
  • the present method is effective for removing various kinds of noise. Filtering of the checkerboard image with correlated noise (gotten by applying a low-pass filter to zero-mean, independent, Gaussian noise) shows a significant improvement in RMS error (see Figure 10(c)), but the reduction in RMS error is not as good as in the examples with uncorrelated noise. Correlated noise modifies the PDF p(x, y) by creating more local maxima and thereby fractures the manifold associated with the original data.
  • the strategy for automatically choosing the Parzen window size, ⁇ , together with the joint entropy minimization scheme, is unable to remove these new feature-space structures. We can verify this by artificially increasing the size of the Parzen window.
  • Figure 11 shows that Parzen windowing, using a finite number of samples, is very sensitive to the value of ⁇ .
  • Many methods/applications with low dimensional features spaces e.g. 2 or 3 operate by manually tuning the scale parameter.
  • the present method relies on a sparsely populated high dimensional space, it is very difficult to manually find values for ⁇ that properly "connect" the data without excessively smoothing the PDF.
  • the best scale parameter changes every iteration, and ⁇ is found via a data-driven approach. Because the goal is to minimize joint entropy, a logical choice is to choose a value for ⁇ that minimizes the same.
  • Figure 12(a) confirms the existence of a unique minimum.
  • Figure 12(b) shows that the choice of ⁇ is not sensitive to the value of ⁇ A ⁇ for sufficiently large ⁇ A ⁇ , which automatically fixes ⁇ A ⁇ to an appropriate value before the filtering begins.
  • Figure 12(c) depicts the decreasing trend for ⁇ as the filtering progresses, which is common to every example and is consistent with entropy-reducing action bringing samples closer in the feature space.
  • the method may terminate when the residual (RMS difference between input and output) equals the noise level.
  • termination can be based on visual inspection of images in the filtered image sequence.
  • An alternative is to use the system and method of the present invention as part of a reconstruction process where entropy minimization acts as a prior that is combined with an image-data or fidelity term. This would allow the method to run to a steady state, and rather than a stopping criterion one can choose the relative weights of the prior and data, a so-called meta parameter. If the noise level is known, one can avoid the meta parameter and treat residual magnitude as a constraint.
  • Rotational invariance does not follow from present method's formulation as illustrated in FIG. 13, because the samples are taken on a rectilinear grid. Square neighborhoods generate results with artifacts exhibiting preferences for grid-aligned features. A solution is to weight the intensities, making neighborhoods more isotropic. Fuzzy weights can also be incorpo ⁇ rated by using an anisotropic feature-space distance metric, ⁇ Z ⁇ M — Vz T Mz, where M is a diagonal matrix. The diagonal elements, TO 1 , . . . , TO n , are the appropriate weights on the influence of the neighbors on the center pixel. To have a sphered RV Z, (that aids in density estimation; Section 4), the weights used can be somewhat homogeneous.
  • Figure 13(a)-(b) shows a disk-shaped mask that achieves this balance.
  • the proposed isotropic mask is a grayscale version of the DUDE strategy of using a binary disc-shaped mask for discrete (half-toned) images. Note that scaling the center-pixel intensity more than its neighbors leads to an elongated space (X', Y') along X' — in the limit, when all neighbors are weighted zero, leading to a thresholding as in the mean-shift method.
  • Typical image boundary conditions e.g. replicating pixels or toroidal topologies, can pro ⁇ cute neighborhoods distorting the feature-space statistics.
  • Boundary neighborhoods can be handled using a strategy similar to that in Appendix C, by collapsing the feature space along the dimensions corresponding to the neighbors falling outside the image. The square regions crossing image boundaries can be cropped and processed in the lower-dimensional subspace. This strategy results in important modifications in two stages of processing. First, the cropped intensity vectors take part in a mean-shift process reducing entropies of the con ⁇ ditional PDFs in the particular subspace where they reside. Second, a Parzen window size can be chosen ⁇ , based only on the regions lying completely in the image interior. E Selecting Random Samples
  • Parzen- window density estimation entails the selection of a set of samples belonging to the density in question, as seen in Section 4. Nominally, this set comprises random samples drawn from a uniform PDF on the sample space.
  • the strategy works well if the image statistics are more or less uniform over the domain, e.g. the fingerprint image in Figure 4. However, it fails if the statistics of different parts of the image are diverse, e.g. the Lena image in Figure 3. This is because distant parts, for such an image, produce samples lying in distant regions of the feature space.
  • p(z(sj)) by selecting random samples S j in a way that favors nearby points in the image domain — using a Gaussian distribution centered at s* with a relatively small standard deviation (10 pixels). This strategy is appropriate for images having locally consistent neighborhood statistics.

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Abstract

Système et procédé de débruitage d'un signal présentant une pluralité d'intensités. Le procédé peut englober les opérations suivantes: identification d'un voisinage de l'image à partir d'une pluralité d'intensités; traitement du voisinage de l'image en tant que variable aléatoire avec espace d'échantillon associé et fonction de densité de probabilité (PDF); et modification de l'intensité des pixels à partir des mesures d'informations appliquées à la fonction PDF dans au moins une dimension.
PCT/US2005/039491 2004-10-28 2005-10-28 Debruitage d'image avec filtrage adaptatif a base d'informations theoriques WO2006050364A2 (fr)

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US62320504P 2004-10-28 2004-10-28
US60/623,205 2004-10-28

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