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Sampling of Planar Curves: Theory and Fast Algorithms
We introduce a continuous domain framework for the recovery of a planar curve from a few samples. We model the curve as the zero level set of a trigonometric polynomial. We show that the exponential feature maps of the points on the curve lie on a low-dimensional subspace. We show that the null-space vector of the feature matrix can be used to uniquely identify the curve, given a sufficient number of samples. The theoretical guarantees show that the the number of samples required for unique recovery depends on the bandwidth of the underlying trigonometric polynomial, which is a measure of the complexity of the curve. We introduce an iterative algorithm that relies the low-rank property of the feature maps to recover the curves when the samples are noisy or when the true bandwidth of the curve is unknown. We also demonstrate the preliminary utility of the proposed curve representation in the context of image segmentation.
The recovery of a curve from finite number of unorganized and noisy points is an important problem, with applications to computer vision [1, 2] and image processing [3, 4, 5]. Since one can find infinite number of curves that pass through the points, this problem is inherently ill-posed. In addition, the recovery is challenging due to the topology and its variation with noise. Popular approaches for the representation of shapes with arbitrary topology include (a) explicit representations using a mesh or graphs [1, 6], and (b) implicit level-set representations [7, 8, 9]. In the first scheme, the shape is constructed from the noisy points as a graph, where the nodes corresponding to adjacent data points are connected. By contrast, level set functions are often constructed from the points as distance functions using the information of normals or by solving a linear system of equations involving radial basis functions . Several methods exist to address the issue of noisy data. For instance, smoothing using spectral graph theory, Laplacian/curvature flow [11, 12] are popular approaches in the context of mesh/graph based representation. Similar iterative strategies do exist in the context of level-set schemes. All of these methods suffer from the inherent parametrization of the curve, which often depends on the sampling density.
The main focus of this paper is to introduce a unified continuous domain perspective for the recovery of points on a curve in a plane. We assume that the points live on a smooth curve, which is the zero level set of a bandlimited function. We note that the bandwidth/support is a measure of the complexity of the curves that can be represented as the zero level set. The support of the function may be large in one dimension and much smaller in the other, which allows for the adaptation of the representation to the shape of the curve. We show that the non-linear features of any arbitrary point on such a curve, obtained by the lifting of the point using an exponential map, can be annihilated by the inner-product with the coefficients of the level set function. We term this property as the annihilation relation. The dimension of the feature maps is dependent on the bandwidth, and hence the complexity of the curve. When multiple points on the curve are available, the coefficients of the level set will annihilate the feature vectors of all of the points. Thus, the coefficient vector is a null-space vector of the feature matrix, whose columns correspond to the exponential features of the points. We use this property to determine the sampling conditions, which guarantees the recovery of the curve from finite number of points. Our results show that the bandlimited function, and hence the curve, can be recovered uniquely, when the number of points exceed a certain number that is dependent on the curve complexity.
We also introduce efficient strategies when the bandwidth of the curve is unknown and the samples are noisy. In practical applications, the precise bandwidth/support of the level set function is usually unknown. We show that when the support is overestimated, there exist multiple linearly independent filters that will annihilate the exponential maps; the common zeros, or equivalently the zero level set of the greatest common divisor of the filters, uniquely specifies the curve in this case. This also implies that the feature matrix has multiple linearly independent null-space vectors and hence is low-rank. In other words, the exponential features of the points on the curve live in a finite dimensional space when the bandwidth is overestimated. Note that the Gram matrix of the exponential features correspond to a kernel matrix, which connects the bandlimited curve model with widely used non-linear low-rank kernel methods . When the curve samples are noisy, we rely on a nuclear norm minimization formulation to denoise the points. Specifically, we seek to find the denoised curve samples such that their feature vectors form a low-rank matrix. We use an iterative reweighted algorithm to solve the above optimization problem, which alternates between the estimation of a weight matrix that approximates the nullspace and a quadratic sub-problem to recover the data. We note that the iterative algorithm bears strong similarity to Laplacian/curvature flows used in graph denoising, which provides the connection between implicit level-set and explicit graph-based curve representations. One can also derive a graph Laplacian matrix from the weight matrix, which will facilitate the smoothing of signals that live on the nodes of the graph. This graph can be viewed as a discrete mesh approximation to the points that live on the curve. Our experiments show that the Laplacian matrix obtained by solving the proposed optimization algorithm is more representative of the graph structure than classical methods , especially when it is estimated from noisy data. This framework reveals links between recent advances in superresolution theory [15, 16, 17], manifold smoothness based regularization, as well as graph signal processing .
This work has connections with Logan’s results  for the recovery of 1-D bandlimited functions from their zero crossings, as well as their extensions to 2-D . The main challenge of these works is the extreme sensitivity of the bandlimited function to the location of the zero-crossings, when no amplitude information of the signal is used ; this has prompted the use of additional information including multi-level contours  and multi-scale edges . By contrast, we focus on the recovery of the curve itself, rather than the bandlimited function, which is considerably simpler. Specifically, we propose to recover the curve as the zero level set of the sum of squares of all band-limited functions that satisfy the sampling conditions. In addition, unlike , our results are also valid for the union of irreducible curves. The proposed work is built upon our prior work on annihilation based super-resolution image recovery [22, 16, 23, 24, 25, 26] that has similarities to algebraic shape recovery  and the recent work by Ongie et al., which considered polynomial kernels . Our main focus is to generalize  to shift invariant kernels, which are more widely used in applications. This approach can also be viewed as the non-linear generalization of the finite rate of innovation theory [28, 29, 30, 31, 32]. We also introduce sampling conditions and algorithms to determine the curve, when the dimension is low. In addition, the iterative algorithm using the kernel trick shows the connections with graph Laplacian based methods used in graph signal processing. The conference version of this work has been published in [33, 34]; this paper provides the proofs of these results, and more elaborate description of details.
Ii-a Parametric level set representation of curves
We model the curve in , as the zero level set
of . We denote the curve specified by the zero level set of by . Note that the level set representation of curves is widely used in segmentation and shape representation . Several authors have proposed to represent as a linear combination of basis functions [8, 9]:
A popular choice is the shift invariant representation  using compactly supported basis functions such as B-splines [9, 8] or radial basis functions . In this case, we have , where is a B-spline or Gaussian function. Here, denotes the number of basis functions, which is equivalent to the number of grid points in the context of shift-invariant representations.
Ii-B Bandlimited level set representation
We now consider the special case of parametric level set curve representation, where the basis functions are chosen as . Then
Here, the Fourier coefficients are supported on a rectangular grid . See Fig. 1 for an illustration. Note that for each index , the basis functions satisfy and hence are not compactly supported as in [8, 9]. The above bandlimited representation has a one-to-one correspondence with curve representation using complex polynomials, as shown in II-C. We hence also refer to the bandlimited function in (3) as a bandlimited polynomial.
We note that the bandlimited level set model can represent closed curves with arbitrary complexity, as illustrated in Fig. 2. The representation cannot represent open curves in the strict sense. However, we note that such an open curve can be approximated with arbitrary accuracy by a bounding closed curve . As can be appreciated from the figure, the complexity of the curve is dependent on the cardinality of the set , which we denote by . This is the number of free parameters in the curve representation. We also assume that is the smallest set of coefficients (minimal set) that satisfies the above relation. We note that in the 1-D setting (), there is a one-to-one correspondence between the number of points satisfying (1) and the bandwidth [16, 27]; this relation enabled the use of in [16, 26] as a surrogate for sparsity in signal recovery. In higher dimensions, can still serve as a complexity measure, when the recovery of isolated Diracs is considered. However, we emphasize that (3) can provide a significantly richer representation, even when the signal is not isolated and consists of points on a curve.
Ii-C Relation between bandlimited functions & polynomials
We now briefly introduce the relation between bandlimited functions and complex polynomials, which we will use in the rest of the paper. We define a one-to-one mapping from trigonometric polynomials to complex polynomials, using the change of variables: . This maps each to the unit circle in the complex plane . This mapping allows us to rewrite (3) as a complex polynomial
We note that is a one-to-one mapping from to . Accordingly, we can say that the two sets
are isomorphic. Thus, we can analyze the properties of (including the number of zeros) to obtain the corresponding properties of the trigonometric polynomial (3).
Definition 1 (Irreducible polynomials).
A trigonometric polynomial is termed as irreducible, if the polynomial specified by is irreducible. A polynomial is irreducible over a field of complex numbers, if it cannot be expressed as the product of two non-constant polynomials with complex coefficients.
Definition 2 (Irreducible curve).
A curve is termed as irreducible, if it is the zero level set of an irreducible polynomial.
Iii Sampling of bandlimited curves
Iii-a Annihilation relations for points on the curve
Note that is a non-linear mapping or lifting of a point to a high dimensional space, whose dimension is given by the cardinality of the set , denoted by . Note that this non-linear lifting strategy is similar to feature maps used in kernel methods. We hence term as the feature map of the point . Note that every point on the curve satisfies (4), which we term as the annihilation relation.
Let us now consider a set of points on the curve, denoted by . Note that the feature maps of each one of the points satisfy the above annihilation relations, which can be compactly represented as:
Here, is the feature matrix of the points and .
When is rank-deficient by one, the coefficient vector can be identified as the unique null-space vector of . This implies that the features lie in an dimensional subspace, whose normal is specified by . This annihilation relation is illustrated in Fig 1, in the context of bandlimited curves considered in the next subsection. We will show that there exists a unique nullspace vector when complex exponential basis functions are chosen as in Section II-B.
In practice, the points are often corrupted by noise. In the presence of noise, the null-space conditions are often not satisfied exactly. In this case, we can pose the least square estimation of the coefficients from the noisy data points as the minimization of the criterion:
where . To eliminate the trivial solution , we pose the recovery as the constrained optimization scheme:
The solution is the minimum eigenvector of .
Iii-B Sampling of an irreducible bandlimited curve
We now focus on the problem of the recovery of the curve (1), given a few points on the curve. Let us take the bandlimited curve representation as (3) for the rest of the section to derive our sampling conditions. We now determine the sampling conditions for the perfect recovery of the curve using (7), when the curve is specified by (3). In this case, the annihilation relation (4) is satisfied with the feature maps defined as
We also assume that is a rectangular neighborhood in of size . We first review some results from algebraic geometry.
There is a one-to-one correspondence between trigonometric polynomials and complex polynomials, as shown in Section II-C. We use the extension of Bézout’s inequality for trigonometric polynomials, which bounds the number of solutions of the system that do not have any common factors.
Lemma 3 (Bézout’s inequality for bandlimited polynomials).
Let and be two bandlimited polynomials, whose Fourier coefficients are support limited to and , respectively. If and have no common factor, then the system of equations
has a maximum of solutions in .
Let be distinct points on the zero level set of an irreducible bandlimited function , whose Fourier coefficients are restricted to a rectangular region with size . Then the curve can be uniquely recovered by (5), when:
The proof is provided in Appendix VI-B.
Note that the above sampling condition does not specify any constraint on the distribution of points on the curve; any set of points are sufficient for the recovery of the curve. This property is similar to well-known results in non-uniform sampling of bandlimited signals , where the recovery is guaranteed under weak conditions on the nonuniform grid and the average sampling rate exceeding Nyquist rate.
We compare this setting with the sampling conditions for the recovery of a piecewise constant image, whose gradients vanish on the zero level set of a bandlimited function . The minimum number of Fourier measurements required to recover the function there is . When , then complex Fourier samples are required, which is far more than real samples required for the recovery of the curve in our setting. Note that the constant values within the regions bounded by the curves also need to be recovered in , which explains the higher sampling requirement.
Iii-C Sampling theorem for union of irreducible curves
We now generalize the previous result to the setting where the composite curve is a union of multiple irreducible curves. Equivalently, the level set function is the product of multiple irreducible bandlimited functions. We have the following result for this general case:
Let be points on the zero level set of a band-limited function , where the bandwidth of is specified by . Assume that has irreducible factors (i.e., ), where the bandwidth of the factor is given by . The curve can be uniquely recovered by (5), when each of the irreducible curves are sampled with
The total number of samples needed for unique recovery is specified by
which is upper bounded by .
We note that the upper bound can be approximated as for small values of , which is the upper bound in Proposition 4. The above result is proved in Appendix VI-C. Note that unlike the case considered in Section III-A, an arbitrary set of samples cannot guarantee the perfect recovery. Each of the irreducible curves need to be sampled proportional to their complexity, specified by to guarantee perfect recovery.
We verify the above proposition in Fig. 3. We consider a curve , where is support limited to a region. We consider the recovery from different number of samples of in the middle row, sampled randomly. The random strategy ensures that the samples are distributed to the factors, roughly satisfying the conditions in Proposition 5. Note that the theory guarantees recovery, when the number of samples exceeds around samples. We observe good recovery of the curve around 50 samples; note that our results are worst-case guarantees, and in practice fewer samples are sufficient for good recovery of most curves.
We further studied the above proposition in Fig. 4. We considered several random curves, each with different bandwidth and considered their recovery from different number of samples. The sampling locations were picked at random. The colors indicate the average reconstruction error between the actual curve and the reconstructed curves. This reconstruction error is computed as the sum of distances between each point on one curve and the closest point to it on the other curve. We have also plotted the upper bound in red, while the number of unknowns in the curve representation is plotted in blue. We note that the curve can be recovered accurately when the number of samples exceed the upper bound. We also note that in general, good recovery can be obtained for most curves, when the number of samples exceed .
Iii-D Curve recovery with unknown Fourier support
Propositions 4 and 5 assume that the true support of the Fourier coefficients of , specified by is known, in addition to the points . However, typically only the points will be known and the filter support will be unknown. We now consider the case where the filter support is over-estimated as . We focus on the recovery of the coefficients from the annihilation relation
The following result shows that the above matrix will have multiple linearly independent null-space vectors. However, if the curves are sampled as described below, the corresponding bandlimited functions satisfy some desirable properties that facilitate the recovery of the curves.
Consider the zero level set of the bandlimited polynomial with irreducible components, as described in Proposition 5. Let the assumed bandwidth of the curve be with and . Then, there exist multiple functions that satisfy . If the irreducible curves of the zero level set of are sampled with
all of the above functions, or equivalently the right nullspace vectors of , will be of the form:
where is an arbitrary function such that .
Note that the minimal function is a special case of (15), with . The above result is proved in Appendix VI-D. Since is the common factor of all the annihilating functions, all of them will satisfy , for any point on the original curve as well as the sampling locations. This also implies that is a common divisor of the above functions . In fact, is the greatest common divisor as we will show it in the next paragraph. We now characterize the number of linearly independent annihilation functions, or equivalently the size of the right null space of .
This set is illustrated in Fig 6 along with and . The above results provide us a means to compute the original curve, even when the original bandwidth/support is unknown. Specifically, Proposition 7 shows that has null-space vectors, each of which satisfies (15). Besides, from the proof of Proposition 7, we can see that any polynomial of the form
is a null-space vector of . Note that the exponentials are linearly independent, and hence the set spans the null space of . Since does not vanish in the domain, the only common zeros of will be the zeros of the minimal polynomial , meaning that is the greatest common divisor of the functions that span the null-space of . Therefore, the common zeros of these functions, or equivalently the zeros of the greatest common divisor, will specify the curve. A cheaper alternative to evaluating the greatest common divisor is to evaluate the sum of squares polynomial, specified by:
which will vanish only on points satisfying . Here is the dimension of the right null-space of . Since is a valid right null-space vector of that only vanishes on the true curve, the sum of squares function specified in (19) will only vanish on the true curve. Thus, if the total number of points sampled are , and are arranged as (14), then the curve can be uniquely recovered.
We demonstrate the above result in Fig. 5. We considered the sampling of the same curve illustrated in Fig. 3, with the exception that we over-estimated the support to be as opposed to the true support of . We considered 100 random samples, which satisfies the sampling conditions in Proposition 6. We obtain a rank of as predicted by Proposition 7. We show three of the annihilating functions in the first three columns of Fig. 5. We note that all of these functions are valid annihilating functions, but possess additional zeros. By contrast, the sum of square polynomial shown on the right uniquely specifies the curve.
We illustrate the utility of the above results in specifying an arbitrary curve from few unorganized points in Fig. 7. In each of the three examples, the user clicked a few points, which are shown by the green circles. The rank of the feature matrix is determined, followed by the evaluation of the sum of squares function as described in (19). The curve specified by the zero set is illustrated by the red curve in these examples.
Iii-E Application of the curve representation in segmentation
The Mumford Shah functional is a popular formulation for segmenting objects into piecewise constant regions. It approximates an image by a piecewise constant function
in the sense, where are the regions and are the constants. The bounding curve is denoted by . Different penalties, including the length of or its smoothness are imposed to regularize the optimization problem. We propose to represent as the zero level set of a bandlimited function specified by (3) as in . In this case, the piecewise constant function satisfies , which can be expressed in the matrix form as
where is a block Toeplitz 2-D convolution matrix and is the vectorized version of . When the bandwidth is over-estimated, has multiple linearly independent null-space vectors and hence the matrix is low-rank. Note that the rank of the matrix can be considered as a surrogate for the complexity of the curve . We hence formulate the segmentation task as the low-rank optimization problem, analogous to .
where is the original image. Note that as , approaches a rank matrix. Once is obtained, the sum of square function of the null space of will specify the curve and is the piecewise constant approximation. We use an alternating minimization strategy as reported in  to solve the above optimization scheme.
We demonstrate the preliminary utility of this scheme in Fig. 8, where the alternating algorithm is initialized with and iterated until convergence. The parameter is set to a high value (e.g. in our experiments) to enforce the rank constraint. The rows correspond to the results with different values of rank. The variation of the shape with rank in Fig. 8 shows that the rank of serves as a good surrogate for the complexity of the curve. We note that these experiments are meant to demonstrate the preliminary utility of the curve representation in segmentation. Several novel image energies including region based formulations are available, which offers significantly better performance than the simple edge based formulation considered in this preliminary work. We note that most of these energies can be incorporated easily into this formulation. However, developing this framework to obtain a state of the art segmentation scheme is beyond the scope of this paper and will be dealt elsewhere.
Iv Recovery from noisy samples
We now consider the case where we have noisy measurements of points lying on the curve , where is represented as a linear combination of basis functions as in (4). When the measurements are noisy, we propose to denoise them using the low-rank property of the features discussed in Proposition 7.
Iv-a Relation to kernel methods
Proposition 7 indicates that we can solve inverse problems by enforcing a low-rank constraint on the feature matrix. However, the size of the feature matrix grows with the dimensionality of the ambient space as well as the size of the over-estimated filter support . Thus, in practice, it might be infeasible to form the feature matrix. However, the Gram matrix given by
is of size , where is the number of points. Note that the complexity of an algorithm that depends on the Gram matrix is independent of the dimension of the ambient space and the chosen filter support . We note that this approach is similar to the “kernel-trick” used in various machine learning applications. Under our assumed model, the entries of this Gram matrix are given by
When is a centered cube in , is a Dirichlet function. Note that the entries of can be evaluated without explicitly evaluating the feature matrix. The width of the Dirichlet function is dependent on the Fourier support . The kernel matrix satisfies , where is given by (16). The relationship (23) implies that the kernel matrix is low-rank, which is a property that is widely used in kernel low-rank methods.
A popular practical choice for kernel is Gaussian functions, or equivalently periodized Gaussian functions when the domain is restricted to . Note that the Gaussian kernel is qualitatively similar to the Dirichlet kernel considered above, where the width of the Gaussian is a parameter similar to the size of the support . We note that the Gaussian kernel function is less oscillatory and is isotropic, which makes it more attractive than Dirichlet kernels in applications. The Gaussian kernel correspond to feature maps of the form
as . Note that this is the Fourier transform of a shifted Gaussian; this setting corresponds to the case where the level set function is being expressed as a shift invariant linear combination of Gaussian functions on a very fine grid. Since the Gaussian kernel matrix is theoretically full rank, the theoretical analysis in the previous section is not directly applicable to Gaussian kernels in the strict sense. However, we observe that the Fourier series coefficients of a Gaussian function may be approximated to be zero outside , which translates to . This implies that the Gaussian kernel matrix can be safely approximated to be low-rank, when is sufficiently high; this corresponds to a more localized kernel in space.
In the next subsection, we will describe how to use proposition 7, without explicitly forming the feature matrix , and computing only it’s Gram matrix instead.
Iv-B Denoising of point clouds using nuclear norm minimization
With the addition of noise, the points deviate from the zero level set of . A high bandwidth potential function is needed to represent the noisy curve. Let the noisy measurements of the matrix be given by . We propose to use the nuclear norm of the feature matrix as a regularizer in the recovery of the points from noisy measurements:
We use an IRLS approach where the nuclear norm is approximated as:
where and is the Gaussian kernel. Here, is a small constant added to ensure that the inverse is well-defined.
The IRLS algorithm alternates between the following two steps:
Here, , and is a constant.
Linearizing the gradient with respect to , we obtain:
where is the entry of a matrix
In matrix form, the gradient can be rewritten as , where is computed from the weight matrix as
Here, is a diagonal matrix with elements defined as .
This results in the following equivalent optimization problem for the estimation of at the iteration, which can be solved analytically:
Thus, we alternate between the estimation of and till convergence to solve (26). We note that this optimization algorithm is non-convex and does not come with any convergence guarantees. However, with reasonable initialization (e.g. ), the algorithm yielded good results in practice.
We illustrate the utility of the kernel low-rank formulation to denoise 2D points in Fig. 9. The first column correspond to noiseless samples, drawn from a TigerHawk logo image. We consider the recovery of this shape from its noisy samples, shown in the second column. Specifically, we added random Gaussian noise to each of the samples. The recovery of the points using conventional Laplacian flow is shown in the third column. Here, the kernel entries are evaluated using the exponential kernel as in , which corresponds to the first iteration of our algorithm. Iterating the optimization process yields improved results as shown in the last column. We also show the ability of the algorithm to yield improved Laplacian matrices, as shown from the eigenvectors of the Laplacian matrices in rows 3-5. Note that the eigenvectors of graph Laplacian matrices are smooth functions on the shapes, analogous to Fourier exponentials on the real plane; this property is exploited extensively in spectral graph theory [41, 42] and the spectral features are often used as fingerprints for shape [43, 44]. We note that the second eigenvector of the Laplacian matrix estimated from the original data, noisy data, data smoothed using exponential kernels (1st iteration of our algorithm) and 50th iteration our our algorithm using (33) agree well. By contrast, the 22nd eigenvector estimated from the noisy data disagrees significantly from the original one; it shows a bright spot, indicating a highly localized function on the shape. We note that the 32nd eigenvector of the proposed scheme is still comparable in smoothness to the corresponding original eigenvector, while both others are more localized. This shows that the Laplacian features estimated using the proposed scheme are more reliable predictors of the original shape.
We introduced a continuous domain framework for the recovery of points on a bandlimited curve. We relied on annihilation relations between complex exponential features of the points on the curve and the coefficient vector of the bandlimited representations to recover the curve. We have introduced sampling conditions that guarantee the recovery of the curve from finite number of its measurements. We also introduce efficient algorithms for the recovery of the points, when they are corrupted by noise. We observe that the Laplacian matrix estimated during the denoising process is a more accurate representation of the shape geometry rather than the ones estimated using conventional methods from noisy data. The preliminary utility of the parametric level set representation is demonstrated in the context of image segmentation.
Vi-a Proof of Lemma 3
We first state the well-known result for complex polynomials, which we extend to the bandlimited setting.
 Let and be two nonconstant polynomials in of degrees and respectively. If and have no common component, then the system of equations
has at most solutions.
Vi-B Proof of Proposition 4
The Fourier coefficients of is support limited within , which is the minimal support. Let be another bandlimited polynomial, whose Fourier coefficients are support limited within and satisfies , for . When the number of samples satisfy (10), this is only possible if is a factor of , according to Bézout’s inequality. Thus, must be a factor of . Since is irreducible, this implies that it is the unique bandlimited irreducible polynomial satisfying . ∎
Vi-C Proof of Proposition 5
The polynomial is represented in terms of its irreducible factors as:
where the bandwidth of is .
Let be another polynomial with bandwidth satisfying , for . Consider one of the irreducible sub-curves , that is sampled on points satisfying (11). According to Lemma 3, both and can be simultaneously zero at these sampling locations only if and have a common factor. Since is irreducible, this implies that is a factor of . Repeating this line of reasoning for all factors , we conclude that divides . Since both and have the same bandwidth, the only possibility is that is a scalar multiple of . This implies that the curve can be uniquely recovered in (11) is satisfied.
The total number of points to be sampled is .
The support of the Fourier coefficients of can be expressed in terms of the supports of . Using convolution properties, we get: and . Thus, and it can be concluded that . ∎
Vi-D Proof of Proposition 6
Following the steps of the proof for Proposition 5, we can conclude that is a factor of . Since , it follows that , where is some arbitrary function such that is bandlimited to . ∎
Vi-E Proof of Proposition 7
Let be the minimal filter of bandwidth , associated with the polynomial . We define the following filters supported in for all .
are the Fourier coefficients of , and are all null-space vectors of the feature matrix . The number of such filters is . Hence, we get the rank bound: .
If the sampling conditions of Proposition 6 are satisfied, then all the polynomials corresponding to null-space vectors of are of the form: . Alternatively, in the Fourier domain, the filters are of the form:
where are the Fourier coefficients of the arbitrary polynomial . Thus, all the null-space filters can be represented in terms of the basis set . This leads to the relation: . ∎
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