Matrix equation representation of the convolution equation and its unique solvability

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.


Introduction
We consider the following convolution equation * = , where = [ ] ∈ R 3×3 and = [ ] ∈ R × are given, and = [ ] ∈ R × is to be determined.The symbol * represents the convolution operator, i.e., the left-hand side of the ( , ) entry of (1) is defined by Since the above expression (2) is only defined for cases when 2 ≤ ≤ − 1 and 2 ≤ ≤ − 1, boundary conditions to define for cases when ∈ {1, } or ∈ {1, } are needed.In other words, the values of 0 , +1, , 0 , , +1 need to be set artificially.In this paper, we consider the following three boundary conditions.
The coefficient matrix F ∈ R × has a special structure that depends on boundary conditions.Under the zero boundary condition, F has a block Toeplitz with Toeplitz blocks (BTTB) structure.Under the periodic boundary condition, F has a block circulant with circulant blocks (BCCB) structure.Under the reflexive boundary condition, F is a sum of BTTB, a Block Toeplitz with Hankel blocks (BTHB), a Block Hankel with Toeplitz blocks (BHTB), and a Block Hankel with Hankel blocks (BHHB), i.e., a Block-Toeplitz-plus-Hankel with Toeplitz-plus-Hankel-Blocks (BTHTHB) matrix.Many algorithms for solving (1) are based on the vectorized form (6).Although the coefficient matrix F can be large, the above structures allow the solution of ( 6) to be computed efficiently without explicitly constructing F .For example, the following algorithms have been proposed: algorithms for the inversion of BTTB matrix [17], numerical algorithms and their preconditioners for solving BTTB systems [2,3,15,18], algorithms for solving BCCB systems by using the fast Fourier transform (FFT) [4,12], and numerical algorithms for solving BTHTHB systems by using discrete cosine transform (DCT) [11].
As described above, representing the convolution equation ( 1) as the linear system (6) leads to prosperous results (including numerical algorithms).This motivates us to find a different representation.In this study, we provide the following representation: the convolution equation ( 1) can be transformed into a generalized Sylvester equation by using special matrices, which is our main contribution.This study may allow one to find mathematical features and numerical solvers of the convolution equation ( 1) by means of procedures for the generalized Sylvester equation.Furthermore, the generalized Sylvester equation can be reduced to simpler forms for some filter matrices that appear in image processing.Using this, we show the necessary and sufficient conditions for the unique solvability of the convolution equation (1) with the filter matrices.
The rest of this paper is organized as follows.In Section 2, we show that the convolution equation ( 1) can be rewritten as a generalized Sylvester equation by using some special matrices.In Section 3, the existence of unique solutions of (1) with some specific filter matrices related image processing is discussed.The concluding remarks are provided in Section 4.

Rewriting the convolution equation to a generalized matrix equation
In this section, we show that the convolution equation ( 1) with the zero, periodic or reflexive boundary condition can be rewritten as a generalized Sylvester equation.
Before describing the main results, let us recall some special matrices.We first introduce shift matrices.The × shift matrices and are defined by The matrices and are called an upper shift matrix and a lower shift matrix respectively.The upper and lower shift matrices represent linear transformations that shift the components of column vectors one position up and down respectively, i.e., it follows that where From ( 8) and ( 9), the following properties hold: Next, we introduce cyclic shift matrices.The × cyclic shift matrices (P) and (P) are defined by A straightforward calculation yields Similarly, we have Here we define the following matrices For products of ( 14) and a matrix ∈ R × , it follows that We can now state our main results.By using the shift matrices (7), the convolution equation with the zero boundary condition can be rewritten as a generalized Sylvester equation. where and , ∈ R × are shift matrices (7).
By using the cyclic shift matrices (11), the convolution equation with the periodic boundary condition can be rewritten as a generalized Sylvester equation.

Theorem 2.2. (Periodic boundary condition)
Then, the convolution equation * = with the periodic boundary condition is equivalent to a generalized Sylvester equation where
By using the tridiagonal matrices ( 14), the convolution equation with the reflexive boundary condition can be rewritten as a generalized Sylvester equation.

Theorem 2.3. (Reflexive boundary condition)
Then, the convolution equation * = with the reflexive boundary condition is equivalent to a generalized Sylvester equation where , (R) ∈ R × are defined by (14).

Existence of unique solutions of the generalized Sylvester equations
This section deals with some realistic filters ∈ R 3×3 arising from image processing and discusses the uniqueness of solutions of the convolution equation (1) for each filter using our main results.Here, we consider the following filter matrices.
Lemma 3.2.Let ∈ R × be a circulant matrix: Then, the eigenvalues of are given by Proof.See [6].
Lemma 3.3.Let ∈ R × be the following tridiagonal matrix: Then, the eigenvalues of are given by Proof.The proof of this Lemma is given in Appendix.
Note that eigenvalues of several tridiagonal matrices, including the matrix in Lemma 3.3, are presented in [9].However, in Appendix, we prove Lemma 3.3 differently from [9].For eigenvalues of more general tridiagonal matrices and their applications, we refer the reader to [5,9].
Proof.From Theorem 2.1 and the box blur filter (19), Eq. ( 1) with the zero boundary condition is equivalent to (26).Therefore, we only need to show necessary and sufficient conditions for the existence of a unique solution of (26).Eq. ( 26) has a unique solution for every if and only if and are nonsingular.From Lemma 3.1, the eigenvalues of are given by Thus, and are nonsingular if and only if , ∉ {3 − 1 : ∈ N}, which completes the proof.
Proof.By an argument similar to Corollary 3.4, it suffices to show necessary and sufficient conditions for (P) and (P) in ( 27) to be nonsingular.From Lemma 3.2, the eigenvalues of (P) are given by , 2, . . ., .
Proof.By a similar argument to the above corollaries, we only need to show necessary and sufficient conditions for (R) and (R) in (28) to be nonsingular.From Lemma 3.3, the eigenvalues of (R) are given by Thus, (R) and (R) are nonsingular if and only if , ∉ {3 : ∈ N}.

Zero boundary condition
When is the Gaussian blur filter, Eq. ( 16) becomes Proof.Since (29) has the same form as one in the case of the box blur filter, to complete the proof, we need only prove that and in (29) are always nonsingular.From Lemma 3.1, the eigenvalues of are given by Thus, it follows that −1 < cos +1 < 1 for ∈ {1, 2, . . ., }, which implies 0 < < 4. Hence, and are always nonsingular.

Periodic boundary condition
When is the Gaussian filter, Eq. ( 17) becomes where (P) := (P) + 2 + (P) ∈ R × .Therefore, the following result holds: Proof.By a similar argument to the corollaries in Subsection 3.1, it is sufficient to provide necessary and sufficient conditions for (P) and (P) in (30) to be nonsingular.From Lemma 3.2, the eigenvalues of (P) are given by , 2, . . ., .
Thus, (P) and (P) are nonsingular if and only if , ∉ {2 : ∈ N}.Hence we complete the proof.

Zero boundary condition
When is the edge detect B filter, Eq. ( 16) becomes the following Lyapunov equation Proof.To complete the proof, it suffices to prove Eq. ( 35) has a unique solution for every .The Lyapunov equation ( 35) has a unique solution if and only if and − have no common eigenvalue.From Lemma 3.1, the eigenvalues of are given by which implies that all eigenvalues of and are positive.Hence, and − have no common eigenvalue, which completes the proof.

Periodic boundary condition
When is the edge detect B filter, Eq. ( 17) becomes where (P) := 2 − (P) − (P) ∈ R × .Therefore, the following result holds: Proof.It is sufficient to show that Eq. (36) does not have a unique solution for every , i.e., (P) and − (P) always have common eigenvalues.From Lemma 3.2, the eigenvalues of (P) are given by which implies = 0 for any ∈ N. Hence, (P) and − (P) always have the common eigenvalue 0.

Reflexive boundary condition
When is the edge detect B filter, Eq. ( 18) becomes Proof.By a similar argument to Corollary 3.14, we only need to show that (R) and (R) in (37) always have at least one eigenvalue in common.From Lemma 3.3, the eigenvalues of (R) are given by Hence, (R) and − (R) always have the common eigenvalue 0.

Emboss
In this subsection, we just characterize the unique solvability of the convolution equation (1) in the case of the zero or periodic boundary condition.

Zero boundary condition
When is the emboss filter, Eq. ( 16) becomes where ˜ := + 2 ∈ R × .Therefore, we have the following results: Proof.The proof is completed by showing that Eq. ( 44) always has a unique solution for every .Applying the vec operator to (44) yields Then, the coefficient matrix of ( 45) is nonsingular because it is a sum of the identity matrix ⊗ and the skew-symmetric matrix 1 2 ˜ ⊗ ˜ − ˜ ⊤ ⊗ ˜ ⊤ .Thus, (44) always has a unique soluton for every , which completes the proof.
Proof.By a similar argument to Corollary 3.22, it is sufficient to show that the coefficient matrix of the linear system obtained by vectorizing (46) is nonsingular.Applying the vec operator to (46) yields Then, the coefficient matrix of ( 47) is nonsingular because it is a sum of the identity matrix ⊗ and the skew-symmetric matrix 1 2 ˜ (P) ⊗ ˜ (P) − ˜ (P) ⊤ ⊗ ˜ (P) ⊤ .

Reflexive boundary condition
When is the emboss filter, Eq. ( 18) becomes where ˜ (R) := + 2 (R) ∈ R × and ˜ (R) := + 2 (R) ∈ R × .Unfortunately, unlike the above cases, we cannot show the necessary and sufficient conditions for a unique solution at present.However, we can infer from the following discussion that Eq. ( 48) has a unique solution for every .Vectorizing Eq. ( 48) yields the linear system (6) with the coefficient matrix Therefore, the uniqueness of solutions of Eq. ( 48) is reduced to the nonsingularity of (49).To support this, we provide some examples of the eigenvalue distribution of F in Figure 1 and check that zero is not an eigenvalue of F .For all cases of matrix sizes shown in Figure 1, the real parts of all eigenvalues are 1.Thus, from the examples, it can be conjectured that all eigenvalues of F for all , can be written in the form = 1 + i (≠ 0), ∈ R.

Conclusion
In this paper, it was shown that the convolution equation (1) under the zero, periodic, or reflexive boundary condition can be represented as a generalized Sylvester equation.Concretely, the result was obtained by using the shift matrices (7), the cyclic shift matrices (11), and the tridiagonal matrices defined by ( 14), for the zero, periodic, and reflexive boundary conditions, respectively.In addition, for some concrete examples arising from image processing, we showed that the generalized Sylvester equation that is equivalent to the convolution equation can be reduced to a simpler form, and we have characterized the unique solvability of the convolution equation.The necessary and sufficient conditions for the convolution equation ( 1) to have a unique solution for every right-hand side are summarized in Table 1.
In the future, we will consider finding numerical algorithms for the convolution equation (1) by utilizing the representation of the convolution equation as a generalized Sylvester equation.Proving the conjecture in Subsubsection 3.7.3 is one of our future work.

Appendix: The Proof of Lemma 3.3
Here we provide a proof of Lemma 3.

Corollary 3 . 7 .
∈ R × .Therefore we have the following result: (Zero boundary condition) Let ∈ R 3×3 be the Gaussian blur filter (20).Then, Eq. (1) with the zero boundary condition has a unique solution for every .

Corollary 3 . 14 .
(Periodic boundary condition) Let ∈ R 3×3 be the edge detect B filter (22).Then, Eq. (1) with the periodic boundary condition does not have a unique solution for every .

Corollary 3 . 15 .
Therefore, the following result holds: (Reflexive boundary condition) Let ∈ R 3×3 be the edge detect B filter (22).Then, Eq. (1) with the reflexive boundary condition does not have a unique solution for every .
× .From this, the following result holds: Corollary 3.19.(Zero boundary condition) Let ∈ R 3×3 be the sharpen filter (24).Then, Eq. (1) with the zero boundary condition has a unique solution for every .