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Circular Radon inversion at the zero truth

Example

Assume Countable Choice. Let $σ$ be the surface measure on $S^{1}$ used in the polar-coordinate formula, let $Σ = S^{1} \times (0, 2)$ with product measure $σ \times d t$ , and for real Schwartz functions $u \in S (R^{2}; R)$ define $(A_{0} u) (z, t) = t \int_{S^{1}} u (z + t ω) d σ (ω), z \in S^{1}, 0 < t < 2.$ The map $A_{0}$ has a unique bounded extension $A : L^{2} (R^{2}; R) \to L^{2} (Σ; R)$ . For full-space $T V_{2}$ , exact datum $f = 0$ , zero truth $u^{†} = 0$ , and certificate $p^{†} = 0$ , the truth is a minimum-TV exact solution satisfying the source condition. For every sequence $α_{n} > 0$ tending to zero, $w_{n} = 0$ , and $u_{n} = 0$ , each $u_{n}$ is a global minimizer of the quadratic TV functional and all nonzero canonical level sets and boundaries of $u_{n}$ and $u^{†}$ are empty; their measure and extended Hausdorff distances are zero.

Example references

A source certificate is a vector whose adjoint image is a TV subgradient.

Full-space TV on L² is the extended distributional variation.

The real $L^{2}$ norm and pairing are defined by the integral.

The circle carries the surface measure used in polar integration.

An $L^{2}$ element is an almost-everywhere equivalence class of measurable representatives.

Schwartz functions are smooth with all polynomially weighted derivatives bounded.

Nonzero levels have canonical density-one representatives and ambient boundaries; equal empty boundaries have Hausdorff distance zero.

Iglesias, Mercier, and Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §5.1

Facts & Assumptions

Given: The circle $S^{1}$ , product data measure on $Σ$ , the circular mean formula in the Statement, and the full-space functional $T V_{2}$ .

[A1]

Countable Choice is required by the polar-coordinate, completed-product integration, Schwartz-density, and $L^{2}$ -completeness suppliers. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

Distributional TV as an extended functional on L²: $T V_{2}$ is the supremum of $\int u div φ$ over compactly supported smooth vector fields with $∥ φ ∥_{\infty} \leq 1$ , hence $T V_{2} \geq 0$ and $T V_{2} (0) = 0$ .

[F2]

Dual source certificates for TV regularization: a source certificate for an exact minimum-TV truth is $p \in L^{2} (Σ)$ with $A^{*} p \in \partial T V_{2} (u^{†})$ .

[F3]

Canonical TV level sets and their boundaries: the positive levels are density-one representatives of ${u > t}$ , negative levels of ${u < t}$ , and equal empty canonical boundaries have extended Hausdorff distance zero.

[F4]

Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma and Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: $σ$ is finite and $∣ \int_{S^{1}} g d σ ∣^{2} \leq σ (S^{1}) \int_{S^{1}} ∣ g ∣^{2} d σ$ .

[F5]

Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma: for nonnegative measurable $h$ , $\int_{R^{2}} h (x) d x = \int_{0}^{\infty} \int_{S^{1}} h (r ω) r d σ (ω) d r$ .

[F6]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: nonnegative measurable functions can be integrated in either order over the circle and Euclidean data variables.

[F7]

Schwartz space is dense in L2: under Countable Choice, complex Schwartz classes are dense in complex $L^{2} (R^{2})$ .

[F8]

$L^{2}$ with the integral pairing is a Hilbert space: under Countable Choice, real $L^{2}$ on every measure space is complete with its real integral inner product.

Verification

technique · direct operator bound followed by completion and zero-objective verification

1.1A1F4F5F6givenalgebra

Put $S = σ (S^{1}) < \infty$ . For each fixed $z \in S^{1}$ , polar coordinates [F5] give $\int_{0}^{2} \int_{S^{1}} ∣ u (z + t ω) ∣^{2} t d σ (ω) d t = \int_{B (z, 2)} ∣ u (x) ∣^{2} d x < \infty,$ so for almost every $t > 0$ the circle section is square-integrable. Cauchy–Schwarz [F4] then gives $∣ (A_{0} u) (z, t) ∣^{2} \leq S t^{2} \int_{S^{1}} ∣ u (z + t ω) ∣^{2} d σ (ω) .$ Integrating in $z, t$ , using polar coordinates [F5] around each center and Tonelli [F6], yields $∥ A_{0} u ∥_{L^{2} (Σ)}^{2} \leq S \int_{S^{1}} \int_{∣ x - z ∣ < 2} ∣ x - z ∣ ∣ u (x) ∣^{2} d x d σ (z) \leq 2 S^{2} ∥ u ∥_{L^{2} (R^{2})}^{2},$ where $∣ x - z ∣ < 2$ and $σ (S^{1}) = S$ . Thus $A_{0}$ is real-linear and bounded on real Schwartz functions with norm at most $\sqrt{2} S$ .

1.2F1F2givenalgebra

The zero vector $u^{†} = 0$ satisfies $A u^{†} = f = 0$ and $T V_{2} (u^{†}) = 0$ . Since $T V_{2} \geq 0$ , it minimizes TV over the exact solution set, regardless of the kernel of $A$ . Its adjoint certificate $p^{†} = 0$ satisfies $A^{*} p^{†} = 0 \in \partial T V_{2} (0)$ because $T V_{2} (v) \geq 0 = T V_{2} (0) + ⟨ 0, v ⟩$ for every $v \in L^{2}$ .

2.1A1F7F8step 1.1chooseconstruct

By [F7], real Schwartz functions are dense in real $L^{2}$ : approximate a real function by complex Schwartz classes and take real parts, which cannot increase the $L^{2}$ error. By [F8], the target real $L^{2} (Σ)$ is complete. For any real $u \in L^{2}$ , choose real Schwartz $u_{k} \to u$ ; step 1.1 makes $(A_{0} u_{k})$ Cauchy, and its limit is independent of the approximating sequence by the same bound. This defines a bounded real-linear extension $A$ with $∥ A ∥ \leq \sqrt{2} S$ . If another bounded extension agrees with $A_{0}$ on the dense Schwartz subspace, continuity forces equality on all of $L^{2}$ .

3.1F1F3givenalgebra∎

For $w_{n} = 0$ , the regularized objective is $J_{n} (u) = \frac{1}{2} ∥ A u ∥_{2}^{2} + α_{n} T V_{2} (u) \geq 0$ , and $J_{n} (0) = 0$ . Hence $u_{n} = 0$ is a global minimizer for every $n$ . If $t > 0$ , ${u_{n} > t} = \emptyset$ ; if $t < 0$ , ${u_{n} < t} = \emptyset$ . The same holds for $u^{†}$ , so all canonical density-one sets and ambient boundaries agree as empty sets; symmetric-difference measures and the extended Hausdorff distances are zero. The noise ratio is $∥ A^{*} ∥ ∥ w_{n} ∥_{2} / α_{n} = 0 < 2 \sqrt{π}$ , so the full-space threshold is met.

Depends on

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Sources

José A. Iglesias, Gwenael Mercier, and Otmar Scherzer, A note on convergence of solutions of total variation regularized linear inverse problems, §5.1 (standard reference, not scraped)