Resolution of the wavefront set using continuous shearlets (2008)

It is known that the Continuous Wavelet Transform of a distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decays rapidly near the points where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S script upper H Subscript psi Baseline f left-parenthesis a comma s comma t right-parenthesis equals mathematical left-angle f psi Subscript a s t Baseline mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mi> ψ </mml:mi> </mml:msub> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the analyzing elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi Subscript a s t"> <mml:semantics> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\psi _{ast}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are dilated and translated copies of a single generating function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi> ψ </mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace psi Subscript a s t Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi> ψ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\psi _{ast}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> form a system of smooth functions at continuous scales <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , locations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of double-struck upper R squared"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t \in \mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>