How to understand the Fourier-Sato transform and microlocalization functors? Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, and I can ask that a sheaf be constant after pulling back to the orbits of this action (i.e. rays in the fibers of $E$). Sheaves that satisfy this property are called conic sheaves. 
There is an operation we can define which takes conic sheaves on $E$ to conic sheaves on the dual bundle $E^*$ called the Fourier-Sato transform, defined as follows: Define $P' = \{(x,y) \in E \times_M E^*: \langle x,y \rangle \leq 0\}$. Given a conic sheaf $G$ we define the Fourier-Sato transform to be $\hat{G} = R\pi^*_!(\pi^{-1}G)_{P'}$, where $\pi^*$ denotes the projection map of the dual bundle.
Now given a sheaf $F$ on an arbitrary manifold $M$, we can define the specialization of $F$ with respect to a given submanifold $N$, which is a conic sheaf on the normal bundle to $N$, denoted by $\nu_N(F)$. The specialization functor is defined by first forming the normal deformation of $N$ in $M-$a sort of a "real blow-up" of the submanifold $N-$ creating a family over $\mathbb{R}$ equipped with a map to $M$ and whose fiber at $0$ is the normal bundle $T_NM$; and then doing some push-pull operations akin to the nearby cycles functor.
Putting the specialization functor $\nu_N$ and the Fourier-Sato transform together gives us the microlocalization functor: $\mu_N(F) = \nu_N(F)$^.
At this point, I have an okay understanding of the specialization functor as something that takes a sheaf $F$ and asks how it behaves infinitesimally close to $N$ in the normal directions to $N$, like we're kind of taking dough and stretching it thin in only normal directions, then asking what our sheaf looks like in this stretched picture. I also have an okay understanding of the Fourier-Sato transform as being defined the way it is to imitate the classical Fourier transform by switching skyscraper and constant sheaves and perhaps have restriction to $P'$ play the role of $e^{-ix \cdot \xi}$.
My question is: how can I see that the microlocalization functor somehow organizes or is related to the microlocal information of my sheaf like its name suggests? I'd appreciate at least some pseudo-geometric intuition in the style of what I've written above  Moreover, is my intuition as written above good or misleading? 
These definitions are all contained in chapters 3 and 4 of Sheaves on Manifolds by Kashiwara, Schapira, but going through their proofs often leaves me, at best, unsure of the ideas behind the proof, and, at worst, hopelessly confused even if I follow individual steps in the proofs. Any help?
I'm including the algebraic geometry tag on this because I'm aware there are related constructions in the algebraic world.
 A: I know comes from far in the future, and you've probably figured it out by now, but I wanted to add to Vivek's comment.
Along a complex hypersurface $\{f=0\}$ that is non-singular at point $p$, there are identifications $\left [ \mu_{\{f=0\}}(F) \right ]_{(p;d_p f)} = (\varphi_f F)_p = R\Gamma_{\{Re(f) \geq 0\}}(F)_p.$ (see Prop 8.6.3 and Ex VIII.13 of The Book (using Kashiwara-Schapira's shift convention for $\varphi_f$). Each of these objects can be interpreted as a sort of "microlocal test functor" for $F$ at the covector $(p,d_p f)$, in that they microlocally detect cohomological changes of $F$.
More geometrically, if $F$ is $\mathbb{C}$-constructible along a complex analytic submanifold $S \subseteq X$, then the stalk $\mu_S(F)_{(p;\xi)}$ at a non-degenerate covector $(p;\xi) \in T_S^*X$ computes the normal Morse data of $F$ along $S$ (in the sense of Goresky-MacPherson's stratified Morse theory), and the isomorphism-type of this stalk is independent of the chosen non-degenerate covector (Jörg Schürmann has a nice discussion of this perspective in section 5.3.3. of "Topology of singular spaces and constructible sheaves")
Also, requiring these microlocal stalks to always be concentrated in cohomological degree $-\dim S$ in $D^b(k)$ along any stratum $S$ is equivalent to the perversity conditions (for the middle perversity), via Theorem 10.3.12 of "sheaves on manifolds".

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