The raw data consists of successive line-scans along the central axis of an arteriole lumen (Fig. 1A). The spatial dimension, x, corresponds to the linear motion of the focal volume along the blood vessel. The required resolution along x is set by the diffraction limit. For a 40-X water-dipping lens, the lateral resolution is about 0.7 μm, yielding a sampling of Δx ≈ 0.3 μm per pixel. For a typical 150 μm field, this requires ~ 500 samples per line. At the ~ 500 μs per line scan rate of high-speed gravimeter mirror scanners (model 6110, Cambridge Technology, Lexington, MA), the sampling time is thus Δt ≈ 1 μs per pixel. An upper limit to the maximum velocity that can be determined with scanning measurements is set by (½)Δx/Δt ≈ 150 mm/s; the factor of ½ results from the use bidirectional scanning to correct for the speed of the scan mirrors. Signal-to-noise constraints may necessitate the use of slower scan rates, as discussed (Tsai and Kleinfeld, 2009 ).
From the continuous data set of Figure 1A, we select a strip along x that contains a straight, planar section of the vessel. We define this width as L ≡ NxΔx, where Nx is the number of spatial samples; typically, L ~ 40 μm. We window the data in time intervals of T = Nt Δt separate windows, where Nt is the number of lines in the sample. We calculate the velocity for successive blocks of the windowed line-scan data, with an overlap of T/4 between blocks. Each block is denoted Fi(x, t) where i is an index that corresponds to time in units of T/4. The Nyquist frequency is (2T)−1, or 10 Hz for the typical choice of T = 50 ms; this is sufficient to capture the heart-rate of the rat without aliasing.
Each windowed data block is first normalized to remove inhomogeneities in illumination in space and to remove the baseline intensity, usually using the first second, 20T of data, so that the data block has a mean value of zero, i.e.,
The Radon transform maps each windowed data block Fi(x, t) from space-time coordinates to space-velocity coordinates, F̃i(r, θ), by
Bright points in the transformed data correspond to lines at particular angles and radii (Fig. 1B). If the original image block Fi(x, t) contains many streaks with a similar orientation, as produced by a line-scan along a blood vessel, the variance of F̃i(r, θ) along r is maximal along the orientation of the streaks.