To generate the Gamma distributed speckle pattern, three fully developed speckle patterns were merged so that M was equal to three.
The resulting PDF is given by Here L is the gray level of fully developed speckle pattern without noise and N is the maximum gray level.
Using the same approach as for the fully developed speckle pattern given above, the quantized PDF of the Gamma distribution is expressed as where M is the number of individual speckle patterns in the speckle image and Γ is the gamma function.
Axial development of bubbly flow structure was examined at 3 axial locations: L/ D = 25, 148 and 253; the first measurement plane represented the entrance region when the flow was still developing while the second and third measuring planes depicted near fully developed flow pattern.
If a SM fiber is used for illumination and collection, and if the sample properties are such that it creates a fully developed speckle pattern, the fundamental mode of the fiber will be excited following a Gaussian probability function in amplitude.
From Eq. (9) and the definition of Δ I, the contrast for a quantized fully developed speckle pattern can be calculated: Therefore the square of the contrast has a negative relationship with the bit depth by means of 2 K − 1 subject to the assumptions made in the preceding sections.
Firstly to compare the influence of the bit depth on the contrast, a fully developed speckle pattern was generated and the intensity of the raw (unquantized) speckle image was scaled by a factor of 0.1 to 1 with an increment of 0.1.
The correlation between the contrast and the signal intensity for a selection of four of the camera bit depths and a fully developed speckle pattern is shown in Fig. 2 under the condition that similar intensities corresponded to a similar fraction of FWD in each case.
Hence, the time needed to reach fully developed Turing patterns on the domain we have explored (≈50 dimensionless units in Figure 4) is estimated to be around 100 h.
Here we use in vitro fertilizations and sperm competition trials to test if this pattern of paternity of fully developed tadpoles reflects patterns of paternity at fertilization and if paternity skews changes during embryonic development.
