P Ryczkowski

M Närhi

C Billet

J.-M Merolla

G Genty

John M Dudley
email: john.dudley@univ-fcomte.fr

Real-time full field measurements reveal transient dissipative soliton dynamics in a mode-locked laser

Dissipative solitons are remarkably localized states of a physical system that arise from the dynamical balance between nonlinearity, dispersion and environmental energy exchange. They are the most universal form of soliton that can exist, and are seen in far-fromequilibrium systems in many fields including chemistry, biology, and physics. There has been particular interest in studying their properties in mode-locked lasers, but experiments have been limited by the inability to track the dynamical soliton evolution in real-time. Here, we use simultaneous dispersive Fourier transform and time lens measurements to completely characterize the spectral and temporal evolution of ultrashort dissipative solitons as their dynamics pass through a transient unstable regime with complex break-up and collisions before stabilization. Further insight is obtained from reconstruction of the soliton amplitude and phase and calculation of the corresponding complex-valued eigenvalue spectrum. These findings show how real-time measurements provide new perspectives into ultrafast transient dynamics in optics.

INTRODUCTION

When operating in their steady-state regime, mode-locked lasers produce highly stable pulse trains which have found widespread applications in photonics and many other areas of science [START_REF] French | The generation of ultrashort laser pulses[END_REF][START_REF] Keller | Recent developments in compact ultrafast lasers[END_REF][START_REF] Hänsch | Lecture: Passion for precision[END_REF]. However, mode-locked lasers can also exhibit complex instabilities when detuned from steady-state, or as stable mode-locking develops from noise [START_REF] Abraham | Transient dynamics in a self-starting passively mode-locked fiber-based soliton laser[END_REF][START_REF] Dudley | Stable and unstable operation of a mode-locked argon laser[END_REF][START_REF] Hönninger | Q-switching stability limits of continuous wave passive mode locking[END_REF]. These instabilities are of particular interest in passively mode-locked fibre lasers, since they reveal a rich landscape of dissipative soliton dynamics [START_REF]Dissipative Solitons[END_REF][START_REF] Grelu | Dissipative solitons for mode-locked lasers[END_REF][START_REF] Grelu | Nonlinear Optical Cavity Dynamics: From Microresonators to Fiber Lasers[END_REF][START_REF] Turitsyn | Dissipative solitons in fiber lasers[END_REF]. Dissipative solitons are the most universal class of localized soliton state in physics, existing for an extended period of time in the presence of nonlinearity and dispersion, even while parts of the structure can experience gain and loss [START_REF]Dissipative Solitons[END_REF]. Although there is an extensive theoretical literature on fibre laser dissipative solitons, experiments have been more limited, and often restricted to measurements using only fast photodetectors [START_REF] Hönninger | Q-switching stability limits of continuous wave passive mode locking[END_REF][START_REF] Grelu | Nonlinear Optical Cavity Dynamics: From Microresonators to Fiber Lasers[END_REF][START_REF] Flynn | An experimental and numerical study of Q-switched mode-locking in monolithic semiconductor diode lasers[END_REF][START_REF] Schlatter | Pulse-energy dynamics of passively mode-locked solid-state lasers above the Q-switching threshold[END_REF][START_REF] Lecaplain | Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser[END_REF]. These experiments have certainly provided insight into the laser operation, but the limited available detector resolution has generally precluded detailed characterisation of the underlying dynamics.

Recent years, however, have seen dramatic advances in the real-time measurement of non-repetitive optical signals [START_REF] Goda | Dispersive Fourier transformation for fast continuous single-shot measurements[END_REF][START_REF] Solli | Optical rogue waves[END_REF][START_REF] Godin | Real time noise and wavelength correlations in octave-spanning supercontinuum generation[END_REF][START_REF] Solli | Fluctuations and correlations in modulation instability[END_REF][START_REF] Wetzel | Real-time full bandwidth measurement of spectral noise in supercontinuum generation[END_REF][START_REF] Runge | Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers[END_REF][START_REF] Runge | Observation of soliton explosions in a passively mode-locked fiber laser[END_REF][START_REF] Herink | Resolving the build-up of femtosecond modelocking with single-shot spectroscopy at 90 MHz frame rate[END_REF][START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Kolner | Temporal imaging with a time lens[END_REF][START_REF] Suret | Single-shot observation of optical rogue waves in integrable turbulence using time microscopy[END_REF][START_REF] Närhi | Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability[END_REF]. The first method to see widespread use was the dispersive Fourier transform (DFT) which has yielded real-time spectral measurements of optical fibre rogue waves, modulation instability and supercontinuum generation [START_REF] Goda | Dispersive Fourier transformation for fast continuous single-shot measurements[END_REF][START_REF] Solli | Optical rogue waves[END_REF][START_REF] Godin | Real time noise and wavelength correlations in octave-spanning supercontinuum generation[END_REF][START_REF] Solli | Fluctuations and correlations in modulation instability[END_REF][START_REF] Wetzel | Real-time full bandwidth measurement of spectral noise in supercontinuum generation[END_REF]. More recently, the DFT has been applied to study the spectral properties of mode-locked lasers, revealing novel decoherence and soliton explosions in a fibre laser [START_REF] Runge | Coherence and shot-to-shot spectral fluctuations in noise-like ultrafast fiber lasers[END_REF][START_REF] Runge | Observation of soliton explosions in a passively mode-locked fiber laser[END_REF], and wavelength evolution dynamics in a Kerr-lens Ti:Sapphire laser [START_REF] Herink | Resolving the build-up of femtosecond modelocking with single-shot spectroscopy at 90 MHz frame rate[END_REF]. In addition, soliton dynamics in the Kerr-lens Ti:Sapphire laser have also been studied using a modified DFT adapted for real-time spectral interferometry [START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF].

In parallel with these spectral measurements, the development of a temporal analogue of a spatial thin lens has resulted in real-time pulse intensity measurements with sub-picosecond resolution [START_REF] Kolner | Temporal imaging with a time lens[END_REF]. This time-lens technique has been used in studies of incoherent soliton propagation in optical turbulence [START_REF] Suret | Single-shot observation of optical rogue waves in integrable turbulence using time microscopy[END_REF], and stochastic breather emergence in modulation instability [START_REF] Närhi | Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability[END_REF].

In this work, we report the first simultaneous combination of real-time spectral DFT and time lens techniques to characterise unstable transient evolution characteristics of dissipative solitons in a mode-locked fibre laser. Our measurements provide a unified snapshot of how solitons evolve from round trip to round trip in the laser cavity, and allow the complex time and frequency domain characteristics of the evolving pulses to be directly correlated. One particular finding is the observation of transient coherent multi-soliton states, a previously experimentallyunobserved feature of soliton fibre lasers associated with short-lived soliton molecules that grow from noise and rapidly decay. In addition, phase retrieval allows reconstruction of the full field (intensity and phase) of the measured pulses and calculation of the corresponding complex nonlinear eigenvalue spectrum. These results provide a unique picture of the internal evolution of dissipative solitons, and we anticipate the wide application of our approach in the optimization and design of lasers with improved stability characteristics. More generally, we believe that our results will stimulate the widespread use of simultaneous temporal and spectral characterization as a standard technique for the real time study of ultrafast optical systems.

RESULTS

Our experiments studied the transient turn-on dynamics of a saturable absorber mode-locked fibre laser configured (in stable operation) to generate pulses of ~4.5 ps duration at 1545 nm with 20 MHz repetition rate (i.e. cavity round trip time of 50 ns). The laser cavity has net anomalous dispersion and in steady-state operation, frequency-resolved optical gating measurements confirmed that the stable output pulses were well-fitted by a hyperbolic secant profile and were transform limited with a uniform temporal phase [START_REF] Billet | Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm[END_REF]. We used a commercial laser system in our experiments to highlight the practical utility of this technique in source development, and we selected the laser turn on phase as a readily-accessible regime that displays rich dynamics.

We characterized the turn-on dynamics using both direct photodiode detection as well as simultaneous DFT and time-lens measurements (see Methods for full details). Figure 1 shows results using direct photodiode detection. Specifically, Fig. 1(a) plots the envelope of the pulses measured with reduced detection bandwidth to capture a time-window of 130 ms, corresponding to 2.6 million cavity round trips. The figure clearly reveals a ~100 ms transient regime of Qswitched mode-locked operation before the appearance of a stable pulse train. Figure 1(b) shows a separate measurement of a portion of the transient regime as indicated, but here using 30 GHz detection bandwidth so that it is possible to resolve a series of ~10 μs bursts separated by a ~70 μs period. The expanded view in Fig. 1(b) shows how under each burst, an irregular train of mode-locked pulses (at 50 ns period) is present. In contrast with this unstable regime, Fig. 1(c) shows measurements in the stable mode-locked regime using 30 GHz detection, to illustrate a regular pulse train of constant intensity. Of course, such complex transient dynamics have been seen in both numerical and experimental studies of a range of other passively-mode-locked lasers [START_REF] Soto-Crespo | Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber[END_REF][START_REF] Schreiber | On the study of pulse evolution in ultrashort pulse mode-locked fiber lasers by numerical simulations[END_REF][START_REF] Sarukura | Pulse evolution dynamics of a femtosecond passively mode-locked Ti:sapphire laser[END_REF][START_REF] Vodonos | Experimental study of the stochastic nature of the pulsation self-starting process in passive mode locking[END_REF][START_REF] Li | Starting dynamics of dissipative-soliton fiber laser[END_REF][START_REF] Zinkiewicz | Witnessing the pulse birth-transient dynamics in a passively mode-locked femtosecond laser[END_REF][START_REF] Ghiu Lee | Transient response of a passively mode-locked Er-doped fiber ring laser[END_REF][START_REF] Wang | Q-switched-like soliton bunches and noise-like pulses generation in a partially mode-locked fiber laser[END_REF], but we include these measurements here for completeness. The key point is that, even with a detection bandwidth of 30 GHz, the temporal width of the pulses seen in the photodiode intensity record is ~30 ps, precluding any detailed study of the underlying soliton dynamics in the transient regime. It is this limitation that we overcome with our simultaneous real-time DFT and time-lens measurements.

Figure 2 shows the setup used. Light from the fibre laser output was split into two paths and sent to the real-time DFT and time-lens acquisition arms. Real time spectral measurements used a standard DFT set up [START_REF] Wetzel | Real-time full bandwidth measurement of spectral noise in supercontinuum generation[END_REF] with spectral resolution of ~0.3 nm. The time lens set-up was similar to that described in Ref. [START_REF] Närhi | Real-time measurements of spontaneous breathers and rogue wave events in optical fibre modulation instability[END_REF] and was capable of real-time measurements of intensity profiles spanning a temporal window up to 60 ps duration and with an effective temporal resolution of 400 fs. Both DFT and time lens signals were recorded using a high storage capacity digital oscilloscope, with data acquisition triggered by the time-lens signal detected after the laser was turned on. We could simultaneously measure spectral and temporal intensity profiles over a maximum of 400 cavity roundtrips (see Methods).

We recorded multiple data sequence to examine the spectral and temporal soliton dynamics both in the Q-switched mode-locked and the stable-modelocking regime. We begin in Fig. 3(a) by showing results for the stable modelocking regime where as expected the (i) spectral, and (ii) temporal intensity profiles are constant with roundtrip. The extracted spectral and temporal widths and the integrated pulse energy are shown as the right subfigures; the small (noise) variations in these figures is of the order of the experimental resolution of the spectral and temporal measurements.

Although the ability to measure both temporal and spectral intensity in real time is itself a highly significant advance, we extend this technique further using Gerchberg-Saxton phaseretrieval [START_REF] Gerchberg | A practical algorithm for the determination of the phase from image and diffraction plane pictures[END_REF][START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF] to recover the corresponding complex electric field of each pulse (see Methods).

These results are shown in Fig. 3(a-iii) where we plot the retrieved intensity (left axis) and phase (right axis) revealing the soliton characteristics of near-uniform phase. Access to the full complex electric field allows us to calculate the associated wavelength-time spectrogram (see Methods), and these results are shown in Fig. 3(a-iv), confirming the localized soliton-like nature of the pulses. Fig 4 shows similar results, but from a data sequence just prior to the regime of stable modelocking. Here, over a relatively small number of round trips, we see significant modulation in both the measured spectral and temporal amplitudes, as well as energy variation of ~30%. This "breathing" of the intracavity pulses just prior to the onset of stability is a known property of dissipative soliton dynamics in mode-locked lasers [START_REF] Herink | Resolving the build-up of femtosecond modelocking with single-shot spectroscopy at 90 MHz frame rate[END_REF], but our results are the first to be able to experimentally characterize it completely in both the time and frequency domains.

Dissipative solitons can display a much richer range of interaction dynamics (7-9), and Fig. 4 shows examples of this behaviour in the evolution of pulses under a transient Q-switched burst. To capture different aspects of these dynamics, we repeated experiments restarting the laser many times, using a variable hold off time for our measurement window to scan various points of the transient regime (see Methods). In fact, we found that whilst the general dynamics were similar for all burst durations (i.e. showing the emergence of multiple pulses from noise followed by subsequent decay) the detailed evolution dynamics varied significantly for different bursts. Indeed, our measurement set-up allowed us to reveal previously unobserved regimes of dissipative soliton propagation and interaction under the transient envelope.

Remarking firstly that the spectral and temporal evolution with round trip is plotted from top to bottom, two typical results are shown in Fig. 4(a) and 4(b) to illustrate the different types of behaviour observed. The figures show (i) spectral and (ii) temporal intensity profiles (with the right subfigure showing energy), as well as (iii) retrieved intensity and phase, and (iv) spectrograms. Also note that energy is not constant in this transient turn-on regime [START_REF] Schlatter | Pulse-energy dynamics of passively mode-locked solid-state lasers above the Q-switching threshold[END_REF]. We first discuss Fig. 4(a) which shows an initial noisy field splitting into 3 distinct pulses (of duration ~5 ps) that propagate coherently together over ~100 roundtrips. Each pulse displays a welllocalized temporal intensity peak, and the pulses are mutually coherent as shown by the distinct modulation in the corresponding spectrum. Significantly, the pulse separation does not vary over ~100 roundtrips as the pulses evolve without seeming to interact before decaying. Figure 4(b) shows a qualitatively different case. Here, two pulses emerge, but rather than propagating without interacting, they undergo attraction and eventually collide, displaying a more complex nonlinear phase profile at this point. Figure 4(c) on the other hand shows another example of burst evolution where a combination of the above dynamics are observed with the emergence of 3 solitons.

From analysis of multiple Q-switched bursts, we have been able to identify that the emergence of multi-pulse states consisting of 2 or 3 distinct soliton-like structures is a typical feature of this transient regime. Such bound coherent soliton states in a laser are referred to as soliton molecules, and are well-known to exhibit a rich range of dynamical behaviour [START_REF] Grelu | Temporal Soliton "Molecules" in Mode-Locked Lasers: Collisions, Pulsations, and Vibrations[END_REF]. Significantly, in contrast to previous studies of stable or periodic dynamics characterized using spectral measurements [START_REF] Herink | Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules[END_REF][START_REF] Krupa | Real-Time Observation of Internal Motion within Ultrafast Dissipative Optical Soliton Molecules[END_REF], our results reveal both time and frequency domain signatures of a regime of short-lived growth and decay that has not previously been observed experimentally. We also note that the interaction dynamics between the pulses within the soliton molecule can be very complex, depending on the pulse frequencies, their temporal separation, as well as their initial amplitudes and phases [START_REF] Grelu | Temporal Soliton "Molecules" in Mode-Locked Lasers: Collisions, Pulsations, and Vibrations[END_REF][START_REF] Tang | Soliton interaction in a fiber ring laser[END_REF][START_REF] Roy | Pulse interactions in the stretched-pulse fiber laser[END_REF][START_REF] Akhmediev | Dissipative soliton interactions inside a fiber laser cavity[END_REF][START_REF] Akhmediev | Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach[END_REF]. In this context, the spectrogram plots in Fig. 4 are useful in illustrating this complexity. In Fig. 4(a) for example, the central frequencies of the constituent pulses change very little with propagation, whereas in Fig 4(b-c) the two pulses that eventually merge initially display a frequency difference of 0.4 and 0.5 nm, respectively, but their frequencies evolve with propagation until they eventually coincide at their superposition.

The ability to completely reconstruct the complex electric field from the measured pulse intensity and phase allows calculation of derived quantities that yield additional and important physical insights. In particular, it is possible to apply numerical techniques from scattering theory to yield spectral eigenvalue portraits of the pulse structures [START_REF] Zakharov | Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media[END_REF][START_REF] Akhmediev | Solitons: Non-linear Pulses and Beams[END_REF][START_REF] Osborne | Nonlinear Ocean Waves and the Inverse Scattering Transform[END_REF][START_REF] Agrawal | Nonlinear Fiber Optics[END_REF][START_REF] Turitsyn | Soliton-based discriminator of noncoherent optical pulses[END_REF]. More specifically, according to inverse scattering theory, any initial condition of the nonlinear Schrödinger equation (NLSE) evolves into a combination of solitons plus non-solitonic (quasi nonlinear) oscillating wave packets, and it is the parameters of the solitons formed that is derived via the inverse scattering transform [START_REF] Turitsyn | Soliton-based discriminator of noncoherent optical pulses[END_REF]. Interestingly, although inverse scattering in fibre optics has generally been used only analytically to obtain closed form solutions to the NLSE, there has also been recent interest in using numerical techniques (often described as a nonlinear Fourier transform) for more arbitrary fields to study rogue wave dynamics [START_REF] Randoux | Inverse Scattering Transform Analysis of Rogue Waves Using Local Periodization Procedure[END_REF] and to overcome channel limits in optical communications [START_REF] Turitsyn | Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives[END_REF].

With this approach applied to the intensity and phase measurements of pulses from the modelocked fibre laser, the calculated eigenspectra for several cases are shown in Fig. 5. Here the red dots in each subfigure show the discrete eigenspectrum for the normalized pulse intensity profiles shown as insets. Here ψ is the normalised pulse envelope that generates the scattering potential, τ is the time variable, and ζ is the calculated eigenvalue (see Methods). In this case, we associate the stable pulse regime in Fig. 3(a) with solitons having eigenvalues at ζ = ±0.5i in the complex plane, physically equivalent to injecting them in optical fibre with initial conditions corresponding to a fundamental soliton. Based on the measurements in Fig. 3(a), we normalize the intensities of other measured pulses with respect to these ideal solitons and perform direct scattering analysis to determine the corresponding more complex eigenvalue spectrum (see Methods and also the Supplementary Information).

We first consider the results in Fig. 5 (a) and (b) which correspond to single pulses at two points of the breather cycle shown in Fig. 3(b-iii). In each case the scattering analysis yields one distinct eigenvalue, and we see that as the pulse intensity varies below and above the unity value of a normalized fundamental soliton (dashed black line), the eigenvalue Im(ζ) also varies below and above the ideal soliton value of ±0.5i. In other words, the variation in the eigenvalue reflects the breathing of the pulse properties in this regime. We also note the residual "eye-like" structure in the eigenvalue spectrum corresponding to the small pedestal in the intensity distribution (see also Supplementary Information).

Figure 5(c) and (d) consider more complex evolution. Figure 5(c) plots the eigenvalue spectrum for the double-soliton pulse shown in Fig. 4(b-iii) where the direct scattering procedure yields two discrete eigenvalues, both with Im(ζ) ~ ±0.5i. Figure 5 (d) shows results for the threesoliton case in Fig. 4(a) and here we see two discrete eigenvalues with Im(ζ )~±0.5i and a third eigenvalue at a slightly lower value. As in the breathing regime of Fig. 3(b) we note the presence of a residual pedestal manifested around the zero of the real axis in the nonlinear spectrum. In all cases, the clear separation of the eigenvalues from the real axis indicates that this particular dynamical regime within the laser bursts prior to steady-state modelocking can be considered as creating discrete pulse complexes, similar to those seen with multiple bound solitons [START_REF]Dissipative Solitons[END_REF][START_REF] Grelu | Dissipative solitons for mode-locked lasers[END_REF][START_REF] Grelu | Nonlinear Optical Cavity Dynamics: From Microresonators to Fiber Lasers[END_REF]. This is a very important physical insight that the nonlinear Fourier transform reveals directly.

DISCUSSION

Dissipative nonlinear systems can be highly complex, but it is clear that recent years have seen tremendous advances in being able to understand their behavior through novel theoretical approaches and numerical modelling. Of course, a full understanding of such complex dynamical phenomena requires that theoretical and numerical results are carefully compared with experiment, and to this end the measurement technique and phase retrieval analysis reported in this paper make a significant contribution in enabling complete characterization of picosecond dissipative soliton systems. The results obtained show a range of ultrafast temporal and spectral dynamics, and in particular we have shown temporal and spectral measurements of a transient phase of soliton dynamics associated with the emergence and decay of coherent multi-soliton states. We anticipate that our results will stimulate many future theoretical studies and numerical modelling to understand the underlying pulse dynamics, building on the extensive existing literature [START_REF] Grelu | Temporal Soliton "Molecules" in Mode-Locked Lasers: Collisions, Pulsations, and Vibrations[END_REF] and including also possible long-range interactions arising from transverse acoustic waves generated through soliton-induced electrostriction [START_REF] Jang | Ultraweak long-range interactions of solitons observed over astronomical distances[END_REF].

More generally, these results also highlight the novelty of the simultaneous use of timelens and DFT measurement technique in providing real-time pulse ultrafast pulse measurements that can yield complete intensity and phase characterisation. When compared to other full field techniques measurement techniques such as Frequency-Resolved Optical Gating [START_REF] Trebino | Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses[END_REF] or Spectral Phase Interferometry for Direct Electric-field Reconstruction (51), simultaneous time-lens and DFT measurements operate at orders of magnitude lower peak power (typically ~1 W peak power is required by the time lens, < 100 mW peak power by the DFT). Moreover, acquisition is possible at the 20 MHz repetition rate of the laser and with asynchronous operation of the time-lens, can be extended to GHz repetition rates. To our knowledge, no other technique can be used to provide real-time characterisation and intensity and phase information of ultrafast instabilities such as those reported here. The temporal resolution in our setup is ultimately limited by the phase matching properties of the silicon waveguide and the duration of the pump pulse in the time-lens system, but resolutions in the ~200 fs range have been reported [START_REF] Salem | Application of space-time duality to ultrahigh-speed optical signal processing[END_REF] and further improvement might be expected with dedicated waveguide dispersion-engineering.

A further area of particular significance concerns our use of the nonlinear Fourier transform to calculate the local eigenspectrum of the pulses generated from the modelocked fibre laser. The calculated spectra clearly show the presence of local soliton content in the complex pulse profiles measured, which provides a new window into the physics of the underlying laser dynamics. We also believe our results will motivate interest in much broader applications of the nonlinear Fourier transform for characterising soliton content of ultrashort pulses [START_REF] Sugavanam | Experimentally Characterized Nonlinear Fourier Transform of a Mode-locked Fibre Laser[END_REF][START_REF] Narhi | Ultrafast Simultaneous Real Time Spectral and Temporal Measurements of Fibre Laser Modelocking Dynamics[END_REF]. In optics, we anticipate particular interest in real-time spectral and temporal studies of nonlinear single-pass propagation dynamics in optical fibres such as rogue wave and modulation instability that also display complex transient noise spikes.

METHODS

Experimental Setup

The fibre laser used in our experiments was a commercial Pritel FFL-500 model using a linear Fabry-Perot cavity configuration shown in Fig. 2, similar to that described in (55) but with a 978 nm pump laser and Er:doped fibre gain medium. Modelocking is sustained by a 2 µm thick bulk saturable absorber (InGaAs on InP substrate) contact bonded to one of the cavity mirrors.

The steady-state modelocking dynamics are dominated by soliton propagation effects because of the net anomalous dispersion in the cavity, with the generation of hyperbolic-secant like pulses of flat phase in stable operation.

We implemented the DFT technique using 850 m of dispersion compensating fibre (DCF) with group velocity dispersion coefficient of 100 ps/nm.km and dispersion slope 0.33 ps/nm 2 .km at 1550 nm. We attenuated the input to the DCF in order to ensure linear propagation, and confirmed the fidelity of the time-stretching technique when the fibre laser was operated in the stable mode-locked regime by comparing the DFT spectrum with that measured using an optical spectrum analyser (Anritsu MS9710B). The real-time DFT signal was measured by a 35 GHz photodiode (New Focus 1474 A) connected to a 30 GHz channel of a real-time oscilloscope (LeCroy 845 Zi-A, 80 GS/s), resulting in a spectral resolution of 0.3 nm.

The time lens measurements used a commercial Picoluz UTM-1500 system described previously in [START_REF] Suret | Single-shot observation of optical rogue waves in integrable turbulence using time microscopy[END_REF] with a temporal magnification factor of 76.4. Total accumulated dispersion for the input and output propagation steps was: D1 = 4.16 ps/nm and D2 = 318 ps/nm respectively, with magnification |M| = D2/D1. The temporal quadratic phase (to reproduce the effect of a thin lens) was imposed through four wave mixing from a pump pulse (100 MHz Menlo C-Fiber Sync and P100-EDFA) with linear chirp accumulated from propagation in a pre-chirping fibre Dp. The imaging condition for magnification is 2/Dp = 1/D1 +1/D2 so that the dispersion for the pump is around twice that of the signal input step. The signal at the time lens output is recorded by a 13 GHz photodiode (Miteq 135GE) connected to the 30 GHz channel of the real-time oscilloscope at a sampling rate of 80 Gs/s, resulting in an effective 400 fs temporal resolution over a (demagnified) time window of 60 ps. The temporal window is determined by the pump pulse duration used in the time lens to impose the required quadratic chirp via four-wave mixing in a highly nonlinear Si waveguide [START_REF] Salem | Application of space-time duality to ultrahigh-speed optical signal processing[END_REF]. In stable mode-locked operation it is possible to synchronize the 20 MHz laser under study with the 100 MHz pump laser, but this is not the case when studying the transient dynamics as there is significant amplitude and phase noise that precludes the detection of a well-defined 20 MHz harmonic for repetition-rate locking. The time lens is therefore operated in asynchronous mode with free-running acquisition triggered by the arrival of the time lens signal, although this limits the number of roundtrips that can be simultaneously measured to ~400 (as there is a walk-off between the Q-switched mode-locked pulses relative to the time lens gate).

To obtain representative data sets at different times of the transient regime evolution we performed multiple measurements with different delays between the switch-on time of the fibre laser and the time lens trigger, restarting the laser between each measurement. We can use this approach to characterize different phases of the pulse evolution in the transient regime because:

(i) the measurement window of 400 roundtrips (20 µs) is longer than that of the typical transient bursts that have lifetimes of only 200-300 roundtrips (10-15 µs), and (ii) we can use a variable hold-off trigger for different measurements and thus capture different phases of the overall transition phase that lasts over many millions of round trips (~100 ms). Although the recorded time window precludes the real-time measurement of the full 100 ms transient regime with subps resolution, this technique allows recording of many key dynamical features. We also emphasize that we performed many hundreds of measurements under conditions of different laser restarts to study different phases of the laser dynamics and obtained consistently results similar to those in Fig. 3(b) (breathing) and Fig. 4 (multiple pulse behaviour).

In more detail, the triggering scheme used was as follows. We employed a two-stage triggering scheme in the ultrafast oscilloscope with an adjustable hold-off time between the two trigger signals. The first trigger was obtained from the first measured DFT pulse (i.e. the first Q-switched mode-locked burst in the full transient evolution cycle) which starts the hold-off countdown time. Once the hold-off time has passed, the actual acquisition of the temporal profile is triggered from the first signal at the time lens output. By performing repeated measurements for different laser restarts with varying hold-off times in the range (1-100 ms), we could readily examine the dynamical properties of bursts throughout the full transition regime.

Finally, we note that because the optical path lengths of the DFT and time lens steps were different, the delay between time-lens and DFT records was calibrated (in a separate measurement) by comparing the arrival times on the oscilloscope of a characteristic intensity pattern imprinted onto a CW laser. This allowed us to match without ambiguity the real-time temporal and spectral intensity profiles of the dissipative solitons during their transient evolution phase.

Phase Retrieval

Phase retrieval was performed with the Gerchberg-Saxton algorithm [START_REF] Gerchberg | A practical algorithm for the determination of the phase from image and diffraction plane pictures[END_REF]. Aside from a few trivial ambiguities, this algorithm is known to be accurate for pulse retrieval when intensity envelopes are measured both in the spectral and time domains. The algorithm constructs an initial guess for the profile of the electric field using the measured temporal intensity IM(t) and an initial random phase φ rand(t): 𝐸𝐸 𝑔𝑔 (𝑡𝑡) = �𝐼𝐼 𝑀𝑀 (𝑡𝑡) 𝑒𝑒 𝑖𝑖𝜙𝜙 rand (𝑡𝑡) which is then Fourier transformed to the spectral domain to yield a corresponding initial guess for the spectral amplitude and phase FT [𝐸𝐸 𝑔𝑔 (𝑡𝑡)] = �𝑆𝑆(ω)𝑒𝑒 𝑖𝑖𝜑𝜑(𝜔𝜔) . The next step involves retaining the calculated spectral phase but replacing the calculated spectral amplitude with the measured spectral amplitude i.e. �𝑆𝑆 𝑀𝑀 (𝜔𝜔)𝑒𝑒 𝑖𝑖𝜑𝜑(𝜔𝜔) . This updated spectral profile is then transformed back into the time domain where we retain the calculated temporal phase but again replace the calculated amplitude with that from experiment �𝐼𝐼 𝑀𝑀 (𝑡𝑡) . This procedure is iteratively repeated until the root mean square error between the measurements and retrieved intensity profiles becomes smaller than a chosen value (3 ×10 -5 in our case). Convergence was improved by applying the measured temporal and spectral constraints only where the measured intensities were well above the noise floor (-20 dB from the maximum). Elsewhere the amplitudes were multiplied with a small constant value of 0.001 forcing them below the noise. If the algorithm was detected to stagnate (no change in the retrieval error before reaching the desired value), a small additional random phase contribution was added. The reliability of the algorithm was tested with simulated pulses with complex properties similar to those seen in experiments, and by performing multiple retrievals on the same experimental data, which all converged to the same results (within the retrieval error). Examples of retrievals using numerical data and additional information can be found in the Supplementary Information.

Spectrogram Calculation

Access to the full field in amplitude and phase allows us to calculate a frequency (or wavelength)-time spectrogram. For a pulse with complex field E(t), the spectrogram can be interpreted as the modulus squared of a short-time Fourier transform, a function that represents the intensity and phase content of the pulse in the time and frequency domains simultaneously.

The spectrogram is defined by:

2 ( , ) ( ) ( ) i t S E t g t e dt ω ω τ τ ∞ - -∞ = - ∫ , (1)
with g(t-τ) a variable delay gate function with delay value τ. The spectrogram thus shows the spectra of a series of time-gated portions of E(t) and provides a highly intuitive and visual means of interpreting the frequency-time content of a complex field. In our calculation of the spectrograms from experimental data, we used a Gaussian pulse gate function of duration (full width at half maximum) of 4 ps.

Nonlinear Fourier Transform

The nonlinear Fourier transform (also known as the direct scattering transform) is a mathematical procedure that identifies and quantifies soliton content in a given pulse structure.

In fact, any initial condition of the NLSE evolves into a combination of solitons plus quasilinear oscillating wave packets and it is the scattering transform that can be used to determine the number of solitons formed as well as their associated parameters. In particular, we consider a system described by the NLSE in normalised form:

(

) 2
where ψ is the normalised pulse envelope that generates the scattering potential, ξ is the propagation variable and τ is the time variable in a co-moving frame. The associated scattering problem yields the following system ( 42):

(

) 3
where v1 and v2 are the amplitudes of the waves scattered by the potential induced by ψ, and ζ is the corresponding complex eigenvalue. For our results, we consider the laser output pulses in stable operation Fig. 3(a) as associated with solitons having eigenvalues at ζ = ±0.5i in the complex plane. This would be physically equivalent to injecting them in optical fibre with initial conditions corresponding to a fundamental soliton, obtained for example by appropriate adjustment of the injected power. This case corresponds to amplitude |ψ | = 1, and the field profiles corresponding to the other pulses analysed were normalised relative to this value. Standard numerical techniques (matrix methods) were used to determine the eigenvalue spectrum (56). and Dp in the time lens are given in Methods. The inputs to both the DFT and time lens were attenuated (Att.) to avoid nonlinear effects and polarization control (PC) was needed to ensure optimal signal from the time-lens. The passively modelocked laser was a commercial Pritel FFL-500 model which was based on a standing wave cavity with a saturable absorber mirror (SA-M) and a tunable spectral filter (TF) of 3dB bandwidth 2 nm (see also Methods).

Figure 1 .

 1 Figure 1. Direct photodetector measurement of transient laser dynamics. (a) Results recorded with reduced bandwidth detection of 20 MHz illustrating the ~100 ms transient regime of Q-switched mode-locked operation before stable mode-locking. The total time record shown corresponds to 2.6 million round trips. (b) Results from separate measurements using 30 GHz bandwidth detection, showing how the laser output during the Q-switched mode-locked regime consists of transient bursts of temporal width ~10 µs separated by ~70 µs period. The expanded view shows how unstable mode-locked pulses at 50 ns period are generated under each burst. (c) Results using 30 GHz detection but in the stable mode-locked regime, showing a regular train of pulses with constant amplitude.

Figure 2 .

 2 Figure 2. Setup used for real-time characterization. Pulses from a 20 MHz passively modelocked fibre laser are split into two paths in a fibre coupler and sent to a DFT real time spectral setup and a time-lens. The DFT used 850 m of dispersion compensating fibre (DCF) to perform time to wavelength conversion, with detection using a 35 GHz bandwidth photodiode (PD2). An optical spectrum analyzer was also used for control measurements of the spectrum. The time lens uses two dispersive propagation steps (D1 and D2), one on each side of a silicon waveguide stage that applies a quadratic temporal phase through four wave mixing (FWM) with linearly-chirped pump pulses generated from a 100 MHz femtosecond pulse fibre laser (TL Pump) after stretching in a dispersive stretching fibre Dp. The time lens signal is extracted from the FWM spectrum by an optical filter and was measured using a 13 GHz bandwidth photodiode (PD1). Both PD1 and PD2 were input to a 30 GHz channel of the real-time oscilloscope. Values for dispersion modules D1, D2

Figure 3 .

 3 Figure 3. Real-time spectral and temporal characterization near stability. Results shown over 200 roundtrips for (a) stable and (b) breathing mode-locking regimes. (i) Measured spectral intensity with the extracted spectral width shown in the right subfigure. (ii) Measured temporal intensity with the extracted temporal width and energy shown in the right subfigures. (iii) Corresponding temporal intensity (left axis) and phase (right axis) extracted using the Gerchberg-Saxton algorithm for pulses at points indicated by arrows. For each of the extracted pulses in (iii), the plots in (iv) show the calculated wavelength-time spectrogram. The colour scale shows spectrogram intensity in linear arbitrary units normalised relative to the maximum.

Figure 4 .

 4 Figure 4. Real-time spectral and temporal characterization of complex dynamics. Results showing real-time spectral and temporal characterization for (a) a non-interacting triplet of three solitons over 170 round trips, (b) more complex break-up and collision dynamics of a soliton double pulse over 200 round trips and (c) a combination of a single non-interacting soliton and a two pulse collision over 170 round trips. (i) Measured spectral intensity. (ii) Measured temporal intensity. The right subfigure shows the integrated energy. (iii) Corresponding temporal intensity (left axis) and phase (right axis) extracted using the Gerchberg-Saxton algorithm for pulses at points indicated by arrows. For each of the extracted pulses in (iii), the plots in (iv) show the calculated wavelength-time spectrogram. The dashed line in the spectrogram plots indicate the central frequencies of the coherent pulses and the colour scale shows spectrogram intensity in linear arbitrary units normalised relative to the maximum.

Figure 5 .

 5 Figure 5. Calculated eigenvalue content of unstable pulses. Results applying the scattering transform to intensity and phase measurements of a selection of measured dissipative solitons. In each case the red dots show the calculated discrete eigenspectrum corresponding to the normalized pulse intensity profiles shown as insets. Here ψ is the normalised pulse envelope that generates the scattering potential, τ is the time variable, and ζ is the calculated eigenvalue. The results in Fig 5(a) and (b) correspond to the two points in the dissipative soliton breather cycle shown in Fig. 3(b). As the pulse intensity varies below and above the unity value of a normalized fundamental soliton, the retrieved eigenvalues Im(ζ) varies below and above ±0.5i. (c) The double soliton case in Fig. 4(b) where we see two discrete eigenvalues with Im(ζ) ~ ±0.5i. (d) The three soliton case in Fig. 4(a) where we see three discrete eigenvalues with Im(ζ) around ±0.5i. (Note that the intensity profiles plotted correspond to results shown in Fig. 3 and 4 but have been smoothed for clarity when shown as insets.) The dashed line in the insets shows the intensity of an ideal soliton.

ACKNOWLEDGEMENTS

Funding

This work was supported by the Agence Nationale de la Recherche project LABEX ACTION ANR11-LABX-0001-01, the Region of Franche-Comté Project CORPS and the Academy of Finland (Grants 267576 and 298463).

Contributions

All authors participated in all the experimental work and data analysis reported, and in the writing and review of the final manuscript. G.G. and J.M.D. planned the research project and provided overall supervision. The authors also thank K. V. Reddy for providing technical details concerning the soliton operating regime of the Pritel laser used in these experiments.

Competing financial interests

The authors declare no competing financial interests.


Luciano Da Fontoura Costa
email: luciano@ifsc.usp.br

Revisiting Colocalization from the Perspective of Similarity

Given two or more concentrations, an interesting and important related issue concerns the quantification of how strongly they are spatially interrelated. The concept of colocalization has been frequently considered as an indication of the tendency of the values of two concentrations to spatially vary together. While this frequently adopted approach presents several interesting characteristics, being a suitable choice for several situations, in the present work we study how multiset similarity indices can be applied for similar purposes, possibly allowing a complementation of the colocalization characterization provided by the Pearson correlation methodology. The problem of colocalization is first addressed in terms of possible underlying mathematical models, and then the Pearson correlation coefficient-based approach, as well as the standardization procedure which is its intrinsic part, are presented and discussed. The particularly important issue of how to define the baseline of the concentrations is also approached and illustrated. The minmax alternative normalization scheme is presented next, followed by the description of the three considered multiset simiarlity indices -namely the interiority, Jaccard, and coincidence similarity approaches. The characteristics of each of these methods is then illustrated respectively to 1D, and then to 2D concentrations under presence of several interesting and relevant effects including spatial displacement, as well as sharpening, presence of unrelated effects. The similarity indices, and in particular the coincidence approach, are found to present some interesting features when applied to the quantification of the colocalization between two or more concentrations, suggesting that it can provided complementary information when performing colocalization analysis.

Introduction

With the advancements in molecular biology experimental resources, in particular fluorescence imaging, it has become possible to mark in accurate and systematic manner gene expression and other biochemical concentrations in tissues and organs. As these advances continue, increasingly more comprehensive and accurate results are obtained, typically in the form of images of densities or concentrations, that demand more and more accurate and effective means for their respective analysis in terms of objective quantifications of the relationship between two or more densities.

The characterization and analysis of relationships between co-existing biochemical densities and concentrations has been approached predominantly through colocalization analysis, often being related to the application of the Pearson correlation coefficient between the respective values (e.g. [START_REF] Bolte | A guided tour into subcellular colocalization analysis in light microscopy[END_REF][START_REF] Dunn | A practical guide to evaluating colocalization in biological microscopy[END_REF][START_REF] Zinchuk | A guided tour into subcellular colocalization analysis in light microscopy[END_REF][START_REF] Oheim | Quantitative colocalisation imaging: concepts, measurements, and pitfalls[END_REF][START_REF] Costes | Automatic and quantitative measurement of protein-protein colocalization in live cells[END_REF]). While these methods are interesting on themselves and are therefore posed to continue to be employed, it becomes interesting to consider alternative approaches that can complement the characteriza-tion of coexisting concentrations and densities. This constitutes the main motivation of the present work, which adopts the concept of multiset (e.g. [START_REF] Hein | Discrete Mathematics[END_REF][START_REF] Knuth | The Art of Computing[END_REF][START_REF] Blizard | Multiset theory[END_REF][START_REF] Blizard | The development of multiset theory[END_REF][START_REF] Mahalakshmi | Properties of multisets[END_REF][START_REF] Singh | Complementation in multiset theory[END_REF]) similarities -more specifically the interiority (or overlap [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF]), Jaccard (e.g. [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF][START_REF] Jaccard | Étude comparative de la distribution florale dans une portion des alpes et des jura[END_REF][START_REF] Samanthula | Secure multiset intersection cardinality and its application to jaccard coefficient[END_REF][START_REF]Jaccard index[END_REF][START_REF] Schubert | A note on the Jaccardized Czekanowski similarity index[END_REF]) and coincidence indices (e.g. [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF]) -as means for quantifying the colocalization of two or more densities.

In the present work, we will consider concentrations as not being necessarily normalized to unit area/volume, white the terms density or distribution is applied to concentrations which have been otherwise normalized.

We start from the principle that the colocalization of two densities means that the respective biological effects tend to take place at the same parts of the considered tissue or organ. This important problem is illustrated in Figure 2, involving two hypothetical densities imaged as (a) and (b). In this case, their combined visualization in (c) readily indicates that, though the two densities are moderately colocalized, they are indeed characterized by a marked spatial shift that was emphasized by the joint visualization. Figure (c) illustrates the image obtained by superimposing the two expressions so that the colocalized portions tend to appear as yellow. In this particular case, it can be readily observed from (c) that the two densities in (a) and (b), though moderately colocalized, also involve a significant relative spatial displacement.

In the present work, we employ the interiority, Jaccard, and coincidence similarity indices in their multiset version (e.g. [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF]) as means for quantifying the colocalization between two or more densities. These multiset similarities, in particular the coincidence index, have been found to present special properties that contribute significantly to their selectivity, sensitivity when comparing similar patterns, as well as robustness to localized features perturbations. These properties have paved the way to effective use as basis for several concepts and applications (e.g. [START_REF] Da | On similarity[END_REF][START_REF] Da | Coincidence complex networks[END_REF][START_REF] Da | Comparing cross correlationbased similarities[END_REF]).

This work starts by addressing the important issue of expressing colocalization more objectively in terms of mathematical models, with emphasis on proportionality relationships. Then, the traditional colocalization approach employing the Pearson correlation coefficient is presented, illustrated and discussed including the standardization of the concentrations that is always implemented as part of the Pearson-based approach. This is followed by the description of the minmax normalization as well as the three multiset similarity indices considered in this work. Several examples are presented illustrating the respective characteristics of the several approaches here considered for colocalization quantification, first respectively to 1D concentrations, and then taking into account several 2D cases.

Colocalization Models

Before proceeding to quantifying colocalization, it is important to have an objective comprehension of what it is. Probably the best approach to this problem consists in identifying some mathematical models that can be used to define, or at least to characterize, what it will be meant by colocalization. This interesting issued is developed in the present section.

Literally, the term colocalization plainly means that something is occupying, or tending to occupy, the same positions than another thing. For instance, in biology, the term is often employed to mean that two protein types are being expressed at the same places. Thus, we should expect that one of the concentrations will tend to follow the other, in the sense that when one is large/small the other will also be or, in other words, that one of the concentrations will be proportional but not necessarily identical to the other, which can be mathematically expressed as:

f (x, y) = a g(x, y) g(x, y) = 1 a f (x, y) (1)
where a ∈ R + is a non-negative generic multiplying factor.

It is also possible, for generality's sake, to incorporate a constant term b as:

f (x, y) = a g(x, y) + b g(x, y) = 1 a f (x, y) -b (2)
Figure 2 illustrates the concept of colocalization in terms of proportional relationship between two onedimensional concentrations f (x) and g(x).

This relationship between the two concentrations, which we will understand as reflecting this concept in the most literal manner as possible, therefore yields a linear relationship between the two concentrations, as illustrated in Figure 3 respectively to two one-dimensional concentrations f (x) and g(x). The proportionality relationship that, in the present work, is understood to underly the concept of colocalization (a). Two linear concentrations f (x) and g(x) are not equal, but any of them can be obtained by multiplying the other by a specific factor a. The different heights indicate that the two concentrations were generated at distinct intensity. Biologically, the two concentrations could correspond to proteins that are being expressed at the same places, but with distinct rates. It may happen that the colocalization is not proportionally observed throughout all the imaged domain, as illustrated in (b), in which there is a region of the domain (larger values of x) at which the concentration f (x) is not being followed by the other concentration g(x). It is interesting to observe that the concept of colocalization as formalized above is not necessarily the same as the idea of identity between two concentrations, given that two colocalized profiles are not necessarily identical. In case the concept of similarity does not allow for concentration scalings, it will also become distinct from the idea of colocalization.

In practice, several effects can eventually influence the basic proportionality assumption in the colocalization.

For instance, we can have:

f (x, y) = a g(x, y) + h(x, y) g(x, y) = 1 a f (x, y) (3)
For instance, h(x, y) can be a uniformly random noise field, or it may be caused by another independent effect not related to g(x, y). The latter possibility is illustrated in Figure 4. Here, we have the same proportional concentrations already seen in Figure 3, but with the difference that an additional peak appears in the concentration f (x). This peak is not related to g(x) in any direct manner, being a consequence of an additional, independent influencing factor. Another interesting situation to be taken into account when analyzing colocalization appears when one of the concentrations corresponds to a shifting of the other, i.e.:

f (x, y) = g(x + δx, y + δy) g(x, y) = f (x -δx, y -δy) (4)
It is also possible that one of the concentrations undergoes modulation by another independent field h(x, y), such as corresponding to a gradient along the concentration domains, e.g.:

f (x, y) = g(x, y) h(x, y) g(x, y) (5)
The concentrations can also be related through function composition, i.e.:

f (x, y) = h (g(x, y)) g(x, y) = h -1 (f (x, y)) (6)
As an example of the above effect, we have the situation in which each of the concentrations correspond to the reciprocal of the other:

f (x) = 1 g(x,y) g(x) = 1 f (x,y) (7)
Needless to say, the several mathematical representation schemes above can appeared combined in their several forms. The planning, quantification, and interpretation of colocalization applications can benefit substantially in case the respectively underlying mathematical model is known or hypothesized (and then subjected to validation).

Standardization and the Pearson Correlation Coefficient

The average value of a 1D continuous concentration X(x) is defined as:

X av = 1 L S ˆS X(x)dx ( 8
)
where S is the support of the concentration, i.e. the interval of x, In = [x m , x M ], for which the concentration is of interest for our analysis. The quantity L S refers to the extension of the interval In.

For 2D concentrations, the following average definition applies

In case the concentration is sampled in terms of N points, sampled at equally spaced intervals, we have that its average value concentration X i , i = 1, 2, . . . , N , can be estimated as:

X av = 1 N N k=1 X i (9)
In the case of a concentration expressed as:

Y (x) = aX(x) + b (10)
its average can be found to be equal to:

A Y = a A X + b (11)
Given N samples X i , i = 1, 2, . . . , N , of a random variable X, they can be standardized (e.g. [START_REF] Johnson | Applied multivariate analysis[END_REF][START_REF] Gewers | Principal component analysis: A natural approach to data exploration[END_REF][START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF]) by making:

Xi = X i -µ X σ X ( 12
)
where µ X and σ X are respectively the mean and standard deviation of X. The standardized variables X have some special properties, including null means and unit standard deviation. In addition, most of them result comprised in the interval [-2, 2]. Though these properties are often useful and adopted, it should be borne in mind that information about the original variable is lost in the process, especially concerning its original average value. We will come back to this respectively to the Pearson correlation coefficient.

Given two sets of observations X and Y of two respective random variables, their Pearson correlation coefficient (e.g. [START_REF] Bertsekas | Introduction to Probability[END_REF][START_REF] Degroot | Probability and Statistics[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF][START_REF] Johnson | Applied multivariate analysis[END_REF][START_REF] Da | Shape Classification and Analysis: Theory and Practice[END_REF])) can be estimated by performing the two following steps: (1) standardize each set separately; and (2) calculate the following average:

P (X, Y ) = P (Y, X) = 1 N N i=1
Xi Ỹi [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF] It can be verified that -1 ≤ P (X, Y ) ≤ 1.

The Pearson correlation coefficient quantifies the tendency of the two variables X and Y to vary together. For instance, when one increases, the other also tends to increase. Similarly, when one decreases, the other tends to accompany the trend. The higher the value of the Pearson correlation coefficient, the strongest this tendency is. Negative values of Pearson correlation coefficient indicate that when one of the variables increases, the other decreases, and vice-versa.

Given the above characterized operation of the Pearson correlation coefficient, it is intrinsically suitable for expressing the colocalization between two concentrations X and Y , being therefore amply adopted. An important aspect of the Pearson correlation coefficient to be considered at all times concerns the fact that its calculation involves the standardization of the two variables, therefore implying the loss of some original information including the averages µ X and µ Y of the two variables. For instance, consider the particularly interesting situation depicted in Figure 5. The Pearson correlation coefficient for these two concentrations is zero because the original averages are not taken into account and also because the respective small scale oscillations correspond to a sine and a cosine. If, instead, we had to sines or cosines of any positive amplitude, the Pearson correlation coefficient would be equal to 1. Thus, we have that small changes in small scale detail of the concentrations can make the Pearson correlation to change from 0 to 1 (or even -1). In case we are interested not only on the joint variation between the concentrations, the original averages can be estimated before standardization in order to be considered later.

Let us now present another example of quantifying the colocalization of the two signals in Figure 6. These two concentrations were defined from a reference concentration Z as: The standardization of the concentrations will remove the respective averages and rescale the results so that they have the same dispersion. These combined processing effectively yields X = Ỹ , therefore implying the respective Pearson correlation coefficient to result equal to its maximum value of 1. This is completely reasonable because the two concentrations were built so as to vary together proportionally.

X = Z + 2 Y = 2Z + 0.5
There is another type of concentration representation that deserves special attention in colocalization studies, namely the situation in which the concentrations have area (in the case of 1D) or volume (2D) equal to one, in which case they can be understood as probability densities. Given any non-null concentration, it can be transformed into a respective probability density function simply by dividing each of its values by the total area A or volume V (assuming unit sampling intervals). The effect of this transformation on the proportional relationship here adopted as a reference for understanding colocalization then becomes:

p f (x, y) = [a g(x, y) + b] V = = a V g(x, y) + b V = a g(x, y) + b (14)
Therefore, the translation of a generic concentration into its respective probability density form does not affect, other than proportional changes on the involved coefficients, the proportionality relationship.

Given the importance of the effect of the average level of the concentrations on the localization, we now discuss another related example illustrated in Figure 7(a). We have both concentrations characterized by a smaller waving pattern on a larger oscillation. More specifically, the concentrations have been defined as:

X(x) = cos(x) + 0.2 sin(8x) + 3 Y (x) = cos(x) + 0.2 cos(8x) + 2.5 (15)
The scatterplot and respectively obtained Pearson correlation coefficient are shown in Figure 7(b). The relatively high value obtained for the Pearson correlation coefficient can be understood to quantify the colocalization of the larger oscillation parts of the two concentrations. However, in case these larger oscillations are understood as a baseline, with the smaller (in terms of height) oscillations patterns, namely sin(8x) and cos(8x), therefore constituting the pattern of interest, a null Pearson correlation value should have been obtained. In the case of the Pearson correlation, this could be achieved by subtracting the larger oscillation terms from both concentrations prior to respective calculation. Though in most practical cases the functional description of the concentrations are not available, there are mathematiccomputational methods that can be used for estimating the baselines of the two concentrations as corresponding to the larger oscillation.

In this work, the baseline is understood to mark the region of interest in the concentrations. More specifically, the portions of the concentrations that are above or around the baseline are to receive special attention in the analysis. For instance, if interest is focused on the smaller variations taking place on larger oscillations or even a constant plateaux, the baseline can be set so as to keep those portions apart from the analysis.

Figure 8 presents the same concentration X(x) as above as well as five possible choices, amongst the infinite possibilities, of baselines. The black baseline coincides with the x-axis, therefore providing a natural reference in case the null concentration is to be taken as reference. The concentration in brown, corresponds to the average of the concentration. The green baseline is defined by the minimum of the concentration. The magenta baseline is an arbitrary choice possibly motivated by specific interests of the research. While all these three baselines correspond to constant values, the baseline in red is a cosine function that has been defined from the larger oscillation in the concentration. Importantly, the Pearson correlation approach can only take into account the brown baseline, as it corresponds to the average of this signal and the standardization shifts the concentration always to this position. The oscillat-ing baseline can also be considered in the Pearson analysis provided it is subtracted from the concentration first. The other constant baselines cannot influence the Pearson correlation coefficient in any way as the standardization always place the concentrations at the respective average.

An important conclusion therefore is that the result of Pearson correlation analysis frequently depends substantially on the definition of the baseline. Two more evident possible baselines in the case of the previous example correspond to: (i) the overall average of the concentrations; and (ii) the larger oscillation itself. However, there is an infinite additional number of choices for baselines, each to be chosen among the infinite possible functions defined in the same interval of x of interest. Importantly, the Pearson correlation cannot cope with constant baselines along that interval, because this term would be otherwise removed by the standardization that is intrinsic to any Pearson correlation coefficient calculation.

We can conclude from the above reasonings that the standardization-Pearson analysis will be particularly suitable for the analysis of concentrations that can be expressed in the following general form:

X = a X Y + b X Y = a Y X + b Y (16)
except for any choice of constant baseline. Some other types of relationships can be eventually transformed into the above representation through mathematical manipulations involving taking logarithms, etc. In addition, the above relationship can be readily generalized to more than two concentrations, e.g.:

   X = a X Y + b X Y = a Y Z + b Y Z = a Z X + b Z
Henceforth, the constant terms in these expressions (e.g. b X , and b Y in Eq. 16), which do not necessarily correspond to the average value of the respective concentrations (as this can also be influenced by the other terms), will be henceforth referred to as constant terms.

Let us now consider a colocalization case, depicted in Figure 9, in which one of the concentrations has a welldefined peak that is not related in any means to the other concentration. More specifically, in this case both concentrations also have the same area, so that we would have half of the area of the concentration X colocalizing with the other concentration. However, the obtained Pearson correlation coefficient resulted in P (X, Y ) = 0.8007, which suggests a colocalization much higher than the half of shared areas.

The reason for this surprisingly high Pearson correlation coefficient can be better understood by referring to Figure 10, which shows the standardized versions of the two concentrations as well as the respectively defined scatterplot. It can be verified from the latter that the peak of X that is unrelated to Y yields a much shorter spoke in the scatterplot that, in addition to the displacement of the scatterplot pattern respectively to its coordinate origin, results not being enough to reduce more substantially the otherwise estimated Pearson correlation coefficient.

All in all, though the standardization of the original concentrations provides a pre-processing that tends to be suitable and compatible with subsequent colocalization analysis by employing the Pearson correlation coefficient, there are situations such as that illustrated above that can be understood as an overestimation of the colocalization. In the following section we consider potential of three multiset similarity indices for possible addressing the overestimation issue.

Minmax Normalization and Multiset Similarity Indices

Another interesting approach to normalizing the two concentrations to be analyzed is by using a method that in this work will be referred as minmax normalization. This can be readily implemented as:

Xi = X i -min(X) max(X) -min(X) (17)
As a consequence of this normalization, the values of X become all comprised between 0 and 1. When preparing the concentrations for subsequent colocalization analysis, 9(a), as well as the respectively obtained scatterplot, which reveals the predominance of the joint variation of just one of the peaks of the concentration X with the colocalized peak along Y . The additional, unrelated peak in X defines a much shorter spoke in the scatterplot that is not sufficient to reduce the Pearson correlation to a smaller value.

the minmax is applied independently on each of the involved concentrations.

From the biological point of view, this normalization can be understood as removing the constant expression term and then normalizing the maximum expression to 1.

Also known as overlap (e.g. [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF]), the interiority index between two concentrations quantifies how much one of the concentrations is contained into the other. It can be expressed as:

I(f (x), g(x)) = ´S min {|f (x)|, |g(x)|} min ´S |f (x)|dx, ´S |f (y)|dy (18)
with 0 ≤ I(f (x), g(x)) ≤ 1 and I(f (x), g(x)) = I(g(x), f (x)).

The real-valued Jaccard similarity index between two concentrations can be written (e.g. [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF]) as:

J(f (x), g(x)) = ´S s f (x) s g(x) min {|f (x)|, |g(x)|} ´S max {|f (x)|, |g(x)|} (19)
where s f (x) = sign(f (x)) and s g(x) = sign(g(x)). We also have that -1 ≤ J(f (x), g(x)) ≤ 1.

The Jaccard index quantifies how similar the two concentrations are while considering their shared area and outline [START_REF] Da | Multiset neurons[END_REF].

Given that the Jaccard index cannot account for the relative interiority between the two concentrations, the coincidence similarity index has been proposed [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF] as the product of the Jaccard and interiority index, i.e.:

C = J(f (x), g(x)) i(f (x), g(x)) (20)
with -1 ≤ J(f (x), g(x)) ≤ 1 and C = J(f (x), g(x)) = J(g(x), f (x)).

These indices, and in particular the real-valued Jaccard and coincidence cases, present some intrinsic interesting features [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF][START_REF] Da | Multiset neurons[END_REF] that make them particularly selective, sensitive and robust to localized features perturbations. These characteristics have allowed successful applications of these indices to several areas and types of problems, including template matching [START_REF] Da | On similarity[END_REF][START_REF] Da | Comparing cross correlationbased similarities[END_REF] and translating datasets into networks [START_REF] Da | Coincidence complex networks[END_REF].

In the present work, we study how these three similarity indices can be applied as an alternative methodology for characterizing the colocalization between two given concentrations. Interestingly, the three similarity indices above can be applied with or without preliminary standardization of the concentrations, depending on the application requirements and type of data to be analyzed. The ability of these indices to cope with constant baselines other than the concentration average constitute on aspect of particular interest for colocalization analysis.

In the case standardization, the obtained similarity indices can be understood, in analogous manner to the Pearson correlation characterization, as quantifying the tendency of joint variation presented by the two concentrations. Another interesting possibility consists of applying the minmax normalization, resulting in two non-negative normalized concentrations with non-null average values, in which case the similarity values will relatively indicate the shared area (for 1D concentrations) or volume (for 2D concentrations) between the two concentrations. A third possibility consists of implementing no normalization, so that generic constant baselines can be specified. These possibilities, which are presented and discussed in the remainder of this section, are summarized in the following for reference purposes: Recall that, in addition to these possibilities, any baseline of interest can be subtracted from the concentrations as a preliminary step.

Figure 11 illustrates the application of the coincidence similarity for the characterization of the two concentrations in Figure 7(a), both of which incorporating respective non-null constant terms. The results in Figure 11(b) refer to the presentation of the two concentrations by ordering the involved pairwise minimum values and averaging the respective maximum around a window centered at each instance. The Jaccard can then be readily approximated visually as corresponding to the quotient between the area below the minimum curve and the area of the average maximum curve (exact Jaccard estimation is obtained by considering the original maximum values, though they tend to be much more jagged). The type of concentration re-ordering shown in (c) is henceforth referred to as sorted representation. The obtained results cannot be directly compared to the respectively obtained Pearson correlation coefficient of P = 0.96, since here the concentrations have not been previously standardized, therefore presenting pronounced constant terms that are duly taken into account by all the considered similarity indices. Figure 12 illustrates the results obtained by the similarity indices respectively to the same concentrations as in Figure 7. However, now both concentrations were preliminary standardized before the estimation of the similarities. All the three similarity indices yielded colocalization indications smaller than the respective Pearson correlation coefficient P = 0.96, which can be understood as reasonable given that the smaller scale oscillations (a sine and a cosine) tend to reduce the colocalization. It is important to recall that none of the presented approaches to the analysis of the concentrations in Figure 7 are absolutely right or more suitable. Indeed, the choice of colocalization normalization and index needs to take into account specific specific research and data demands and constraints.

Colocalization of Noisy in 1D Concentrations

As an illustration of the effect of uniformly distributed noise in the concentrations, we consider the situation in Figure 15, which was obtained from the concentrations in Figure 14 by adding relatively strong uniform noise.

In this example, the concentrations were preliminary minmax normalized. As it could have been expected, the noisy concentrations yielded smaller colocalization values as gauged not only by the Pearson correlation coefficient, but also by the three similarity indices. By being uncorrelated, uniform noise incorporates similar amounts of joint and opposite variations which tend to cancel one another.

Colocalization of Hetetogeneous 1D Concentrations

As a example of colocalization quantification in presence of independent expression, we now return to the situation addressed in Figure 10. We had seen that the application of the Pearson correlation coefficient to these concentra- tions led to a substantial overestimation of the colocalization as a consequence of the additional to one of the concentrations of a peak that had no relationship with the other concentration. Figure 16 depicts the results obtained by using the multiset similarity index with both concentrations being first minmax normalized. Interestingly, the colocalization value estimated by the coincidence similarity yielded exactly the ratio of the areas of the concentration profile with two peaks with the other concentration. That is a consequence of the fact that the coincidence similarity is based on the shared area between the concentrations, and not only their tendency to vary together.

Colocalization of 2D Concentrations

Having presented and illustrated the possibility to apply the three multiset similarity indices to gauge the colocalization of 1D concentrations, in the following sections we extend these possibilities to several situations involving 2D concentrations. All subsequent results refer to the two concentrations neither having being preliminary normalized before application of the multiset similarity indices. In order to allow complete control on the properties of the concentrations, and also to provide a reference, all 2D concentrations used in this section were synthesized through mathematic-computational methods. More specifically, they correspond to uniformly random spatial distributions of points with given densities that are subsequently smoothed via a gaussian kernel with specific dispersion. All concentrations are then normalized to volume 1, and thus can be understood as probability densities.

Figure 17 presents two distinct 2D concentrations that, though being randomly synthesized, still present some considerable overlap, so that some moderate colocalization could be expected. However, the Pearson correlation coefficient yielded the markedly small value of P = 0.101. Larger values were obtained from the Jaccard and coincidence similarity values, but with the latter, as could be expected, resulting in the smallest colocalization value between them.

An interesting situation involving relative spatial displacement, or shift, between the two concentrations is depicted in Figure 18. The two original concentrations were identical prior respectively imposed displacement.

A particularly interesting colocalization situation arises when one of the concentrations corresponds to a sharpening of the other, as illustrated in Figure 19. The green concentration is so much sharper than the red that it completely predominates in the joint visualization shown in (c), with a good deal of the red concentration not being strongly related to the green. Still, a exceedingly high correlation coefficient of P = 0.933 has been obtained. A substantially smaller coincidence value was obtained which seems to better express the colocalization relationship between the two concentrations in this situation. This interesting result stems from the interesting characteristics of the similarity indices in being more accurate when comparing relatively similar patterns [START_REF] Da | Multiset neurons[END_REF].

The important situation in which one of the concentration profiles incorporates a markedly distinct part that does not relate whatsoever with the other concentration, as illustrated in Figure 20. Other than the additional spot in the red concentration in (a), both concentrations are identical. Thus, a substantially high colocalization value could be expected, which was the case for the Jaccard and coincidence similarity indices. However, at the same time the Pearson correlation coefficient yielded the result P = 0.734, which seems to be too small given that most of the concentrations are nearly perfectly identical other than in the additional spot region.

Figure 21 (a-c)illustrates the colocalization analysis respectively to a situation where one of the concentrations is the reciprocal of the other (Eq. 7). In their own ways, all considered colocalization approaches indicate that the two concentrations are not related: while a negative Pearson correlation was obtained, the Jaccard and coincidence indices resulted very nearly zero. That is mostly a consequence of the standardization adopted in the Pearson correlation case. Indeed, in case standardization is implemented prior to the application of the two similarity indices, we get J = -0.426 and C = -0.320. Indeed, the similarity indices start behaving in similar qualitative manner to the Pearson when the colocalizations are preliminary standardized, which allows portions of the concentrations to become negatively interrelated (negative joint variations). These results again illustrate that quite distinct results can be obtained depending on the adopted normalization.

Concluding Remarks

Several real-world situations, especially in biology, involves signals that present similar spatial characteristics. The quantification of the colocalization between two or more concentrations corresponds to an important and frequently approach that has received substantial attention in the literature. Basically, the colocalization of two concentrations has been typically understood as relating to the tendency of the concentrations to vary together along space. By measuring a directly related tendency, the Pearson correlation coefficient has been frequently adopted for quantifying the colocalization in biological applications such as gene expression and protein synthesis.

Based on multiset theory, similarity indices have been described that are capable of quantifying the relationship between two generical mathematical structures, including concentrations, in terms of the relative shared area. The present work addressed the possibility to use the interiority, Jaccard and coincidence indices as a means to provide additional information about the colocalization of concentrations that can therefore complement the more traditional characterization in terms of the Pearson correlation coefficient.

We started by addressing the colocalization problem from a mathematical perspective of possible respective models, with emphasis on a model where a proportional iterrelationship is observe between the two concentrations. Several other modeling variations and possibilities were also discussed. These models can help not only better understanding the meaning of colocalization, but also help in choosing between normalizations as well as defining a more adequate manner to quantify the respective colocalization. The often critical issue of defining a baseline compatible with the poised questions was also characterized and discussed, given that this choice can have strong influence on the results obtained by the several considered methods of colocalization analysis.

One particularly important aspect that has been presented is the tendency of the Pearson correlation colocalization to be strongly biased by the presence of portions (peaks) of one of the concentrations that has no basic relationship whatsoever with the other concentration. In the considered respective example in this work, this effect resulted in a strong overestimation of the interrelationship (colocalization) between the two analyzed concentrations. This effect is mostly related to the displacement of the concentrations as implemented by the standardization. Indeed, the characterization of this type of concentrations by using the coincidence approach, which allows generic baselines (in this case that defined by the minmax normalization) to be taken into account. The zero baseline could then be considered, leading to a more meaningful colocalization quantification in this case.

We then presented the concepts of concentration average, which is essentially important respectively to the important concept of concentration baseline, as well as the standardization method, which is actually an intrinsic part of the Pearson correlation coefficient. Several examples of colocalization characterization by using this approach were the presented and discussed respectively to1D concentrations, which allow a more direct observation of the situations and effects.

The minmax normalization was then presented, followed by a description of the multiscale-based similarity indices.

It should be kept in mind that the choice of baseline, normalization, and indices for quantification of colocaliza- tion all can have strong influence on the obtained results. In addition, neither of them are absolute adequate or inadequate, as illustrated by several of the discussed examples. The choice between the several existing alternative should take into account how the data was generated, the specific ways in which the concentrations are intrinsically produced and interrelated by the systems of interest, as well as the questions implied by each respective research.

A particularly interesting general approach is to consider several combinations of normalization and colocalization quantification approaches so that they can provide respectively complementary information about the problem of interest, but even so the respective interpretation should take into account the above observed requirements and constraints.

By presenting new concepts and methods to be possibly applied for colocalization quantification, the present work paved the way to a large number of possible future developments. These include but are not limited to evaluating the proposed approaches respectively to more types of concentrations, considering alternative normalization methods, and extending the several described approaches to situations involving more than two concentrations.

Figure 1 :

 1 Figure 1: An illustration of the colocalization task. Two distinct biochemical densities are acquired as two respective images (a) and (b).Figure(c) illustrates the image obtained by superimposing the two expressions so that the colocalized portions tend to appear as yellow. In this particular case, it can be readily observed from (c) that the two densities in (a) and (b), though moderately colocalized, also involve a significant relative spatial displacement.

Figure 2 :

 2 Figure2: The proportionality relationship that, in the present work, is understood to underly the concept of colocalization (a). Two linear concentrations f (x) and g(x) are not equal, but any of them can be obtained by multiplying the other by a specific factor a. The different heights indicate that the two concentrations were generated at distinct intensity. Biologically, the two concentrations could correspond to proteins that are being expressed at the same places, but with distinct rates. It may happen that the colocalization is not proportionally observed throughout all the imaged domain, as illustrated in (b), in which there is a region of the domain (larger values of x) at which the concentration f (x) is not being followed by the other concentration g(x).

Figure 3 :

 3 Figure 3: The basic understanding of colocalization adopted in the current work consists in assuming that one of the concentrations, f (x) is proportionally related to another concentration g(x) by a proportionality factor a, as illustrated in this figure for the case a > 1.

Figure 4 :

 4 Figure 4: Two concentrations characterized by being generally proportional except for the peak p appearing on the right portion of concentration f (x).

Figure 5 :

 5 Figure5: Two one-dimensional concentrations X(x) and Y (y) have average values approximately equal to 2.0, but both these concentrations incorporate some small scale waving. Despite the overall similarity between these two concentrations, their respective Pearson correlation coefficient can be verified to be zero.

Figure 6 :

 6 Figure 6: Two concentrations that have markedly distinct averages and magnitudes, but which have shapes proportional one to the other. As a consequence of the standardization implemented in the Pearson correlation coefficient, its value results 1 for this specific example.

Figure 7 :

 7 Figure 7: Two concentrations presenting two scales of oscillations (a) and the respective scatterplot and Pearson correlation coefficient (b), which can be understood to characterize the colocalization at the largest spatial scale of the two cosines cos(x). The null colocalization existing between the two smaller oscillations cannot be characterized directly by the Pearson correlation coefficient, demanding the prior subtraction of the larger cosine baseline.

Figure 8 :

 8 Figure 8: A given concentration X(x) (in blue) and five possible choices of baselines. Only the brown baseline is effectively considered by the Pearson correlation approach. The red baseline can also be considered for the Pearson analysis provided it is subtracted beforehand from the concentration.

Figure 9 :

 9 Figure 9: The Pearson correlation coefficient applied to the quantification of the colocalization between these two concentrations, one of which incorporating a peak unrelated to the other, yields a substantially high value. Indeed, one half of the area of the concentration X undergoes joint variation with Y .

Figure 10 :

 10 Figure10: The standardized versions of the two concentrations in Figure9(a), as well as the respectively obtained scatterplot, which reveals the predominance of the joint variation of just one of the peaks of the concentration X with the colocalized peak along Y . The additional, unrelated peak in X defines a much shorter spoke in the scatterplot that is not sufficient to reduce the Pearson correlation to a smaller value.

  A -No standardization: The concentrations are used as given; B -No standardization, given baseline: Specific baselines of interest (depending on the data and research questions) are subtracted from the concentrations and no normalization is applied to them. A particularly interesting baseline would correspond to the average of the concentrations; C -Standardization: The concentrations are standardized prior to colocalization quantification; D -Minmax normalization: The concentrations are minmax-normalized.

Figure 11 :

 11 Figure 11: Quantification of colocalization between two concentrations X and Y involving respective non-null constant terms (a). The respective Pearson is P = 0.96. The results respective to the two other indices, namely interiority and Jaccard, are also shown in (b).

Figure 12 :

 12 Figure 12: Similarity-based colocalization analysis of the concentrations in Figure 7. Both concentrations were preliminary standardized. All these three indices led to colocalization values smaller than the previously obtained P = 0.96, which seem to have overestimated the colocalization.

Figure 13

 13 Figure13presents the same situation as before, but now the two concentrations underwent minmax normalization. As expected, the implied higher baseline implied the similarity indices to take into account the shared areas as well as the joint variation tendencies, resulting in moderately larger colocalization values.

Figure 13 :

 13 Figure 13: Similarity analysis of colocalization of the the concentrations in Figure 7 adopting minmax normalization.

Figure 14

 14 Figure 14 presents a situation similar to that in Figure 7, but now the two larger oscillation components to both concentrations consist of a sine and a cosine, which

Figure 14 :

 14 Figure 14: Similarity analysis of colocalization of the two concentrations presenting little net joint variations, obtained while adopting standardization. As could be expected, very small colocalization values has been obtained.

Figure 15 :

 15 Figure 15: The colocalization analysis of the concentrations in Figure 14 with relatively intense added noise. The colocalization values obtained from all the considered aproaches were substantially reduced.

Figure 16 :

 16 Figure 16: The sorted representation respective to the concentrations in Fig. 10 as employed in the similarity coincidences, which led to more reasonable indications of colocalization in this specific case.

Figure 17 :

 17 Figure 17: Colocalization of two 2D concentrations randomly synthesized. Though moderate overlap can be observed between these concentrations, the Pearson correlation coefficient resulted markedly small. More reasonable quantifications seem to have been provided by the Jaccard and coincidence similarity indices, with the latter yielding the smallest colocalization value between them.

Figure 18 :

 18 Figure 18: Two identical, but relatively displaced, concentrations. Though a relatively strong displacement can be observed, a Pearson correlation coefficient P = 0.686 was respectively obtained. The coincidence-based approach led to a smaller colocalization value.

Figure 19

 19 Figure 19

Figure 20 :

 20 Figure 20: Two identical concentrations, except for the additional spot in the red concentration, which bears no relationship whatsoever with the other concentration. The presence of this localized discrepancy strongly influenced the obtained Pearson correlation coefficient of P = 0.734, which resulted markedly small despite the fact that all other portions of the concentrations are in nearly perfect match one another. A more consistent quantification of the colocalization in this particular case seems to have been provided by the Jaccard and coincidence similarity indices.

Figure 21 :

 21 Figure 21: Example of colocalization analysis where one of the concentrations is the reciprocal of the other. Near zero results are observed for the similarity indices, with the Pearson correlation coefficient indicated negative relationship between the concentrations.

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).