source: sasview/Invariant/invariant.py @ 3244cbe1

"""
    This module implements invariant and its related computations.
    @author: Gervaise B. Alina/UTK
    
    
    TODO: 
        - intro / documentation
        - add unit tests for sufrace/volume computation with and without extrapolation.
        - replace the get_extra_data_* methods
"""
import math 
import numpy

from DataLoader.data_info import Data1D as LoaderData1D

# The minimum q-value to be used when extrapolating
Q_MINIMUM  = 1e-5

# The maximum q-value to be used when extrapolating
Q_MAXIMUM  = 10

# Number of steps in the extrapolation
INTEGRATION_NSTEPS = 1000

class Transform(object):
    """
        Define interface that need to compute a function or an inverse
        function given some x, y 
    """
    
    def linearize_data(self, data):
        """
            Linearize data so that a linear fit can be performed. 
            Filter out the data that can't be transformed.
            @param data : LoadData1D instance
        """
        # Check that the vector lengths are equal
        assert(len(data.x)==len(data.y))
        if data.dy is not None:
            assert(len(data.x)==len(data.dy))
            dy = data.dy
        else:
            dy = numpy.ones(len(data.y))
            
        # Transform the data
        data_points = zip(data.x, data.y, dy)

        output_points = [(self.linearize_q_value(p[0]),
                          math.log(p[1]),
                          p[2]/p[1]) for p in data_points if p[0]>0 and p[1]>0 and p[2]>0]
        
        x_out, y_out, dy_out = zip(*output_points)
       
        # Create Data1D object
        x_out = numpy.asarray(x_out)
        y_out = numpy.asarray(y_out)
        dy_out = numpy.asarray(dy_out)
        linear_data = LoaderData1D(x=x_out, y=y_out, dy=dy_out)
        
        return linear_data
    
    def get_allowed_bins(self, data):
        """
            Goes through the data points and returns a list of boolean values
            to indicate whether each points is allowed by the model or not.
            
            @param data: Data1D object
        """
        return [p[0]>0 and p[1]>0 and p[2]>0 for p in zip(data.x, data.y, data.dy)]
        
    def linearize_q_value(self, value):
        """
            Transform the input q-value for linearization
        """
        return NotImplemented

    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
        """
            set private member
        """
        return NotImplemented
     
    def evaluate_model(self, x):
        """
            Returns an array f(x) values where f is the Transform function.
        """
        return NotImplemented
    
    def evaluate_model_errors(self, x):
        """
            Returns an array of I(q) errors
        """
        return NotImplemented
    
class Guinier(Transform):
    """
        class of type Transform that performs operations related to guinier 
        function
    """
    def __init__(self, scale=1, radius=60):
        Transform.__init__(self)
        self.scale = scale
        self.radius = radius
        ## Uncertainty of scale parameter
        self.dscale  = 0
        ## Unvertainty of radius parameter
        self.dradius = 0
        
    def linearize_q_value(self, value):
        """
            Transform the input q-value for linearization
            @param value: q-value
            @return: q*q
        """
        return value * value
    
    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
        """
           assign new value to the scale and the radius
        """
        self.scale = math.exp(constant)
        self.radius = math.sqrt(-3 * slope)
        
        # Errors
        self.dscale = math.exp(constant)*dconstant
        self.dradius = -3.0/2.0/math.sqrt(-3 * slope)*dslope
        
    def evaluate_model(self, x):
        """
            return F(x)= scale* e-((radius*x)**2/3)
        """
        return self._guinier(x)
             
    def evaluate_model_errors(self, x):
        """
            Returns the error on I(q) for the given array of q-values
            @param x: array of q-values
        """
        p1 = numpy.array([self.dscale * math.exp(-((self.radius * q)**2/3)) for q in x])
        p2 = numpy.array([self.scale * math.exp(-((self.radius * q)**2/3)) * (-(q**2/3)) * 2 * self.radius * self.dradius for q in x])
        diq2 = p1*p1 + p2*p2        
        return numpy.array( [math.sqrt(err) for err in diq2] )
             
    def _guinier(self, x):
        """
            Retrive the guinier function after apply an inverse guinier function
            to x
            Compute a F(x) = scale* e-((radius*x)**2/3).
            @param x: a vector of q values 
            @param scale: the scale value
            @param radius: the guinier radius value
            @return F(x)
        """   
        # transform the radius of coming from the inverse guinier function to a 
        # a radius of a guinier function
        if self.radius <= 0:
            raise ValueError("Rg expected positive value, but got %s"%self.radius)  
        value = numpy.array([math.exp(-((self.radius * i)**2/3)) for i in x ]) 
        return self.scale * value

class PowerLaw(Transform):
    """
        class of type transform that perform operation related to power_law 
        function
    """
    def __init__(self, scale=1, power=4):
        Transform.__init__(self)
        self.scale = scale
        self.power = power
   
    def linearize_q_value(self, value):
        """
            Transform the input q-value for linearization
            @param value: q-value
            @return: log(q)
        """
        return math.log(value)
    
    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
        """
            Assign new value to the scale and the power 
        """
        self.power = -slope
        self.scale = math.exp(constant)
        
        # Errors
        self.dscale = math.exp(constant)*dconstant
        self.dradius = -dslope
        
    def evaluate_model(self, x):
        """
            given a scale and a radius transform x, y using a power_law
            function
        """
        return self._power_law(x)
    
    def evaluate_model_errors(self, x):
        """
            Returns the error on I(q) for the given array of q-values
            @param x: array of q-values
        """
        p1 = numpy.array([self.dscale * math.pow(q, -self.power) for q in x])
        p2 = numpy.array([self.scale * self.power * math.pow(q, -self.power-1) * self.dradius for q in x])
        diq2 = p1*p1 + p2*p2        
        return numpy.array( [math.sqrt(err) for err in diq2] )
       
    def _power_law(self, x):
        """
            F(x) = scale* (x)^(-power)
                when power= 4. the model is porod 
                else power_law
            The model has three parameters: 
            @param x: a vector of q values
            @param power: power of the function
            @param scale : scale factor value
            @param F(x)
        """
        if self.power <= 0:
            raise ValueError("Power_law function expected positive power, but got %s"%self.power)
        if self.scale <= 0:
            raise ValueError("scale expected positive value, but got %s"%self.scale) 
       
        value = numpy.array([ math.pow(i, -self.power) for i in x ])  
        return self.scale * value

class Extrapolator:
    """
        Extrapolate I(q) distribution using a given model
    """
    def __init__(self, data, model=None):
        """
            Determine a and b given a linear equation y = ax + b
            
            If a model is given, it will be used to linearize the data before 
            the extrapolation is performed. If None, a simple linear fit will be done.
            
            @param data: data containing x and y  such as  y = ax + b 
            @param model: optional Transform object 
        """
        self.data  = data
        self.model = model
       
        # Set qmin as the lowest non-zero value
        self.qmin = Q_MINIMUM
        for q_value in self.data.x:
            if q_value > 0: 
                self.qmin = q_value
                break
        self.qmax = max(self.data.x)
             
    def fit(self, power=None, qmin=None, qmax=None):
        """
           Fit data for y = ax + b  return a and b
           @param power: a fixed, otherwise None
           @param qmin: Minimum Q-value
           @param qmax: Maximum Q-value
        """
        if qmin is None:
            qmin = self.qmin
        if qmax is None:
            qmax = self.qmax
            
        # Identify the bin range for the fit
        idx = (self.data.x >= qmin) & (self.data.x <= qmax)
        
        fx = numpy.zeros(len(self.data.x))

        # Uncertainty
        if type(self.data.dy)==numpy.ndarray and len(self.data.dy)==len(self.data.x):
            sigma = self.data.dy
        else:
            sigma = numpy.ones(len(self.data.x))
            
        # Compute theory data f(x)
        fx[idx] = self.data.y[idx]
        
        # Linearize the data
        if self.model is not None:
            linearized_data = self.model.linearize_data(LoaderData1D(self.data.x[idx],
                                                                fx[idx],
                                                                dy = sigma[idx]))
        else:
            linearized_data = LoaderData1D(self.data.x[idx],
                                           fx[idx],
                                           dy = sigma[idx])
        
        ##power is given only for function = power_law    
        if power != None:
            sigma2 = linearized_data.dy * linearized_data.dy
            a = -(power)
            b = (numpy.sum(linearized_data.y/sigma2) \
                 - a*numpy.sum(linearized_data.x/sigma2))/numpy.sum(1.0/sigma2)
            
            
            deltas = linearized_data.x*a+numpy.ones(len(linearized_data.x))*b-linearized_data.y
            residuals = numpy.sum(deltas*deltas/sigma2)
            
            err = math.fabs(residuals) / numpy.sum(1.0/sigma2)
            return [a, b], [0, math.sqrt(err)]
        else:
            A = numpy.vstack([ linearized_data.x/linearized_data.dy,
                               1.0/linearized_data.dy]).T        
            (p, residuals, rank, s) = numpy.linalg.lstsq(A, linearized_data.y/linearized_data.dy)
            
            # Get the covariance matrix, defined as inv_cov = a_transposed * a
            err = numpy.zeros(2)
            try:
                inv_cov = numpy.dot(A.transpose(), A)
                cov = numpy.linalg.pinv(inv_cov)
                err_matrix = math.fabs(residuals) * cov
                err = [math.sqrt(err_matrix[0][0]), math.sqrt(err_matrix[1][1])]
            except:
                err = [-1.0, -1.0]
                
            return p, err
        

class InvariantCalculator(object):
    """
        Compute invariant if data is given.
        Can provide volume fraction and surface area if the user provides
        Porod constant  and contrast values.
        @precondition:  the user must send a data of type DataLoader.Data1D
                        the user provide background and scale values.
                        
        @note: Some computations depends on each others. 
    """
    def __init__(self, data, background=0, scale=1 ):
        """
            Initialize variables
            @param data: data must be of type DataLoader.Data1D
            @param background: Background value. The data will be corrected before processing
            @param scale: Scaling factor for I(q). The data will be corrected before processing
        """
        # Background and scale should be private data member if the only way to
        # change them are by instantiating a new object.
        self._background = background
        self._scale = scale
        
        # The data should be private
        self._data = self._get_data(data)
      
        # Since there are multiple variants of Q*, you should force the
        # user to use the get method and keep Q* a private data member
        self._qstar = None
        
        # You should keep the error on Q* so you can reuse it without
        # recomputing the whole thing.
        self._qstar_err = 0
        
        # Extrapolation parameters
        self._low_extrapolation_npts = 4
        self._low_extrapolation_function = Guinier()
        self._low_extrapolation_power = None
   
        self._high_extrapolation_npts = 4
        self._high_extrapolation_function = PowerLaw()
        self._high_extrapolation_power = None
        
    def _get_data(self, data):
        """
            @note this function must be call before computing any type
             of invariant
            @return data= self._scale *data - self._background
        """
        if not issubclass(data.__class__, LoaderData1D):
            #Process only data that inherited from DataLoader.Data_info.Data1D
            raise ValueError,"Data must be of type DataLoader.Data1D"
        new_data = (self._scale * data) - self._background   
        
        # Check that the vector lengths are equal
        assert(len(new_data.x)==len(new_data.y))
        
        # Verify that the errors are set correctly
        if new_data.dy is None or len(new_data.x) != len(new_data.dy) or \
            (min(new_data.dy)==0 and max(new_data.dy)==0):
            new_data.dy = numpy.ones(len(new_data.x))  
        
        return  new_data
     
    def _fit(self, model, qmin=Q_MINIMUM, qmax=Q_MAXIMUM, power=None):
        """
            fit data with function using 
            data= self._get_data()
            fx= Functor(data , function)
            y = data.y
            slope, constant = linalg.lstsq(y,fx)
            @param qmin: data first q value to consider during the fit
            @param qmax: data last q value to consider during the fit
            @param power : power value to consider for power-law 
            @param function: the function to use during the fit
            @return a: the scale of the function
            @return b: the other parameter of the function for guinier will be radius
                    for power_law will be the power value
        """
        extrapolator = Extrapolator(data=self._data, model=model)
        p, dp = extrapolator.fit(power=power, qmin=qmin, qmax=qmax) 
        
        return model.extract_model_parameters(constant=p[1], slope=p[0], dconstant=dp[1], dslope=dp[0])
    
    def _get_qstar(self, data):
        """
            Compute invariant for pinhole data.
            This invariant is given by:
        
                q_star = x0**2 *y0 *dx0 +x1**2 *y1 *dx1 
                            + ..+ xn**2 *yn *dxn 
                            
            where n >= len(data.x)-1
            dxi = 1/2*(xi+1 - xi) + (xi - xi-1)
            dx0 = (x1 - x0)/2
            dxn = (xn - xn-1)/2
            @param data: the data to use to compute invariant.
            @return q_star: invariant value for pinhole data. q_star > 0
        """
        if len(data.x) <= 1 or len(data.y) <= 1 or len(data.x)!= len(data.y):
            msg =  "Length x and y must be equal"
            msg += " and greater than 1; got x=%s, y=%s"%(len(data.x), len(data.y))
            raise ValueError, msg
        else:
            n = len(data.x)- 1
            #compute the first delta q
            dx0 = (data.x[1] - data.x[0])/2
            #compute the last delta q
            dxn = (data.x[n] - data.x[n-1])/2
            sum = 0
            sum += data.x[0] * data.x[0] * data.y[0] * dx0
            sum += data.x[n] * data.x[n] * data.y[n] * dxn
            
            if len(data.x) == 2:
                return sum
            else:
                #iterate between for element different from the first and the last
                for i in xrange(1, n-1):
                    dxi = (data.x[i+1] - data.x[i-1])/2
                    sum += data.x[i] * data.x[i] * data.y[i] * dxi
                return sum
            
    def _get_qstar_uncertainty(self, data):
        """
            Compute invariant uncertainty with with pinhole data.
            This uncertainty is given as follow:
               dq_star = math.sqrt[(x0**2*(dy0)*dx0)**2 +
                    (x1**2 *(dy1)*dx1)**2 + ..+ (xn**2 *(dyn)*dxn)**2 ]
            where n >= len(data.x)-1
            dxi = 1/2*(xi+1 - xi) + (xi - xi-1)
            dx0 = (x1 - x0)/2
            dxn = (xn - xn-1)/2
            dyn: error on dy
           
            @param data:
            note: if data doesn't contain dy assume dy= math.sqrt(data.y)
        """          
        if len(data.x) <= 1 or len(data.y) <= 1 or \
            len(data.x) != len(data.y) or \
            (data.dy is not None and (len(data.dy) != len(data.y))):
            msg = "Length of data.x and data.y must be equal"
            msg += " and greater than 1; got x=%s, y=%s"%(len(data.x),
                                                         len(data.y))
            raise ValueError, msg
        else:
            #Create error for data without dy error
            if data.dy is None:
                dy = math.sqrt(y) 
            else:
                dy = data.dy
                
            n = len(data.x) - 1
            #compute the first delta
            dx0 = (data.x[1] - data.x[0])/2
            #compute the last delta
            dxn= (data.x[n] - data.x[n-1])/2
            sum = 0
            sum += (data.x[0] * data.x[0] * dy[0] * dx0)**2
            sum += (data.x[n] * data.x[n] * dy[n] * dxn)**2
            if len(data.x) == 2:
                return math.sqrt(sum)
            else:
                #iterate between for element different from the first and the last
                for i in xrange(1, n-1):
                    dxi = (data.x[i+1] - data.x[i-1])/2
                    sum += (data.x[i] * data.x[i] * dy[i] * dxi)**2
                return math.sqrt(sum)
        
    def _get_extrapolated_data(self, model, npts=INTEGRATION_NSTEPS,
                              q_start=Q_MINIMUM, q_end=Q_MAXIMUM):
        """
            @return extrapolate data create from data
        """
        #create new Data1D to compute the invariant
        q = numpy.linspace(start=q_start,
                           stop=q_end,
                           num=npts,
                           endpoint=True)
        iq = model.evaluate_model(q)
        diq = model.evaluate_model_errors(q)
         
        result_data = LoaderData1D(x=q, y=iq, dy=diq)
        return result_data
    
    def get_qstar_low(self):
        """
            Compute the invariant for extrapolated data at low q range.
            
            Implementation:
                data = self._get_extra_data_low()
                return self._get_qstar()
                
            @return q_star: the invariant for data extrapolated at low q.
        """
        # Data boundaries for fitting
        qmin = self._data.x[0]
        qmax = self._data.x[self._low_extrapolation_npts - 1]
        
        # Extrapolate the low-Q data
        self._fit(model=self._low_extrapolation_function,
                         qmin=qmin,
                         qmax=qmax,
                         power=self._low_extrapolation_power)
        
        # Distribution starting point
        q_start = Q_MINIMUM
        if Q_MINIMUM >= qmin:
            q_start = qmin/10
        
        data = self._get_extrapolated_data(model=self._low_extrapolation_function,
                                               npts=INTEGRATION_NSTEPS,
                                               q_start=q_start, q_end=qmin)
        
        # Systematic error
        # If we have smearing, the shape of the I(q) distribution at low Q will
        # may not be a Guinier or simple power law. The following is a conservative
        # estimation for the systematic error.
        err = qmin*qmin*math.fabs((qmin-q_start)*(data.y[0] - data.y[INTEGRATION_NSTEPS-1]))
        return self._get_qstar(data), self._get_qstar_uncertainty(data)+err
        
    def get_qstar_high(self):
        """
            Compute the invariant for extrapolated data at high q range.
            
            Implementation:
                data = self._get_extra_data_high()
                return self._get_qstar()
                
            @return q_star: the invariant for data extrapolated at high q.
        """
        # Data boundaries for fitting
        x_len = len(self._data.x) - 1
        qmin = self._data.x[x_len - (self._high_extrapolation_npts - 1)]
        qmax = self._data.x[x_len]
        q_end = Q_MAXIMUM
        
        # fit the data with a model to get the appropriate parameters
        self._fit(model=self._high_extrapolation_function,
                         qmin=qmin,
                         qmax=qmax,
                         power=self._high_extrapolation_power)
        
        #create new Data1D to compute the invariant
        data = self._get_extrapolated_data(model=self._high_extrapolation_function,
                                           npts=INTEGRATION_NSTEPS,
                                           q_start=qmax, q_end=q_end)        
        
        return self._get_qstar(data), self._get_qstar_uncertainty(data)
    
    def get_extra_data_low(self, npts_in=None, q_start=Q_MINIMUM, nsteps=INTEGRATION_NSTEPS):
        """
           This method generates 2 data sets , the first is a data created during 
           low extrapolation . its y is generated from x in [ Q_MINIMUM - the minimum of
           data.x] and the outputs of the extrapolator .
            (data is the data used to compute invariant) 
           the second is also data produced during the fit but the x range considered
           is within the reel range of data x.
           x uses is in [minimum of data.x up to npts_in points]
           @param npts_in: the number of first points of data to consider  for computing
           y's coming out of the fit.
           @param q_start: is the minimum value to uses for extrapolated data
           @param npts: the number of point used to create extrapolated data 
           
        """
        # Create a data from result of the fit for a range outside of the data
        # at low q range
        q_start = max(Q_MINIMUM, q_start)
        qmin = min(self._data.x)
        
        if q_start < qmin:
            data_out_range = self._get_extrapolated_data(model=self._low_extrapolation_function,
                                                         npts=nsteps,
                                                         q_start=q_start, q_end=qmin)
        else:
            data_out_range = LoaderData1D(x=numpy.zeros(0), y=numpy.zeros(0))
            
        # Create data from the result of the fit for a range inside data q range for
        # low q
        if npts_in is None :
            npts_in = self._low_extrapolation_npts

        x = self._data.x[:npts_in]
        y = self._low_extrapolation_function.evaluate_model(x=x)
        data_in_range = LoaderData1D(x=x, y=y)
        
        return data_out_range, data_in_range
          
    def get_extra_data_high(self, npts_in=None, q_end=Q_MAXIMUM, nsteps=INTEGRATION_NSTEPS ):
        """
           This method generates 2 data sets , the first is a data created during 
           low extrapolation . its y is generated from x in [ the maximum of
           data.x to  Q_MAXIMUM] and the outputs of the extrapolator .
            (data is the data used to compute invariant) 
           the second is also data produced during the fit but the x range considered
           is within the reel range of data x.
           x uses is from maximum of data.x up to npts_in points before data.x maximum.
           @param npts_in: the number of first points of data to consider  for computing
           y's coming out of the fit.
           @param q_end: is the maximum value to uses for extrapolated data
           @param npts: the number of point used to create extrapolated data 
           
        """
        #Create a data from result of the fit for a range outside of the data
        # at low q range
        qmax = max(self._data.x)
        if  q_end != Q_MAXIMUM or nsteps != INTEGRATION_NSTEPS:
            if q_end > Q_MAXIMUM:
               q_end = Q_MAXIMUM
            elif q_end <= qmax:
                q_end = qmax * 10
                
            #compute the new data with the proper result of the fit for different
            #boundary and step, outside of data
            data_out_range = self._get_extrapolated_data(model=self._high_extrapolation_function,
                                               npts=nsteps,
                                               q_start=qmax, q_end=q_end)
        else:
            data_out_range = LoaderData1D(x=numpy.zeros(0), y=numpy.zeros(0))
        
        #Create data from the result of the fit for a range inside data q range for
        #high q
        if npts_in is None :
            npts_in = self._high_extrapolation_npts
            
        x_len = len(self._data.x)
        x = self._data.x[(x_len-npts_in):]
        y = self._high_extrapolation_function.evaluate_model(x=x)
        data_in_range = LoaderData1D(x=x, y=y)
        
        return data_out_range, data_in_range
          
     
    def set_extrapolation(self, range, npts=4, function=None, power=None):
        """
            Set the extrapolation parameters for the high or low Q-range.
            Note that this does not turn extrapolation on or off.
            @param range: a keyword set the type of extrapolation . type string
            @param npts: the numbers of q points of data to consider for extrapolation
            @param function: a keyword to select the function to use for extrapolation.
                of type string.
            @param power: an power to apply power_low function
                
        """
        range = range.lower()
        if range not in ['high', 'low']:
            raise ValueError, "Extrapolation range should be 'high' or 'low'"
        function = function.lower()
        if function not in ['power_law', 'guinier']:
            raise ValueError, "Extrapolation function should be 'guinier' or 'power_law'"
        
        if range == 'high':
            if function != 'power_law':
                raise ValueError, "Extrapolation only allows a power law at high Q"
            self._high_extrapolation_npts  = npts
            self._high_extrapolation_power = power
        else:
            if function == 'power_law':
                self._low_extrapolation_function = PowerLaw()
            else:
                self._low_extrapolation_function = Guinier()
            self._low_extrapolation_npts  = npts
            self._low_extrapolation_power = power
        
    def get_qstar(self, extrapolation=None):
        """
            Compute the invariant of the local copy of data.
           
            @param extrapolation: string to apply optional extrapolation    
            @return q_star: invariant of the data within data's q range
            
            @warning: When using setting data to Data1D , the user is responsible of
            checking that the scale and the background are properly apply to the data
        """
        self._qstar = self._get_qstar(self._data)
        self._qstar_err = self._get_qstar_uncertainty(self._data)
        
        if extrapolation is None:
            return self._qstar
        
        # Compute invariant plus invariant of extrapolated data
        extrapolation = extrapolation.lower()    
        if extrapolation == "low":
            qs_low, dqs_low = self.get_qstar_low()
            qs_hi, dqs_hi   = 0, 0
            
        elif extrapolation == "high":
            qs_low, dqs_low = 0, 0
            qs_hi, dqs_hi   = self.get_qstar_high()
            
        elif extrapolation == "both":
            qs_low, dqs_low = self.get_qstar_low()
            qs_hi, dqs_hi   = self.get_qstar_high()
            
        self._qstar     += qs_low + qs_hi
        self._qstar_err = math.sqrt(self._qstar_err*self._qstar_err \
                                    + dqs_low*dqs_low + dqs_hi*dqs_hi)
        
        return self._qstar
       
    def get_surface(self, contrast, porod_const, extrapolation=None):
        """
            Compute the surface of the data.
            
            Implementation:
              V=  self.get_volume_fraction(contrast, extrapolation)
        
              Compute the surface given by:
                surface = (2*pi *V(1- V)*porod_const)/ q_star
               
            @param contrast: contrast value to compute the volume
            @param porod_const: Porod constant to compute the surface 
            @param extrapolation: string to apply optional extrapolation
            @return: specific surface 
        """
        # Compute the volume
        volume = self.get_volume_fraction(contrast, extrapolation)
        return 2 * math.pi * volume *(1 - volume) * float(porod_const)/self._qstar
        
    def get_volume_fraction(self, contrast, extrapolation=None):
        """
            Compute volume fraction is deduced as follow:
            
            q_star = 2*(pi*contrast)**2* volume( 1- volume)
            for k = 10^(-8)*q_star/(2*(pi*|contrast|)**2)
            we get 2 values of volume:
                 with   1 - 4 * k >= 0
                 volume1 = (1- sqrt(1- 4*k))/2
                 volume2 = (1+ sqrt(1- 4*k))/2
           
            q_star: the invariant value included extrapolation is applied
                         unit  1/A^(3)*1/cm
                    q_star = self.get_qstar()
                    
            the result returned will be 0 <= volume <= 1
            
            @param contrast: contrast value provides by the user of type float.
                     contrast unit is 1/A^(2)= 10^(16)cm^(2)
            @param extrapolation: string to apply optional extrapolation
            @return: volume fraction
            @note: volume fraction must have no unit
        """
        if contrast <= 0:
            raise ValueError, "The contrast parameter must be greater than zero"  
        
        # Make sure Q star is up to date
        self.get_qstar(extrapolation)
        
        if self._qstar <= 0:
            raise RuntimeError, "Invalid invariant: Invariant Q* must be greater than zero"
        
        # Compute intermediate constant
        k =  1.e-8 * self._qstar/(2 * (math.pi * math.fabs(float(contrast)))**2)
        # Check discriminant value
        discrim = 1 - 4 * k
        
        # Compute volume fraction
        if discrim < 0:
            raise RuntimeError, "Could not compute the volume fraction: negative discriminant"
        elif discrim == 0:
            return 1/2
        else:
            volume1 = 0.5 * (1 - math.sqrt(discrim))
            volume2 = 0.5 * (1 + math.sqrt(discrim))
            
            if 0 <= volume1 and volume1 <= 1:
                return volume1
            elif 0 <= volume2 and volume2 <= 1: 
                return volume2 
            raise RuntimeError, "Could not compute the volume fraction: inconsistent results"
    
    def get_qstar_with_error(self, extrapolation=None):
        """
            Compute the invariant uncertainty.
            This uncertainty computation depends on whether or not the data is
            smeared.
            
            @param extrapolation: string to apply optional extrapolation
            @return: invariant, the invariant uncertainty
        """    
        self.get_qstar(extrapolation)
        return self._qstar, self._qstar_err
    
    def get_volume_fraction_with_error(self, contrast, extrapolation=None):
        """
            Compute uncertainty on volume value as well as the volume fraction
            This uncertainty is given by the following equation:
            dV = 0.5 * (4*k* dq_star) /(2* math.sqrt(1-k* q_star))
                                 
            for k = 10^(-8)*q_star/(2*(pi*|contrast|)**2)
            
            q_star: the invariant value including extrapolated value if existing
            dq_star: the invariant uncertainty
            dV: the volume uncertainty
            
            The uncertainty will be set to -1 if it can't be computed.
            
            @param contrast: contrast value 
            @param extrapolation: string to apply optional extrapolation
            @return: V, dV = volume fraction, error on volume fraction
        """
        volume = self.get_volume_fraction(contrast, extrapolation)
        
        # Compute error
        k =  1.e-8 * self._qstar /(2 * (math.pi* math.fabs(float(contrast)))**2)
        # Check value inside the sqrt function
        value = 1 - k * self._qstar
        if (value) <= 0:
            uncertainty = -1
        # Compute uncertainty
        uncertainty = math.fabs((0.5 * 4 * k * self._qstar_err)/(2 * math.sqrt(1 - k * self._qstar)))
        
        return volume, uncertainty
    
    def get_surface_with_error(self, contrast, porod_const, extrapolation=None):
        """
            Compute uncertainty of the surface value as well as the surface value.
            The uncertainty is given as follow:
            
            dS = porod_const *2*pi[( dV -2*V*dV)/q_star
                 + dq_star(v-v**2)
                 
            q_star: the invariant value
            dq_star: the invariant uncertainty
            V: the volume fraction value
            dV: the volume uncertainty
            
            @param contrast: contrast value
            @param porod_const: porod constant value 
            @param extrapolation: string to apply optional extrapolation
            @return S, dS: the surface, with its uncertainty
        """
        # We get the volume fraction, with error
        #   get_volume_fraction_with_error calls get_volume_fraction
        #   get_volume_fraction calls get_qstar
        #   which computes Qstar and dQstar
        v, dv = self.get_volume_fraction_with_error(contrast, extrapolation)

        s = self.get_surface(contrast=contrast, porod_const=porod_const)
        ds = porod_const * 2 * math.pi * (( dv - 2 * v * dv)/ self._qstar\
                 + self._qstar_err * ( v - v**2))

        return s, ds