The general approach in SI is to illuminate the object with a sinusoidal illumination pattern of the form [

2]

at each of three spatial phases

*ϕ*_{1} = 0,

*ϕ*_{2} = 2

*π*/3, and

*ϕ*_{3} = 4

*π*/3. Although other structured forms are possible [

19], this one is particularly easy to implement. The quantity

*m* is the modulation contrast (a number varying from 0 to 1), and

*ν* is the modulation spatial frequency. The factor of 1/2 placed in front, not present in previous work, is used here in order to represent the fact that half of the illumination light is absorbed or reflected by the grid placed in the illumination path. If we take the limit

*m* → 0, we obtain standard widefield illumination of half the intensity that one would obtain without the grid in place. Note that

*s* represents a

*normalized* illumination amplitude, ranging from 0 to 1.

Ignoring the effects of optical blurring, the resulting modulated images

*g*_{i}(

*x, y*) are given by

for a planar object distribution

*f*(

*x,y*) and out-of-focus light

*d*(

*x,y*), both scaled to what one would obtain with standard widefield imaging. For fluorescence imaging, the absolute brightness

*f* of the object contains the illumination irradiance

*I*, the fluorophore quantum yield

*q*, and a factor Ω resulting from integrating the angular distribution of fluorescence emission over the numerical aperture of the imaging optics:

*f*_{fluor} =

*Iq*Ω. For brightfield imaging,

*f* is simply the illumination

*I* multiplied by the object reflectance

*R: f*_{bright} =

*IR*. Thus, the expression for

*g*_{i} is valid for the cases of both fluorescence imaging and brightfield imaging, but with a subtle difference in what

*f* means for each case.

In order to obtain an optically sectioned image

*i*(

*x,y*) at the focal plane, the most common algorithm used is [

1,

2,

8,

20–

31]

based on square law detection. Alternative illumination patterns allow for different processing approaches [

19]. SI for superresolution, for example, relies on the Moire effect to detect light emitted outside the conventional bandwidth limit.

Since the algorithm (3) operates on each pixel independently, we have dropped the spatial arguments (

*x,y*) as unnecessary. (These can be added back into each equation at any point.) Inserting

Eqs. (1) into

(2) and applying trigonometric identities, we obtain the result

Thus, the sectioned image is a copy of the object distribution, as we expect, but scaled by the factor

. The quantitatively-scaled algorithm is obtained by multiplying

Eq. (3) by the inverse of this factor:

Note that the algorithm assumes that the out-of-focus light

*d*(

*x,y*) does not change with a shift in the illumination pattern. A consequence of this result is that in order to obtain quantitative results for the sectioned image, one must estimate the modulation contrast

*m*. A further assumption required is that of linearity, which in fluorescence imaging is limited to weakly fluorescent structures [

1].

The scale factor in front of the square root differs from that given in previous studies. The factor of 2 in the numerator appears as a result of the 1/2 scaling introduced into our definition of

*s*_{i}(

*x, y*) and thus is new. All previous authors have also assumed ideal modulation (

*m* = 1). This is an assumption which introduces a large error into the quantitative result. Moreover, since the modulation

*m*(

*x,y*) is in general spatially varying, the error introduced is generally not a simple scalar factor for the whole image. As a whole, the literature shows wide disagreement over the appropriate scale factor to place in front of the square root. Refs [

20,

22,

26,

28,

32,

33] use

, which is appropriate when

*m* = 1 and the factor of 1/2 in

*s*(

*x,y*) is not used. Refs [

2,

23–

25,

27,

29] use a scale factor of 1, which is the most appropriate choice for a non-quantitative approach, while other authors use alternative factors such as

[

8],

[

1,

31], or

[

21] without explanation.

In practice, one finds that even for ideal samples *m* cannot achieve the maximum value of 1. The modulation contrast, however, remains excellent (*m* > 0.5) in thin samples in which the sectioned plane is taken near the surface, but poor (*m* < 0.1) in dense tissue samples (in which multiple scattering is present) and in deeper layers of thinly scattering media.