## Abstract

With the advent of modern-day computational imagers, the phase of the optical transfer function may no longer be summarily ignored. This study discusses some important properties of the phase transfer function (PTF) of digital incoherent imaging systems and their implications on the performance and characterization of these systems. The effects of aliasing and sub-pixel image shifts on the phase of the complex frequency response of these sampled systems are described, including an examination of the specific case of moderate aliasing. Key properties of this function in aliased imaging systems are derived and their potential treatment to a range of diverse applications encompassing traditional and computational imaging systems is discussed.

©2011 Optical Society of America

## 1. Introduction

A digital optically incoherent imaging system may be characterized using its optical transfer function (OTF) which represents the transformation of the spatial frequency content of the object onto the image plane. The OTF may be decomposed into its magnitude, represented by the modulation transfer function (MTF) and its phase, given by the phase transfer function (PTF). The MTF represents the contrast reduction at each spatial frequency whereas the PTF represents the spatial shift of these frequencies [1]. While the MTF has been used extensively to characterize imaging systems, the PTF has long been ignored since it was deemed not to contain valuable information in well-designed traditional imaging systems. In this context, the term “well-designed system” implies one in which the point spread function (PSF) is small and symmetric. In 1980, Rosenbruch & Gerschler argued that in such systems, the MTF alone would be sufficient to characterize imaging performance and the PTF could therefore be neglected [2,3]. In [2] it was claimed that the PTF never exceeds π/4 as long as the MTF is higher than 0.2 and the PTF could therefore generally be neglected while evaluating image quality. While this insight was largely valid for the imaging systems of the time, it does not apply to the new breed of digital imagers that exploit computational imaging techniques. In light of the major advances presently being witnessed in the evolution of computational imagers, two arguments can be made against this claim, namely

- (1) The criterion that the normalized MTF (0 ≤ MTF ≤ 1) should be greater than 0.2 for extracting good quality images from a scene is no longer a prerequisite for many of today’s imaging systems. For instance, computational imaging systems often deliberately sacrifice contrast at the optical sub-system end, but then employ reconstruction algorithms as an integral part of their imaging chain to recover this contrast, within the limits imposed by SNR. These algorithms are intended to be used to enhance the performance of the overall system while reducing the burden of requiring that the optics yield high MTF values [4].
- (2) Computational imaging systems have expanded the dimensions of their design space resulting in imaging scenarios which violate Rosenbruch and Gerschler’s observation that the PTF never exceeds π/4 as long as the MTF remains higher than 0.2.

It is well-known that the PSF contains all the information in the spatial domain required to describe the response of an imaging system to a point object or impulse [9]. When transformed to the spatial frequency domain, this information manifests itself in the complex OTF, which consists of both the MTF and the PTF. In traditional well-corrected systems, the nature of this transformation is such that information present in the PSF results in a real OTF which is nothing but the MTF itself. However, when the PSF is asymmetric (sometimes by design), the PTF is non-zero and therefore, a singular reliance on the MTF is no longer appropriate. Recent years have seen the emergence of a variety of computational imaging architectures that more often than not, do not conform to the traditional notions of PTF characteristics of a well-designed imaging system. The optics in these architectures exhibit a substantial PTF presence and therefore this PTF must be accounted for in order to optimize system performance. On the other hand, the design, manufacture and assembly of traditional imaging systems could also benefit from the knowledge of PTF in areas such as alignment compensation and wavefront correction. In view of the above insight, it becomes increasingly important to understand and account for the PTF of digital incoherent imaging systems in order to achieve optimal performance of such imagers.

The effects of diffraction and aberrations on the PTF of an incoherent imaging system in the absence of aliasing have been analyzed in detail [1,10,11]. However, with the advent of digital imaging systems, the effect of aliasing on the overall system behavior also becomes an important consideration. In this paper, we provide a mathematical description of the phase transfer function of sampled (digital) incoherent imaging systems, both in the absence and in the presence of aliasing. Furthermore, we herein present some general properties of the phase transfer function of a digital incoherent imaging system. We believe that knowledge of these properties would be useful in developing new methods for performance characterization of digital imaging systems and help develop new techniques for information extraction in various computational imaging architectures.

The remainder of this paper is organized as follows: In Section 2, we derive an expression for the PTF of sampled incoherent imaging systems and discuss how aliasing and the spatial sampling phase (i.e., sub-pixel shift) affect the PTF. A detailed analysis of the sampled PTF in case of moderate aliasing is presented, which occurs when the optical cut-off frequency is larger than the detector Nyquist frequency but smaller than the detector sampling frequency. In Section 3, we present some general properties of the PTF of such imaging systems along with a mathematical proof of these properties in the Appendices. Section 3 also discusses some of the implications of these properties. We further show in Section 3 that the first derivative of the PTF is a useful tool for estimating the optical cut-off frequency and determining the extent of aliasing present in the system in scenarios where this information may not be readily available. We finally conclude in Section 4 with a summary of our findings.

## 2. The phase transfer function of sampled incoherent imaging systems

Most optical image formation models generally assume that the optical system response is space-invariant. On the other hand, this notion of space-invariance does not readily extend to sampled imaging systems on account of intensity averaging caused by the finite pixel areas, as well as aliasing due to inadequate sampling. However, Wittenstein *et al* showed that the sampled imaging system may be approximated as a linear space-invariant system by modifying the OTF to include the effects of sampling and the spatial offset [12]. Although an approximation, the notion of sampled OTF has proven to be very useful in the characterization, design and optimization of digital imaging systems [13]. While the effects of sampling on the MTF have been previously analyzed and well understood [12,13], the effects of sampling on the PTF have not previously been similarly explored. The objective of this section is therefore to investigate the effects of aliasing and sampling phase on the PTF. The following analysis begins with a forward image formation model of sampled imaging systems and proceeds to provide some useful insights on the behavior of the sampled PTF with and without the presence of aliasing in the system. Particular emphasis is given here to the imaging condition that results in moderate aliasing.

In the sampled image formation model, the input signal is blurred by the optical PSF and then sampled by the detector as illustrated in Fig. 2
. In this figure, the detector sampling grid is represented by the green boxes (image pixels), while the blue dashed lines on the right hand side of the schematic represent the corresponding conjugate region in the object plane (object pixels). The dots at the center of each of these boxes represent the spatial locations of each sample. The object spatial coordinates are specified as (*x*′, *y*′) and the detector plane coordinates as (*x*, *y*), where *x*′, *y*′, *x* and *y* are continuous variables. In the event of sampling, information from the input scene is collected at discrete locations on a uniformly spaced sampling grid marked by the green dots on the detector array in Fig. 2, whose conjugate locations are given by the blue dots in the object plane.

Consider an object point at an arbitrary location within an object pixel whose center is at (*x*′* _{s}*,

*y*′

*). Let the lateral displacement of this point from (*

_{s}*x*′

*,*

_{s}*y*′

*) be (Δ*

_{s}*x*′, Δ

*y*′) so that its exact coordinates are given by (

*x*′

*– Δ*

_{s}*x′*,

*y*′

*– Δ*

_{s}*y′*). The corresponding imaging pixel will then be centered at (

*x*,

_{s}*y*) and the pre-sampled image point will be centered about (

_{s}*x*– Δ

_{s}*x*,

*y*– Δ

_{s}*y*), where (Δ

*x*, Δ

*y*) represents the lateral displacement (sub-pixel shift) of the image from the center of the imaging pixel. The image plane coordinates are scaled with respect to the object plane coordinates by the magnification

**M**of the system so that

The pre-sampled image may then be mathematically expressed as

*i*(

*x*– Δ

_{s}*x*,

*y*– Δ

_{s}*y*) is the captured image,

*o*(

*x*′

*– Δ*

_{s}*x′*,

*y*′

*– Δ*

_{s}*y′*) is the input signal prior to magnification,

*h*(

_{o}*x*,

*y*) is the optical PSF,

*h*(

_{d}*x*,

*y*) is the detector PSF due to the finite pixel area,

*p*is the pixel pitch and ⊗ denotes a convolution operation. For simplicity, a square pixel is assumed, whose pitch

*p*is equal to its size (i.e., 100% fill factor). Here,

*h*(

_{o}*x*,

*y*) is considered to be space-invariant and the system is assumed to have negligible noise.

Additionally, the maximum extent of the sub-pixel shift (Δ*x*, Δ*y*) along the horizontal and vertical directions is one-half the pixel pitch *p*, such that –*p*/2 ≤ (Δ*x*, Δ*y*) ≤ *p*/2. The sub-pixel shift is related to the angular sampling phase (Φ* _{x}*, Φ

*) by the relationship*

_{y}*, Φ*

_{x}*) ≤ π. For the sake of simplicity, we refer to (Δ*

_{y}*x*, Δ

*y*) as the sampling phase in the rest of this paper. It is noted that this analysis may be extended to cases where the fill factor is less than 100%, with the condition that the maximum extent of the sampling phase Δ

*x*now reduces to less than ±

*p*/2. We further restrict the analysis to one dimension and unit magnification (i.e.,

**M**= 1) for the sake of brevity. A point of note is that the intent of the work presented herein is to keep the analysis agnostic to the reconstruction filter or the end user. It is also noted that the underlying 1D analysis and approach presented in this paper may be similarly extended to the 2D case. Equation (2) may then be rewritten as

*comb*function is given by [9]

*k*is the pixel index. It must be noted that while the definition of the

*comb*function has infinite extent, the image plane itself has a finite support that results in a truncation of the spatial extent of this

*comb*function. This finite support can be modeled by the inclusion of a

*rectangle*function as discussed in [9]. For purposes of this analysis, the effect of this function is not critical and is therefore ignored.

If *o*(*x _{s}* – Δ

*x*) is an ideal point source, then

*i*(

*x*– Δ

_{s}*x*) would be the overall pre-sampled impulse response

*h*(

*x*– Δ

_{s}*x*) of the imaging system such that

However, when sampling occurs, information regarding the precise location of the image of the point source is lost within the confines of the detecting pixel and its positional information is now transferred to the center of the pixel. In mathematical terms, the process of sampling modifies Eq. (6) to yield the sampled impulse response *h _{s}*(

*x*) as

_{s}The Fourier transform of *h _{s}*(

*x*) then yields the OTF, namely

_{s}*H*(

_{s}*u*) of the system as

*u*is the non-normalized spatial frequency and

*u*= 1/(2

_{n}*p*) is the detector Nyquist frequency [14]. Equation (8) may be further simplified as

*H*=

*H*×

_{o}*H*is the overall pre-sampled OTF whose magnitude and phase are given by

_{d}*M*(

*u*) and

*Θ*(

*u*) respectively. From Eq. (9), it is seen that the process of sampling creates replicas of the OTF centered about integer multiples of the sampling frequency 2

*u*.

_{n}In order to gain a better insight into this process, we now proceed to analyze the sampled PTF for different aliasing scenarios. To categorize these different aliasing scenarios, we herein recall the relationship between Nyquist, sampling and optical cutoff frequencies. For a detector with a pixel pitch *p*, the Nyquist and sampling frequencies are given by *u _{n}* = 1/(2

*p*) and 2

*u*= 1/

_{n}*p*respectively. The optical cutoff frequency is given by

*u*= 1/(λF/#), where λ is the wavelength of light and F/# denotes the F-number of the imaging system, i.e., the ratio of its focal length to aperture width. The ratio of the sampling frequency to the optical cutoff frequency is then given by 2

_{o}*u*/

_{n}*u*= (λF/#)/

_{o}*p*[14]. When (λF/#)/

*p*≥ 2, we have

*u*≤

_{o}*u*, which indicates an absence of aliasing. When (λF/#)/

_{n}*p*< 2, we have

*u*>

_{o}*u*, which indicates the presence of aliasing. In this work, we also analyze a special case of aliasing, namely that of moderate aliasing, where the ratio (λF/#)/

_{n}*p*is between 1 and 2. This case is of particular interest since a large number of modern imaging systems fall into this category.

#### 2.1. The sampled PTF in the absence of aliasing

The well-known effect of sampling on the spatial frequency spectrum is to create replicas of the complex frequency response of the imaging system. Mathematically, these replicas manifest as individual terms within the summation operator of Eq. (9). In the event that the optical cutoff spatial frequency *u _{o}* is less than or equal to the Nyquist frequency

*u*, there is no aliasing in the system due to the low pass filtering nature of the optics. The only term within the summation operator in Eq. (9) that then contributes to system performance is the baseband replica of the OTF (i.e., for

_{n}*k*= 0) centered at

*u*= 0. The expression for the sampled OTF

*H*(

_{s}*u*) may be rewritten in terms of the magnitude

*M*(

*u*) and phase

*Θ*(

*u*) of the pre-sampled OTF as

As expected, the left hand side of Eq. (10) is simply the pre-sampled OTF of the imaging system plus an additional linear phase shift component that is proportional to the sampling phase Δ*x*. This linear phase does not affect the unaliased sampled MTF, but results in a global shift of the image in the spatial domain, as reflected in the sampled PTF expression given by Eq. (11). Figure 3
shows an example of the sampled PTFs for various values of the sampling phase in the absence of aliasing. Furthermore, it is seen that the sampled PTF *Θ _{s}*(

*u*) is quite sensitive to the sampling phase Δ

*x*.

#### 2.2. The sampled PTF in the presence of aliasing

In the presence of aliasing, several higher-order replicas of the OTF fold back onto the baseband spectrum and these replicas must be taken into account in determining the sampled OTF of the system. The expression for the sampled OTF *H _{s}*(

*u*) given in Eq. (9) may then be rewritten in terms of the magnitude

*M*(

*u*) and phase

*Θ*(

*u*) of the pre-sampled OTF as

Euler’s identity may then be applied to the complex exponential term to yield

The sampled MTF of the system in the presence of aliasing is then given by

The sampled PTF may similarly be expressed as

The extent of aliasing in an imaging system may either be severe or moderate. Severe aliasing is taken to be an imaging scenario where *u _{o}* > 2

*u*, i.e., the optical cutoff frequency is in excess of twice the Nyquist frequency. In such cases, it is evident from Eq. (14) and Eq. (15) that the spectral overlap of several copies of the OTF centered about integer multiples of twice the Nyquist frequency would corrupt the entire spatial frequency spectrum. Obtaining reliable PTF estimates from captured images would then be difficult and require some form of digital super-resolution techniques. On the other hand, moderate aliasing is considered as the scenario when a portion of the spatial frequency response remains uncorrupted by the folding of the higher spatial frequencies onto the lower frequencies, i.e., when

_{n}*u*<

_{n}*u*≤ 2

_{o}*u*. This most commonly occurring case of aliasing is examined next.

_{n}#### 2.3. The sampled PTF in the case of moderate aliasing

The case of moderate aliasing warrants particular consideration because, as will be shown subsequently, it yields special properties of the sampled PTF that are applicable only to this case. Moderate aliasing is said to occur when the optical frequency cutoff lies between one and two times the Nyquist frequency (i.e., *u _{n}* <

*u*≤ 2

_{o}*u*). In this case, the only relevant values of

_{n}*k*that would come into play are

*k*= 0 and

*k*= ± 1. The corresponding replicas of the OTF are centered about

*u*= 0 and

*u*= ± 2

*u*. The sampled OTF as described by Eq. (9) may then be expressed as

_{n}Using Eq. (14), the sampled MTF may then be written as

The above result has been previously reported in [12,13]. An important point of note here is that even though the individual MTF replicas are independent of Δ*x*, the overall sampled MTF is a function of the sampling phase. Extending this analysis to the domain of the phase transfer function, the sampled PTF for the case of moderate aliasing is directly obtained from Eq. (15) as

Figure 4
shows an example of the sampled PTF for the case of moderate aliasing and zero sampling phase, i.e., Δ*x* = 0. It is seen from this figure that *Θ _{s}*(

*u*) has been corrupted in the aliased region of the spatial frequency spectrum and is therefore reliable only in the unaliased region, namely for frequencies where 0 ≤

*u*< |2

*u*–

_{n}*u*|. Furthermore, Eq. (18) indicates that the sampled PTF

_{o}*Θ*(

_{s}*u*) is very sensitive to the sampling phase Δ

*x*, and even more so in the aliased region of the spectrum. The relationship between

*Θ*(

_{s}*u*) and Δ

*x*is also nonlinear in the aliased region, as seen in Fig. 5 .

The results shown in Fig. 3 and Fig. 4 are for a cubic phase mask wavefront coding imager with low phase mask strength α [5,7,8]. This type of imager was selected as an example because of the presence of both linear and non-linear components in its optical PTF (the former on account of defocus and the latter due to the anti-symmetric cubic nature of its phase mask element), thus affording a model case study. Additionally, the cubic nonlinearity is representative of coma aberration in traditional imagers. A low value of α was chosen to highlight the differences between *Θ*(*u*) and *Θ _{s}*(

*u*).

An important point of note is that the sampled PTF at the Nyquist frequency is an integer multiple of π radians regardless of the magnitude of the sampling phase. In the next section, it is shown that this property is valid irrespective of the degree of aliasing present in the system. Furthermore, in traditional imaging systems, a linear optical PTF is often ignored since it merely results in a global shift of the image – an effect that is often easily compensated. However, as may be seen in Fig. 5, the overall system PTF in the event of sampling may not be summarily ignored since aliasing and sampling phase could cause the sampled PTF to be nonlinear even when the optical PTF is linear. Compensating for such non-linear effects would require additional knowledge of the behavior of the sampled PTF in the event of aliasing.

In light of these aforementioned insights, a formal study of the properties of the phase transfer function of sampled imaging systems is undertaken with a view of understanding the effects of sampling on the PTF of such imagers. In the following section, we present some of these properties and discuss their implications to a number of imaging applications.

## 3. Properties of the sampled PTF

A mathematical evaluation of Eq. (9) offers some key insights into the behavior of the observed PTF of a digital incoherent imaging system, based on which we herein present some general properties of the sampled PTF *Θ _{s}*(

*u*) and its derivative

*Θ′*(

_{s}*u*) at

*u*= 0 and

*u*=

*u*. A formal mathematical proof of each of these properties is presented in the Appendices. The first two properties hold true irrespective of the extent of aliasing present in the system. The third property is applicable to the case of no aliasing or moderate aliasing, while the fourth property applies to the special case of moderate aliasing. These properties may be mathematically stated as follows:

_{n}- 1. ${\text{\Theta}}_{s}\left(0\right)=0.$
- 2. $\begin{array}{cccc}{\Theta}_{s}\left({u}_{n}\right)& =& \{\begin{array}{r}\begin{array}{cc}2m\pi & ;\mathrm{Re}\left\{{H}_{s}\left({u}_{n}\right)\right\}\ge 0\end{array}\\ \begin{array}{cc}\left(2m-1\right)\pi & ;\mathrm{Re}\left\{{H}_{s}\left({u}_{n}\right)\right\}<0\end{array}\end{array}& ,\forall \mathrm{int}egerm\end{array}.$
- 3. ${{\Theta}^{\prime}}_{s}\left(0\right)={\Theta}^{\prime}\left(0\right)-2\pi \Delta x;\forall {u}_{o}:0<{u}_{o}\le 2{u}_{n}.$
- 4. ${{\Theta}^{\prime}}_{s}\left({u}_{n}\right)={\Theta}^{\prime}\left({u}_{n}\right)-2\pi \Delta x+\frac{{M}^{\prime}\left({u}_{n}\right)}{M\left({u}_{n}\right)}\mathrm{tan}\left\{\Theta \left({u}_{n}\right)-2\pi {u}_{n}\Delta x\right\};\forall {u}_{o}:{u}_{n}<{u}_{o}\le 2{u}_{n}.$

While it is well-known that the OTF of an incoherent imaging system exhibits Hermitian symmetry, it can be shown that this symmetry also extends to sampled incoherent imaging systems, regardless of aliasing. Indeed, given that the sampled signal in incoherent imaging systems is real and non-negative, it naturally follows that its sampled OTF should exhibit such a behavior. An important implication of this result is that the sampled PTF is anti-symmetric in nature and hence always zero at DC, irrespective of whether or not the imaging system suffers from aliasing. This fact is evident in the plots of Fig. 4 and Fig. 5.

Property 2 asserts that the sampled OTF at the Nyquist frequency is strictly real and therefore its phase *Θ _{s}*(

*u*) is an integer multiple of π radians. Again, this property is true regardless of the extent of aliasing in the imaging system. Whether this integer multiple is even or odd relies on whether the real part of

_{n}*H*(

_{s}*u*) is non-negative or negative, respectively. Two key variables that influence the sign of Re{

_{n}*H*(

_{s}*u*)} are the pre-sampled PTF values at

_{n}*Θ*((1 – 2

*k*)

*u*) and the sampling phase, namely Δ

_{n}*x*.

The third and fourth properties deal with the first derivatives of the sampled PTF in the special case of moderate aliasing. The third property asserts that when aliasing is absent or if the extent of aliasing in the system is moderate, then the difference between the first derivatives of the pre-sampled and sampled PTFs at zero-frequency equals the sampling phase scaled by a constant value of 2π. Property 3 as stated above is specific to the case of moderate or no aliasing; however a formal relationship between the first derivatives of the pre-sampled and sampled PTFs as a function of sampling phase, regardless of extent of aliasing, is given in Eq. (C8) in Appendix C and may be employed in extending its application to severely aliased situations. The fourth property indicates that the relationship between the derivatives of the pre-sampled and sampled PTFs at the Nyquist frequency is a function of the pre-sampled MTF and PTF, the sampling phase and the derivative of the pre-sampled MTF. While Property 4 is presented for the special case of moderate aliasing, it may be extended to severe aliasing cases by utilizing Eq. (D7) in Appendix D.

In section 1, it was argued that the PTF is important and should be taken into consideration in both computational imaging systems as well as traditional imagers. An overwhelming majority of today’s imaging systems are digital in nature and hence sampling is an inherent part of the imaging chain. Unlike the MTF, the PTF is very sensitive to the sampling phase even in the absence of aliasing. The process of measuring the PTF would then inevitably be subject to registration artifacts which could easily result in ambiguous data collection. When aliasing is absent, a non-zero sampling phase would still introduce a measurement error so that the measured PTF would not always be a true representation of the pre-sampled PTF. Understanding the nature of this error and quantifying its presence is important because this error can then be accounted for, in say, image reconstruction algorithms, which today are a staple of computational imaging systems.

In addition to the sampled PTF and its slope, useful information can also be gleaned from higher-order derivatives of the former. While an extended treatise on such derivatives is beyond the scope of this paper, we nonetheless discuss preliminary observations on the behavior of these functions. For instance, in the presence of moderate aliasing, the sampled PTF begins to deviate from its pre-sampled counterpart in a non-linear fashion starting at the spatial frequency location where the spectral replica adds on to the baseband frequency response, i.e., at *u _{a}* = 2

*u*–

_{n}*u*. This effect is not only apparent in the sampled PTFs as seen in the plots of Fig. 4 and Fig. 5, but also in the first and second derivatives as shown in Fig. 6 and Fig. 7 respectively. When the pixel size

_{o}*p*and hence

*u*is known, identifying the value of

_{n}*u*helps to determine the optical cutoff

_{a}*u*. While the pixel size could be easily deduced from the knowledge of pixel count and the sensor format, most off-the-shelf lens datasheets typically do not include optical cut-off frequency information. Determining

_{o}*u*in such instances helps in turn determine the effective value of

_{a}*u*, which could then be valuable for example, in improving edge-based MTF testing algorithms or digital super-resolution techniques. The above approach may also be beneficial in practical scenarios such as metrology and imager characterization where the extent of aliasing in a system may not be known a priori. In some applications such as adaptive super-resolution, it is possible to envision situations where the optical properties of the system could be changing in real-time and hence adaptive feedback mechanisms or post-capture analysis of images might warrant determination of instantaneous properties of the imaging system. Another example is a system exhibiting space-variant imaging or aberrations, where the optical cutoff could vary within a single image frame. Furthermore, knowledge of the F-number only helps identify the upper limit of

_{o}*u*; the true optical cut-off frequency could in fact be lower due to factors such as aberrations. A case in point is that many digital imaging systems include anti-aliasing filters to reduce the value of

_{o}*u*. Here, knowledge of

_{o}*u*could then yield better estimates of the effective optical cutoff frequency.

_{a}In systems with moderate values of optical PTF which are also a slowly varying function of spatial frequency, the second derivative of the sampled PTF exhibits an extremum at *u _{a}*, whose location does not shift along the frequency axis when the sampling phase is varied. This behavior is clearly evident upon inspecting Fig. 7. In the presence of measurement noise, a low-pass smoothing filter or an appropriate curve-fitting technique may be employed on the sampled PTF prior to evaluating its derivatives. Judicious selection of filters or curve-fitting parameters could isolate the random noise variations from the effects of aliasing to ensure that the second derivative would correctly identify the value of

*u*. Even for systems whose sampled PTF is rapidly varying with respect to

_{a}*u*, measurements with two or more sampling phases can be made to identify

*u*. It this case, the difference in the first derivatives would be constant in the region where aliasing is absent, given that this difference is made up of merely the linear slope differentials due to different sampling phases. Beyond

_{a}*u*, aliasing induces non-linearity in the sampled PTF and this difference is no longer constant. Figure 6 illustrates this point for several sampling phase values. A similar signature also extends to the second derivative of the sampled PTF, wherein the second derivatives are identical for

_{a}*u*<

*u*and diverge thereafter as seen in Fig. 7. An absence of such signatures in the derivatives of the sampled PTF would then be an immediate indicator of severe aliasing in the system. Moreover, local extrema of the second derivatives induced by an oscillating PTF would shift along the spatial frequency axis when the sampling phase is modified, whereas a similar signature marking the beginning of aliasing remains stationary. Again, Fig. 7 demonstrates this feature wherein the spikes marking

_{a}*u*do not vary with Δ

_{a}*x*, whereas local maxima within the aliased region shift along the

*u*-axis as Δ

*x*is modified. It is similarly observed that for moderate aliasing, the second derivative becomes zero at the Nyquist frequency for a wide range of imaging systems including those that exhibit substantial optical PTFs with both linear and nonlinear characteristics.

It is envisioned that the properties presented in this section could be used independently or in conjunction with one another in the development of a range of algorithms for sub-pixel shift estimation and registration, super-resolution, slanted-edge PTF measurement techniques [15,16], wavefront coding imager characterization, lens misalignment detection and sparse aperture alignment. For instance, Properties 2, 3 and 4 may be useful in isolating the sampling phase contribution from the measured PTF which would be of great practical significance for sub-pixel shift estimation, super-resolution and OTF measurement. Even though differences between sub-pixel-shifted PSFs may not be readily apparent to visual inspection alone, this information is more discernable when observed via the PTF. Given that applications that rely on Property 3 would depend only on the frequency bins at and adjacent to DC, the ensuing algorithms would remain robust even in the presence of noise and moderate aliasing. At the same time, Property 4 could provide a supporting data point when used in tandem with Property 3.

Many of today’s computational imaging systems such as cubic-pm wavefront coding imagers are characterized by non-traditional OTFs where the PTF is not only non-zero but also highly non-linear. Such PTFs therefore impart perceptible shifts to various spatial frequencies contained within the image resulting in visible artifacts even under the most optimal of imaging conditions [8]. For instance, the depth dependent nature of the PTF in wavefront coding systems could be leveraged for focusing and characterizing such imagers [17]. Knowledge of the PTF could also be used to create better filters [8] in the image reconstruction chain to minimize adverse effects of these artifacts. In addition, real-time knowledge of the PTF obtained through image-based measurements could in principle be employed as a performance metric for adaptive, closed-loop-feedback imaging systems. On the other hand, ignoring the PTF in traditional systems that exhibit odd-order aberrations could result in erroneous image quality information. For example, the presence of misalignment in optical elements of a traditional imager could manifest itself as tilt and coma aberrations in addition to defocus [18]. These signatures often tend to be more pronounced in the PTF than in the MTF of these systems [19]. In short, leveraging the knowledge of the PTF as an additional metric could be valuable in advancing the performance of modern digital incoherent imaging systems.

## 4. Summary

The analysis presented in this paper is a first step in understanding how sampling affects the PTF at different spatial frequencies and how this influence varies with the severity of aliasing. In this paper, some of the properties of the phase transfer function of digital incoherent imaging systems were reviewed, and the use of the PTF as an important tool to enhance performance and characterization of imaging systems was advocated. An expression for the PTF of a sampled imaging system was presented and the effects of aliasing and sampling phase on this function were investigated. It is envisioned that the properties of the sampled PTF obtained via this study could be exploited in several applications including wavefront coding, image reconstruction algorithms, sub-pixel shift estimation, super-resolution, adaptive computational imaging, sparse-aperture imaging and optical alignment in a manufacturing environment.

## Appendix A. The sampled OTF is Hermitian symmetric

To prove that the sampled OTF exhibits Hermitian symmetry, i.e., *H _{s}*(–

*u*) =

*H**(

_{s}*u*) as per Property 1, Eq. (9) is rewritten for –

*u*as

Let *l* = –*k*, then the above equation becomes

## Appendix B. The sampled OTF is real at the Nyquist frequency

Proof that the sampled OTF is real at the Nyquist frequency is obtained by showing that *H _{s}*(

*u*) =

_{n}*H**(

_{s}*u*), thereby establishing that

_{n}*H*(

_{s}*u*) is a strictly real quantity. The phase of the sampled OTF would then be an integer multiple of π radians. The complex conjugate of the sampled PTF is derived from Eq. (9), as

_{n}Given that the pre-sampled OTF is Hermitian-symmetric, it follows that

Incorporating the above relationship into the expression for *H _{s}**(

*u*) yields

_{n}Upon changing the indices of summation such that *k* = 1 – *l*, we obtain

*H*(

_{s}*u*)} = 0. The sampled PTF

_{n}*Θ*(

_{s}*u*) at the Nyquist frequency is then either an even or an odd integer multiple of π radians depending on whether Re{

_{n}*H*(

_{s}*u*)} is non-negative or negative respectively. Property 2 is thus proved.

_{n}## Appendix C. The first derivative of the sampled PTF at DC

Evaluating Eq. (15), which is applicable to all cases of sampling regardless of aliasing, let **α*** _{s}*(

*u*) = Re{

*H*(

_{s}*u*)} and

**β**

*(*

_{s}*u*) = Im{

*H*(

_{s}*u*)} such that

The sampled MTF and PTF of the imaging system may then be expressed as

Furthermore, the first derivative of the sampled PTF is given by

**α**′

*and*

_{s}**β**′

*are the first derivatives respectively of*

_{s}**α**

*and*

_{s}**β**

*, with respect to*

_{s}*u*. From Property 1 it is seen that

**β**

*(0) = 0. Equation (C3) may then be reduced to the form*

_{s}Evaluating for **β**′* _{s}*(

*u*), we have

In the above equation, *M′*(*u*) and *Θ′*(*u*) are the first derivatives respectively of *M*(*u*) and *Θ*(*u*), with respect to the spatial frequency *u*. At this point, the Hermitian symmetry of the pre-sampled OTF is recalled as

Evaluating for *Θ*′* _{s}*(0) yields

Upon recognizing that the expression within the summation in the first term on the right hand side of Eq. (C7) is nothing but **α*** _{s}*(0) as per Eq. (C1), the above expression may be expanded as

*Θ′*(0) under all sampling scenarios. However, in the special case of moderate aliasing, the only replica of the OTF that comes into play at

_{s}*u*= 0 is the original baseband spectrum (i.e., for

*k*= 0) since

*H*(

*u*) = 0 for

*u*≥

*u*. In such a situation,

_{o}**α**

*(0) =*

_{s}*M*(0) from Eq. (C1), and Eq. (C8) reduces to

*u*< 2

_{o}*u*. Considering that in moderate or no aliasing, none of the replicas contribute to

_{n}*Θ′*(0), the sampled PTF at DC is then given by Eq. (11). Taking the derivative of Eq. (11) and evaluating it at

_{s}*u*= 0 would then directly yield Property 3.

## Appendix D. The first derivative of the sampled PTF at the Nyquist frequency

A general expression for the sampled PTF at the Nyquist frequency in the case of aliasing is obtained by evaluating Eq. (C3) at *u* = *u _{n}* and utilizing Property 2, which implies that the sampled OTF at the Nyquist frequency is real and hence

**β**

*(*

_{s}*u*) = 0. Equation (C3) then reduces to

_{n}At *u* = *u _{n}*, we also have

*u*– 2

*ku*= (1 – 2

_{n}*k*)

*u*. Now, let

_{n}Upon splitting up the right hand side of the above equation into two parts, one for *k* ≤ 0 and the other for *k* > 0, we have

Performing a change of variables on the index of summation of the first term on the right hand side of Eq. (D4), in the form of *k* = 1 – *l* yields

Now, inspecting Eq. (D2) in conjunction with Eq. (C6) reveals that the functions *g*_{1} and *g*_{2} are symmetric about *k* = 0. In other words, the two summation terms on the right hand side of Eq. (D5) are identical, save for the variable names of the indices of summation. The above equation then may be written as

Substituting for *g*_{1} and *g*_{2} and evaluating for *Θ*′* _{s}*(

*u*), yields the result

_{n}Equation (D7) yields the derivative of the sampled PTF at the Nyquist frequency under all aliased sampling conditions, irrespective of the extent of aliasing. In the special case where aliasing is moderate, the only terms within the summation expression in Eq. (D7) that would come into effect would be those corresponding to *k* = 1. In such a case, we also have

## Acknowledgments

The authors would like to thank Prasanna Rangarajan, Indranil Sinharoy and Predrag Milojkovic for their insightful discussions and constructive suggestions. The authors also thank the reviewers for their valuable feedback which greatly helped strengthen this paper.

Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-06-2-0035. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation heron.

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