## Abstract

We introduce and analyze the concept of space-spectrum uncertainty for certain commonly used designs of spectrally programmable cameras. Our key finding states that, it is not possible to simultaneously acquire high-resolution spatial images while programming the spectrum at high resolution. This phenomenon arises due to a Fourier relationship between the aperture used for resolving spectrum and its corresponding diffraction blur in the spatial image. We show that the product of spatial and spectral standard deviations is lower bounded by $\frac {\lambda }{4\pi \nu _0}$ femto square-meters, where *ν*_{0} is the density of groves in the diffraction grating and *λ* is the wavelength of light. Experiments with a lab prototype validate our findings and its implication for spectral programming.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Spectrum is often a unique feature of materials and is used for identification and classification across diverse fields such as geology [1], bio-imaging [2,3] and material identification [4,5]. Tools such as the hyperspectral camera capture the spectrum of a scene which is subsequently used for identification and classification purposes. Capturing the full spectrum, while useful, is also wasteful especially if we are only interested in measuring similarity of the spectral profile at each pixel to a small collection of reference spectra. It is hence useful to have cameras that can *optically* perform this comparison. Such cameras, called *spectrally-programmable cameras*, have been demonstrated [6,7] with compelling applications in computer vision. This paper analyzes a popular design for enabling spectral programmability, and derives a fundamental relationship between its achievable spatial and spectral resolutions.

#### 1.1 Problem setting

The analysis in this paper is for the optical setup shown in Fig. 1, commonly used in prior art for spectral programming [4,7–9]. The optical system consists of a series of lenses of focal length $f$, each subsequent pair separated by $2f$. The setup relays the image plane from plane P1 to P3 with a *pupil code* in P2. A diffraction grating placed on P3 provides a spectral dispersion of the light. The dispersed light is focused on plane P4 to form the so-called *rainbow plane*, where each point corresponds to the average intensity of light of the whole scene for a single wavelength. The image on P3 is simply relayed on to plane P5. Arbitrary spectral programming can then be performed by placing an (SLM) on the rainbow plane (P4) and measuring image on plane P5. Intuitively, planes P1 to P3 is a simple camera with an aperture in its Fourier plane, and planes P2 to P4 is a spectrometer with the aperture replacing a slit. The setup provides an insight into the tradeoff between spatial and spectral resolutions. While a camera requires a large and open aperture for a compact spatial blur, this would lead to severe loss in spectral resolvability, as a spectrometer requires a narrow opening. Our goal is to formalize the role played by the shape of the pupil code in deciding spatial and spectral resolution.

#### 1.2 Main result

We show that the pupil code $a(x, y)$ introduces a spectral and spatial blur, $h_\lambda (\lambda )$ and $h_x(x)$ respectively with standard deviations $\sigma _\lambda$ and $\sigma _x$ (detailed expression in Eqs. (7) and (8)). Our main contribution is in the form of a lower bound on the *space-spectrum bandwidth product* that relates the spectral resolution at which light can be programmed and the spatial resolution of captured image. This is encapsulated in the following theorem.

**Theorem 1** *For the spectrally-coded imaging architecture shown in Fig. 1, the product of spatial and spectral standard deviations* $\sigma _x$ *and* $\sigma _\lambda$*, respectively, is bounded as*

*where*$\nu _0$

*is the density of slits in the diffraction grating*.

This result was first explored in [7] and [9] where the authors demonstrated that the spatial and spectral resolutions were related to the choice of pupil code. Our paper builds on their results by providing a concise expression for the tradeoff. We prove that a Gaussian-shaped pupil code achieves the lower bound and leads to most compact spatial blur for a targeted spectral blur.

#### 1.3 Implications

The space-spectrum bandwidth product introduces an uncertainty in spectrally-programmable cameras, stating that one cannot arbitrarily program spectrum at high resolution without loss in spatial resolution. We demonstrate the impact of uncertainty by building a spectrally-programmable camera and showing that blocking one of two closely-spaced narrowband sources cannot be done without severe loss in spectral resolution. We also show that for narrowband filtering, the spatial blur is affected by the pupil code as well as the shape of the narrowband filter, and that a slit, a commonly used narrowband filter shape leads to a spectrally-varying spatial blur. Instead, using a Gaussian-shaped narrowband filter achieves *spectrally-independent* spatial blur, thereby being the optimal candidate for spectral programming.

**Hyperspectral imagers.** Apart from spectral programming, several hyperspectral imaging architectures [8–10] rely on obtaining spectrally-programmed images. Our findings impact such setups, as a key requirement of such setups is to capture high resolution images without sacrificing spectral resolution. Hence, the space-spectrum bandwidth product can serve as a design guide to carefully choose the pupil code to obtain desired spatial and spectral resolutions.

**Spatially-coded cameras.** We note that the analysis in the paper is targeted specifically at spectrally-programmable cameras and does not apply to many hyperspectral cameras where there is no pupil plane coding. Cameras such as the pushbroom camera and the coded aperture snapshot spectral imager (CASSI) [11,12] which scan the full HSI only perform spatial coding and, and as such, are not affected by this result. Since such systems code space and then measure its spectrally-sheared image, the spatial code only affects the spectral resolution and not the spatial resolution.

## 2. Prior work

We start our discussion by talking about capturing images with arbitrary spectral filters and then briefly state its applications. We then state the fundamental tradeoffs based on system parameters.

**Measurement model.** Consider a scene’s hyperspectral image (HSI) represented by $H(x, y, \lambda )$, where $(x, y)$ represent spatial coordinates and $\lambda$ represents wavelength. Our goal is to optically obtain a spectrally-programmed image. Specifically, given a spectral filter $f(\lambda )$, our aim is to implement a camera that captures the following grayscale image,

**Applications of spectral programming.** The ability to arbitrarily program spectrum enables a wide gamut of applications. This includes adaptive color displays [6], programmatically blocking illuminants [7], and detecting materials [4,5]. The key advantage in all these applications is to not measure the complete HSI, but only the desired spectrally-programmed images; this leads to fewer measurements at higher signal to noise ratio (SNR). Such a system can also be used for compressively sensing the complete HSI [8,9] which relies on capturing projection of a scene’s HSI on random or designed spectral filters.

**Spectrally-programmable camera architecture.** Spectral programming is a technique that is often used in imaging applications, such as Bayer filters for RGB cameras or narrowband spectral filters for fluorescence microscopy [3]. Static filters offer arbitrarily high spectral resolution, but are not tailored for applications that require changing filters rapidly; while this can be achieved with filter wheels, the speed of such devices is constrained by the speed at which the filters can be changed. Electronically tunable filters, in part can be achieved by using a tunable filter [13] where liquid crystal (LC) cells are used to obtain a combination of narrowband spectral filters. LC filters however are typically slow as they require large settling times.

The most practical way of implementing programmable spectral filters that can be changed electronically and at high speeds is to rely on the setup shown in Fig 1. Here, a dispersion element such as a grating or prism is used to create the so-called *rainbow plane* [6] where each point corresponds to intensity of a single wavelength of the whole scene. By placing a spatial modulator (SLM) on this plane, one can achieve arbitrary spectral programming. This approach is similar to replacing a sensor in a spectrometer with an SLM, and has been the *defacto* way of spectral programming in some of the past works [7,8].

The SLM-based approach for spectral programming has certain advantages. Since SLMs are fast, one can achieve high frame rates (often in excess of 60fps), which is crucial for imaging dynamic scenes, and in applications that require rapidly switching spectral filters. Two, the system is *potentially* capable of high spectral resolution without sacrificing capture time. However, as we will see next, a high spectral resolution leads to a severe loss in spatial resolution. The focus of this paper is on the fundamental trade-off of spectral and spatial resolutions.

**Time-frequency bandwidth product.** Our main result is based on the time-frequency bandwidth product [14–16], which we state here for completeness. Let $x(t)$ be a centered time-domain signal, and let $X(\nu )$ be its (centered) continuous-time Fourier transform. We define the spread of time-domain and frequency-domain signals as,

**Space-spectrum resolution tradeoff.** In order to understand the impact of the pupil code shape on spatial and spectral resolutions, let us consider the design of a spectrometer, which consists of a narrow opening, a dispersive element, and a sensing element. The spectral resolution of the measurements is a function of width of the opening slit; a narrower opening leads to high resolution, while a broad slit leads to blurred spectrum. Similarly, a programmable camera would also necessitate a narrow slit to ensure that spectrum can be modulated at high resolution. However, such a narrow slit leads to a severe loss in spatial resolution, since imaging at high resolution requires a large and open aperture. In [6], it is noted that a large slit leads to loss of spectral resolution, but they do not mention what happens to the spatial resolution. The authors in [7] identified this tradeoff and stated an approximate relationship between spectral and spatial tradeoff for fully open aperture and demonstrated that high spectral resolution lead to blurry images. We formalize the result and show that such a tradeoff applies to any pupil code shape and can be concisely stated as a space-spectrum bandwidth product. In the upcoming sections, we will formalize the spatial and spectral resolutions that result from the choice of a pupil code.

## 3. Fundamental limits of spatial/spectral resolution

We now derive a concise lower bound on product of spatial and spectral spreads due to a spectrally-programmable camera.

**Spectral and spatial blurs.** Let us revisit the optical setup in Fig. 1, where we placed a pupil code $a(x)$ in plane P2 and obtained the rainbow plan on P4 and image on P5. We wish to study the effect of $a(x)$ on the blur it introduces in spectral and spatial measurements. We present the expressions for spatial and spectral blurs here and refer the interested readers to Appendix A. for a detailed derivation. For brevity, we show the blur along $x$-axis alone, as there is no spectral dispersion along $y$-axis. Let $a(x)$ be the shape of the aperture function and let $A(u)$ be its Fourier transform. Without loss of generality, we assume that $a(x)$ and $A(u)$ are both centered such that

#### 3.1 The space-spectrum uncertainty principle

Our main result, stated in Theorem 1, suggests that the spatial and standard deviations are related by the inequality, $\sigma _x \sigma _\lambda \ge \frac {\lambda }{4\pi \nu _0}$. We now outline the proof of our theorem.

**Proof.** The spectral and spatial standard deviations are,

**Implication.** We make some observations about the uncertainty principle here.

- •
*Invariance to scaling.*The bandwidth product does not change even if the aperture is stretched or squeezed. If the aperture $a(x)$ is replaced by $a(sx)$, then spectral blur changes to $h_\lambda (\lambda ) = |a(-s\lambda f \nu _0)|^2$ and the spatial blur changes to $\left |A\left (-\frac {x}{s\lambda f}\right )\right |^2$. This changes the spectral and spatial variances to $s^2 \sigma ^2_\lambda$ and $\sigma ^2_x/s^2$, thereby keeping the product a constant. - •
*Invariance to power of lenses.*The bandwidth product is independent of focal length of the system, implying that one cannot expect any increase in product of standard deviations by changing the lenses. - •
*Dependence on groove density.*The bandwidth product inversely depends on the groove density $\nu _0$. In theory, one can achieve arbitrarily low space-bandwidth product by having high groove density, but the limiting factor becomes the aperture size of lenses. - •
*Dependence on wavelength.*The bandwidth product is directly proportional to wavelength. This is expected, as the limiting case of our statement, where $\sigma _\lambda$ is several hundreds of nanometers is just a normal grayscale imager, and in that case, the expression looks very similar to Abbe’s diffraction limit [20]. However, one may make the expression independent of wavelength by using lower bound of the spectral range,

**Achievability of lower bound.** As in the case of time-frequency uncertainty, there exists a pupil code function that has its space-spectrum bandwidth product *equal* to $\frac {\lambda }{4\pi \nu _0}$. This is achieved by the family of Gaussian windows:

## 4. Experiments

Armed with our theoretical insights, we next verify the results with some real experiments.

**Optical setup.** We built an optical setup shown in Fig. 3 with relevant components marked. The setup is a minor modification of the schematic shown in Fig. 1. We placed a spatial light modulator (SLM) on plane P2 which enabled display of various coded apertures, and a diffraction grating in plane P3. The spectral measurement camera is on plane P4. Instead of placing spatial camera on P5, we place it on P3 (using beamsplitter BS1). Since we do not code the rainbow plane P4, image on P3 and P5 are equivalent. Focal length of all our lenses was $75$ mm and the diffraction grating had $300$ grooves/mm. List of components can be found in appendix B.

**Visualization of spectral and spatial resolutions.** To illustrate our hypothesis, we placed a USAF resolution chart on the image plane P1. The scene was illuminated with a cool white compact fluorescent lamp (CFL) which is comprised of several narrow peaks. This setup enabled us to simultaneously visualize sharp spectrum as well as sharp spatial features. Figure 4 shows images and spectra for some representative cases. Each row shows results for a specific coded aperture, whereas each column shows results for a fixed spectral resolution. The trend of decreasing spectral resolution with increasing spatial resolution is clearly visible. Further, a Gaussian aperture is superior to slit in terms of greater spatial resolution for the same spectral resolution, which agrees with our theoretical findings.

**Quantitative verification.** We illuminated a pinhole with a spectrally-narrowband light source with a central wavelength of 670 nm and an FWHM of 3 nm. We then captured both spectrum of the light source and image of the pinhole, which we then used for computing the corresponding standard deviations. Figure 5 compares reciprocal of spatial resolution against spectral resolution. The two plots show a straight line, thereby verifying that the product of spatial and spectral resolutions is a constant. We also observe that the line for Gaussian aperture is very close to the theoretically optimal line, thereby confirming that the lower bound is tight, even in practice.

#### 4.1 Spectral programming

Next, we discuss the impact of our findings for various scenarios of spectral programming.

**Effect of edge-pass filter.** The tradeoff between spectral and spatial resolution affects how well the spectrum can be coded. To test this, we illuminate two closely spaced spatial points in a scene with two narrowband light sources (520 nm and 532 nm). We then attempt to block the 520 nm laser with various coded apertures and observe the spatial image. Figure 6 shows the schematic and our lab prototype for spectral programming, which is similar to the schematic shown in Fig. 1 with a spatial mask placed in plane P4. The results are shown in Fig. 7. With a broad aperture, it is not possible to effectively block one of the two lasers, shown in fourth column. A narrow aperture can lead to effective blocking (compare first and last columns) but with loss in resolution.

**Effect of the shape of a narrowband filter.** We now show that a slit has unintended implication, when used as a narrow band filter. In order to perform narrowband spectral programming, an intuitive choice is to place a narrow slit on the rainbow plane P4 and a camera on plane P5. This results in a spectra that looks similar to the example in Fig. 8(a). While such a mask works well for the target wavelength, the spatial images corresponding to adjacent wavelengths have severe loss in spatial resolution.

To understand the effect of a narrowband filter, consider a scene illuminated by a monochromatic light source of wavelength $\lambda _1$. The resulting field on rainbow plane,

Now let a spatial mask $\widehat {a}(x)$ centered around $\lambda _2$ be placed on the rainbow plane. Then the output just after the mask,## 5. Discussions and conclusion

We formalized the tradeoff between spectral and spatial resolution associated with a spectrally-programmable camera of the type shown in Fig. 1 and stated the space-spectrum uncertainty principle. We showed through theory, simulations and real experiments that one can finely resolve space or spectrum, but not both. Our analysis then showed that a Gaussian-shaped aperture achieves the theoretical lower bound, and that a Gaussian-shaped narrowband filter introduces a wavelength-independent spatial blur. We believe our findings impacts scientific imaging at large by providing insights and design guidelines for settings which rely on spectral programming.

**Application to fundamental limits of hyperspectral imaging.** We note that our analysis *does not* limit the spatial and spectral resolution of hyperspectral cameras. In case of cameras which utilize tunable filters, the spatial resolution is a function of the grayscale camera, while the spectral resolution is *independently* dictated by the tunable filter. For spatially-coded cameras such as pushbroom and CASSI, the sensing process does not rely on pupil coding, which does not produce diffractive blur. To understand this, consider a spatially coded camera that has only one opening in the spatial plane, $\delta (x-x_0, y-y_0)$. The measurement on the camera sensor after propagating through a dispersive element is,

## Appendix A Spatio-spectral blur due to pupil code

We note that all our analysis is for spatially-incoherent light; any phase component is hence irrelevant. We rely on Fourier transform property of a thin lens [19] as well as the derivation in [9]. Assume that the complex field distribution on plane P1 is $i_1(x, y, \lambda )$. Then the field distribution on P2 that is $2f$ away is given by the scaled Fourier transform relationship, $i_2(x_2, y_2, \lambda ) = \frac {1}{j\lambda f}I_1\left (\frac {x_2}{\lambda f}, \frac {y_2}{\lambda f}, \lambda \right )$, where $I_1(u, v)$ is the Fourier transform of $i_1(x, y)$. Propagating the signal through the optical setup simply requires us to perform such operations iteratively.

Consider a single spatial point on P1 of the form $i_1(x_1, y_1, \lambda ) = s(x_0, y_0, \lambda )\delta (x_1 - x_0, y_1 - y_0)$, where $s(x_0, y_0, \lambda )$ is the complex amplitude of the point as a function of wavelength. Any arbitrary image can then be treated as infinite such point sources. The amplitude distribution on plane P2 is the scaled Fourier transform of amplitude on plane P1 and is given by,

**Intensity measurements.** Consider cameras placed on planes P4 and P5 with a spectral response of $c(\lambda )$. The intensity measurement on P4,

**Spectral and spatial blurs.** For brevity and ease of understanding, we drop the $y$ axis as it does not affect the spectral blur. From (26) and (29), we get the following expressions for spectral and spatial blurs,

## A.1 Verification using simulations

We provide a validation of our theory with simulations. We specifically compared a box aperture that simulates a slit or fully open aperture, and a Gaussian aperture. For the purpose of exposition, we used $f=75$ mm and a diffraction grating of $300$ groves/mm. Figure 9(a) shows a plot of spatial and spectral standard deviations and (b) shows a plot of spatial and spectral modulation transfer function (MTF) at $30\%$ contrast ratio. The plots show a clear trade off between the two resolutions, independent of resolution metric.

We also observe that Gaussian codes achieve the theoretical limit for standard deviation. Hence we conclude that the space-spectrum bandwidth product is a tight bound.

## Appendix B List of components

Figure 10 shows list of components for the setup in Fig. 3, and Fig. 11 shows the list of components for the setup in Fig. 6.

## Funding

National Geospatial-Intelligence Agency's Academic Research Program (Award No. HM0476-17-1-2000); NSF CAREER grant (CCF-1652569); NSF Expeditions award (1730147); Prabhu and Poonam Goel graduate fellowship.

## Disclosures

The authors declare no conflicts of interest.

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