2.1.

Design and Fabrication of Hadamard Masks

Instead of using $M$ random masks one at a time, a single extended Hadamard mask, $H_e$, may be constructed that encompasses all possible Hadamard patterns as submatrices. The procedure to construct an extended Hadamard mask is as follows. A cyclic $S$-matrix of size $pq\times pq$, where $p$ and $q$ are prime numbers and $q=p+2$, is formed using the twin prime construction method by which a row of $pq$ elements of value 0 or 1 is produced by a set of rules on the $i$’th element based on the remainders of $i/p$ and $i/q$.¹⁷ Once the first row of $S$ is formed, the $S$-matrix can be completed by shifting the row by one element to the left and filling the missing rightmost element in a circular pattern. Taking $p=5$ and $q=7$ as an example, Fig. 2(a) gives the $S$-matrix in which black represents 0 and white 1. To form the extended Hadamard mask, the elements of the first row of the $S$-matrix are folded into a $p\times q$ matrix that is concatenated to the right and below to form a $(2p-1)\times (2q-1)$ matrix as shown in Fig. 2(b). From the extended Hadamard mask, one can randomly choose $pq$ Hadamard acquisition matrices $H_i$, $i=1$ to $pq$ by matching the top left corner of a template of size $p\times q$ to the location $(k,l)$ in $H_e$, where $i=(k-l)q+l$ corresponds to the $i$’th row of the $S$-matrix.

Fig. 2

The elements of cyclic $S$ matrix (a) and extended Hadamard matrix $H_e$ shown by the box (b) for $p=5$ and $q=7$. The dark shades represent 0 and the light shades 1. The matrix elements are indentified by numbers 1 to 35 in (b).

The design of mask size and pixel resolution is based on the following considerations:

We used $p=41$ and $q=43$ with a pixel size of 1.24 mm and produced an extended cyclic mask of $81\times 85$ pixels of size $10.04\times 10.53\mathrm{cm}$. Each $p\times q$ mask is of size $50.84\times 53.32\mathrm{mm}$. The extended Hadamard mask was fabricated using chrome coating on a millimeter-wave transparent quartz plate, as shown in Fig. 3. The colored boxes show two of the 1763 possible Hadamard patterns that can be created by exposing parts of the mask.

Fig. 3

An extended Hadamard mask of size $81\times 85$ pixel fabricated on a quartz plate with chrome coating. A $41\times 43$ pixel mask area such as the ones indicated by the colored boxes is exposed for each acquisition.

3.1.

Imager Setup

Figure 4 shows a two-lens passive millimeter-wave setup for CS implementation using Hadamard masks. The extended Hadamard mask is placed at the image plane of a six-in. lens, where the image of a distant target is formed. A metal plate (template) with a hole of size $p\times q$ is placed in front of the mask, which defines the exposure window. The extended mask is controlled by a two-axis translation stage to expose different mask patterns for compressive data collection, one for each measurement. A second lens of 1-inch diameter collects the modulated radiation field through the Hadamard mask and focuses it onto the multichannel radiometer. The positions of the lenses and the mask are governed by the lens equation: $1/f_r=1/d_i+1/d_o$, where $f_r$ is the focal length of either of the lenses, and $d_i$ and $d_o$ are the image and object distances, respectively. Because the targets used in these experiments do not have spectral features, we averaged all 16 spectral channels, which offers an increase of SNR by a factor of four.

Fig. 4

Compressive sensing setup for passive MMW imaging.

3.2.

SNR Analysis of CS Imager Setup

It is important to compare the SNR between the raster scanning and CS systems to ensure that the reduction in imaging time obtained with the latter does not come at the expense of the radiometer sensitivity. A reduction in sensitivity by a factor $x$ in the CS setup is equivalent to a reduction in the integration time by a factor of $1/x^2$ in the raster scanning setup, as the sensitivity of a Dicke-switched radiometer is given by: $\mathrm{\Delta }\mathrm{T}=2T_N/\sqrt{B\tau }$, where $T_N$ is the noise temperature of the receiver, $B$ is the predetection bandwidth, and $\tau$ is the postdetection integration time. For example, if the CS setup offers a reduction in imaging time by a factor of 10 at a cost of 3.16 times less in sensitivity ($3.16\mathrm{\Delta }T$), the same saving in imaging time can be obtained with conventional imaging by reducing the integration time by ten times for an equivalent sensitivity of $3.3\mathrm{\Delta }T$. However, if the raster scanning system uses a stop-and-go implementation for data collection at each pixel, the total acquisition time including the translation time of the raster scanning system can be substantially longer than that of the CS system.

Figure 5 shows ray tracing diagrams of raster-scanned and CS-based signal acquisitions. While the raster scanning system collects the radiation from each pixel of the target, the CS system in principle sums up the radiation from $\sim N/2\mathrm{pixels}$ (as approximately half of the pixels in the mask are open) of the target. If there were to be no loss due to the mask or in the radiation collection from the target to the radiometer, the SNR of the CS system would be $\sqrt{N/2}$ times the raster scanned system as the noise is averaged while the signal related to the target temperature remains the same. In practice, however, radiation losses occur due to (a) diffraction effects from the subwavelength-sized mask apertures and (b) inefficient radiation collection by the finite-size lenses. The diffraction loss due to the mask may be analyzed as follows. Each aperture in the mask can be treated as a dipole antenna¹⁸ radiating over a spherical area and only part of this radiation is collected by the 1-in. diameter lens. The actual intensity of each pixel collected by the radiometer can be expressed as: ${\tilde{I}}_{\text{pixel}}=w{I}_{\text{pixel}}$, where $I_{\text{pixel}}$ is the intensity of a single pixel, and $w$ is the fraction of the intensity collected by the radiometer. For the experimental setup in Fig. 4, $w$ is the ratio of the area of the focusing lens and that of the spherical surface at a distance $d_o$. Accordingly, $w=A_{\text{lens}}/4\pi d_o^2=4\pi r_{\text{lens}}^2/4\pi d_o^2={(r_{\text{lens}}/d_o)}^2$. For $r_{\text{lens}}=1.27\mathrm{cm}$ and $d_o=12.7\mathrm{cm}$, $w={(r_{\text{lens}}/d_o)}^2\approx 1\%$ and ${\tilde{I}}_{\text{pixel}}=w{I}_{\text{pixel}}\approx 1\%\times I_{\text{pixel}}$. The total intensity from all pixels collected by the radiometer is thus given by $\tilde{I}=0.5wpq{\tilde{I}}_{\text{pixel}}\approx 0.5\%\times p(p+2)\times I_{\text{pixel}}$, where we have assumed uniform intensity from all the pixels. For $p=41$, we get $\tilde{I}\approx 0.5\%\times p(p+2)\times I_{\text{pixel}}\approx 9I_{\text{pixel}}$. Hence, the corresponding improvement in SNR is estimated as three times that of the raster scanned imagers. The two-lens setup we have used is not an optimal design for radiation collection since a large portion of the radiation from the target does not reach the radiometer. With better collection efficiency and higher pixel count $N$, the SNR of the CS system is expected to be higher than that of the raster-scanned setup. Furthermore, there is a tradeoff between pixel size (image resolution) and SNR; with a larger aperture (lower resolution), the diffraction losses will be less, resulting in a higher $w$ and, in turn, a higher SNR.

Fig. 5

Signal acquisition by (a) raster scanned camera and (b) CS-based single-pixel camera.

4.1.

Image Reconstruction from Full Hadamard Acquisitions

If a complete set of $N$ acquisitions is obtained by fully raster scanning the extended Hadamard mask, the image reconstruction consists of simple matrix manipulations as given below. The measured intensity vector ${\mathit{I}}_m$ using Hadamard masks may be expressed as¹⁹

For a given Hadamard mask size, the $\mathit{S}$ matrix in Eq. (2) can be predetermined from the Hadamard sequence, so the image reconstruction is very fast.

To test image formation and reconstruction, we used a single lens imaging setup as in Fig. 6, with a light bulb (thermal light source) illuminating an object and the Hadamard mask placed behind the object in close proximity. The relative sizes of the object (at the mask) and images for the single-lens setup may be determined using the ray tracing diagram in Fig. 6. According to the lens equation, the object and image distances $d_0$ and $d_i$ are given by $1/d_o+1/d_i=1/f$, where $f$ is the focal length. With $d_o=12.7\mathrm{cm}$ and $f=2.54\mathrm{cm}$, $d_i=3.175\mathrm{cm}$. The magnification is given by $M\equiv b/a=d_i/d_o=1/4$, where $b$ and $a$ are the object and image sizes, respectively. For an antenna of radius $r_{\text{antenna}}=0.3175\mathrm{cm}$, the field of view (FOV) is $\mathrm{FOV}\equiv 2a=2b/M=2r_{\text{antenna}}/M=8r_{\text{antenna}}=2.54\mathrm{cm}$. The imaging setup we used provides only 2.54 cm FOV; as a result, we covered the imaging area for this setup with a metal plate having a 2.54 cm diameter hole. In order to expand the FOV to the full extent so that it covers the size of the Hadamard template ($50.84\times 53.32\mathrm{mm}$), the distances $d_0$ and $d_1$ may be changed to 22.86 and 2.8575 cm, respectively, keeping the lens diameter the same.

Fig. 6

Proof-of-principle test with a thermal light source illuminating an object consisting of a 2.54 cm diameter circular hole with a 3-mm wide rectangular metal strip.

We first simulated the Hadamard transform and image reconstruction process for an object geometry consisting of a metal plate with a circular hole of diameter 2.54 cm and a 3-mm wide rectangular metal strip across the middle. Figure 7(a) gives the binary coded image of the object with 1 representing the hole and 0 the metal portion, and Fig. 7(b) is its Hadamard transformed image according to Eq. (1) after folding the $pq\times 1$ vector into a $p\times q$ matrix. The reconstructed image from Hadamard transformed data using Eq. (2) was exact and identical to Fig. 8(a) as there was no measurement noise for this ideal case.

Fig. 7

Simulation of target geometry: (a) digitized image of circular hole with a strip in the middle and (b) simulated Hadamard transformed image.

Fig. 8

Experimental data: (a) Hadamard-transformed image and (b) reconstructed image.

We next obtained a full set of $pq=1763$ Hadamard acquisitions by raster scanning the Hadamard mask. Figure 8(a) gives the Hadamard transformed image, and Fig. 8(b) shows the reconstructed image using Eq. (2). An excellent agreement is seen between the simulated [Fig. 7(b)] and experimental [Fig. 8(a)] Hadamard transformed images. The reconstructed image of a circular hole with a horizontal metal strip shows the feasibility of Hadamard imaging at millimeter wavelengths with subwavelength resolution (1.24 mm pixel size for 2 mm wavelength).

4.2.

Image Reconstruction from Partial Hadamard Acquisitions

Instead of fully scanning the extended Hadamard mask, one may sample the mask randomly or sequentially every $n$’th pixel in the horizontal and vertical directions. Figure 9 gives a flowchart of data acquisition and image reconstruction steps as the data acquisition proceeds. The measured data with the Hadamard matrices are in the Hadamard transform space. If the Hadamard transform space is complete with all $pq$ acquisitions, the reconstruction is simply multiplication of the $S^{-1}$ matrix with the unwrapped Hadamard transform data as presented before. To reconstruct from an incomplete data set in the Hadamard transform space (compressive sampling), we introduce an iterative method of estimating the missing elements in the Hadamard transform space: the relaxation method, used in the numerical electrodynamics field.²⁰

Fig. 9

Flowchart of Hadamard matrix acquisition and image reconstruction from compressively sampled data.

The relaxation method is based on iteratively estimating the two-dimensional (2-D) functional value $\langle \langle F(x,y)\rangle \rangle$ at the coordinate ($x,y$) from the nearest neighbors (straight) and next nearest neighbors (cross),

The iterative procedure consists of: 1. Enter the known values, 2. guess the missing elements, 3. apply the filter B and estimate the average value for ($x,y$), 4. reassert the known values and iterate on step 3 until desired convergence is reached. If the function is well behaved, convergence can be shown by a Taylor series analysis.²⁰

To test the relaxation technique, we sampled every third column and third row of the Hadamard space, providing $1/9$ of the full acquisitions. Figure 10(a) gives the recovered image in the Hadamard space, and Fig. 10(b) shows the reconstructed image of the object. The recovered Hadamard transformed image [Fig. 10(a)] from partial data compares well with the full Hadamard transformed image in Fig. 8(a). The reconstructed image of the object clearly shows the circular hole with a strip in the middle; however, the geometry looks somewhat distorted around sharp edges, which gets improved with additional samples.

Fig. 10

Reconstructed image from $1/9$th of samples: (a) relaxation method-based reconstruction of Hadamard space and (b) reconstructed image.

4.3.

Progressive Compressive Sensing and Real-Time Image Reconstruction

We developed a progressive sampling and image reconstruction method in which the Hadamard acquisition starts at every $n$’th row and $n$’th column in the Hadamard space. The relaxation technique is applied to fill the Hadamard space from which the image is reconstructed after every sample in real time by Eq. (2), since the $S$-matrix is predetermined for a given $p$ and $q$. If the image is not satisfactory, we continue sampling the Hadamard space in between the sampled points and reconstruct with ($2N/n$) data, and so on. The sample space is progressively increased until satisfactory image quality is obtained. The complete image acquisition and reconstruction software is implemented in LabVIEW.

The reconstructed image quality using the progressive sampling method was compared against the Bayesian random sampling method that we had developed in the past.²¹ A normalized mean squared error (NMSE) metric was adopted for comparison of reconstructed images from partial and full sets of samples. $\mathrm{NMSE}={\mathrm{\Sigma }}_{i=1}^N{(I_p(i)-I_f(i))}^2/{\mathrm{\Sigma }}_{i=1}^N{(I_f(i))}^2$, where $I_p(i)$ and $I_f(i)$ are the intensities of the $i$’th pixel corresponding to the reconstructed images from partial and full acquisitions, respectively.

Figure 11 provides the comparison of NMSE versus the percent completion of the full acquisitions for the case of (a) relaxation method with random Hadamard patterns, (b) progressive sampling after each acquisition starting at $1/64$ of full samples, (c) progressive sampling after each complete cycle in the Hadamard space, and (d) Bayesian reconstruction from random Hadamard acquisitions. The reconstructed image obtained after every complete cycle (e.g., $1/32,1/16,\dots ,1$) showed comparable or better performance than the random sampling methods. In addition, the computational time of the Hadamard transform-based reconstruction is significantly less than that of the conventional nonlinear minimization algorithms used in traditional CS reconstruction methods.

Fig. 11

Comparison of normalized mean square error (MSE) for progressive sampling and conventional random sampling methods.

4.4.

Imaging with Two-Lens Setup

The single lens imager setup shown in Fig. 6 was used for proof-of-principle testing of compressive sensing. To extend it to a full imaging system, we used a two-lens CS setup as shown in Fig. 4. To ensure high thermal contrast under indoors, a 60 W incandescent lamp (thermal source) was used as the target to be imaged. In outdoor conditions, however, such an artificial hot source is not needed as the cold sky reflected radiation would offer excellent thermal contrast. Figure 12 gives the reconstructed image of the lamp with (a) full and (b) 11% samples, and Fig. 13 gives that with one quarter of the lamp blocked by a metal plate with (a) full and (b) 11% samples. The reconstructed images with 11% samples compare well with those from full samples; however, there is a slight distortion in the object geometry due to the smoothing nature of the algorithm around sharp edges.

Fig. 12

Reconstructed image of lamp with (a) full and (b) 11% samples.

Fig. 13

Reconstructed image of lamp with one quarter of it blocked by a metal plate with (a) full and (b) 11% samples.