Analysis of Keller Cones for RF Imaging
Anurag Pallaprolu, Belal Korany and Yasamin Mostofi
UC Santa Barbara, California, USA, 93106

2023 IEEE Radar Conference (RadarConf23) | 978-1-6654-3669-4/23/$31.00 ©2023 IEEE | DOI: 10.1109/RADARCONF2351548.2023.10149785

apallaprolu@ucsb.edu, {belalkorany, ymostofi}@ece.ucsb.edu
Abstract—Imaging still objects with the received signal powerof off-the-shelf WiFi transceivers is considerably challenging. Theinteraction of object edges with the incoming wave, dictatedby the Geometrical Theory of Diffraction and the resulting
Keller cones, presents new possibilities for imaging with WiFi
via edge tracing. In this paper, we are interested in bringing acomprehensive understanding to the impact of several different
parameters on the Keller cones and the corresponding edgebased imaging, thereby developing a foundation for a methodical
imaging system design. More specifically, we consider the impactof parameters such as curvature of a soft edge, edge orientation,
distance to the receiver grid, transmitter location, and otherparameters on edge-based WiFi imaging, via both analysis andextensive experimentation. We finally show that Keller cones canbe used for imaging objects that lack visibly-sharp edges, aslong as the curvature of the edge is small enough, by imaging a
number of such daily objects.

I. I NTRODUCTION
There has been a considerable interest in recent years
in using WiFi signals for sensing and learning about theenvironment. WiFi signals, for instance, have been used for localization, tracking, person identification, occupancy analytic,
and smart health, among other applications. There has also
been work on using WiFi signals for imaging still objects.
However, imaging still objects with WiFi signals has been considerably challenging due to the lack of motion (as comparedto the aforementioned applications). In fact, work on imaging
still objects either focused on using expensive customized
mmWave radar equipment [1], or focused on imaging basic
scenes with WiFi [2]. Other work has also attempted machinelearning for this purpose but with limited generalizability [3].
In a recent work [4], we proposed a different way of
thinking about the challenging problem of imaging objects
with WiFi, by focusing on how the edges of an object interact
with the incoming wave. More specifically, when a wave isincident on an edge, it results in a cone of outgoing rays,
dictated by the Keller’s Geometrical Theory of Diffraction(GTD) [5]. Thus, the scattered WiFi signals off of objects
carry much more information about the edges than the surface
points, since edges diffract the incoming wave into many
directions (i.e., the diffraction cone), while surface points canappear quasi-specular (near mirror-like) at WiFi frequencies.
In [4], we then proposed Wiffract, a new approach toimaging still objects by edge tracing, which uses the Keller
cones and the corresponding conic sections to infer the edge
angles, via building proper projection kernels. This approachproduced promising results, considerably better than whenattempting to image object surfaces with WiFi, showing its
potential for imaging complex objects. The main focus of[4] was to develop the underlying theories and show theThis work was funded in part by ONR award N00014-20-1-2779, in partby NSF award 2215646 and in part by NSF RI award 2008449.

Imag

ing p
lane
at

RX
G

rid

Dz

TX at pt
Dx

RX at prFig. 1. Imaging setup: a 2D RX grid (at y=0 plane) captures wireless signals
emitted by a TX located at pt = [r cos θ, 0, r sin θ]. The TX illuminates
objects in object space Ψ, a 2D plane parallel to the RX (at y = yI ).

performance, mainly for objects with visibly-sharp edges.
However, many objects that do not have such sharp edges still
have edges with small curvatures. We then have the following
question: If the curvature is small enough as compared to thewavelength, can the edge effectively appear as a sharp edge tothe incoming wave, producing a similar Keller cone that canbe exploited for imaging? Moreover, several other parameterssuch as edge orientation, distance to the RX grid, TX location,and size of the RX grid affect the exiting Keller cones. In
this paper, we then present a comprehensive analysis of the
impact of several key parameters on edge-based WiFi imaging,and further support it by extensive experimentation. We finallyshow the performance of edge-based imaging when imaging
daily objects that lack visibly-sharp edges. The analysis andexperimentation can provide a foundation for designing edgebased imaging systems, thus motivating future work in thisarea.
II. P ROBLEM F ORMULATION
Consider an imaging system, shown in Fig. 1. A TX locatedat pt = [r cos θ, 0, r sin θ] illuminates the area. The aperture
RX is a two dimensional grid of dimensions Dx × Dz ,located at y = 0 plane, composed of uniformly-spaced receiverantennas (indexed by pr ) that capture radiation scattered off ofobjects in its surroundings. Each antenna is an off-the-shelfdevice that can only measure the received signal powerreflected by its surroundings. We are interested in imaging theobjects in Ψ, which is a 2D plane at y = yI , parallel to the RX
grid. We denote the actual object locations by po ∈ Θ ⊂ Ψ.
Using Born approximation [6], the power of the received signal
at pr (after subtracting the direct path from the TX to the RX)
can be approximated as follows [4]:XP (pr ) ≈ 2R
Λ(po , pt , pr )g ∗ (pt , pr )g(po , pr ) , (1)


po ∈Θ

where Λ(po , pt , pr ) = α(pt , po )g(pt , po )α(po , pr )α∗ (pt , pr ),Authorized licensed use limited to: Univ of Calif Santa Barbara. Downloaded on October 04,2023 at 20:59:31 UTC from IEEE Xplore. Restrictions apply.

R{.} is the real part operation, g(pi , pj ) is the Green’s
2πfunction for the path from pi to pj given by e−j λ (||pi −pj ||) ,λ is the wavelength, ||.|| is the norm and α(pi , pj ) is thecorresponding amplitude attenuation factor of the path.
The traditional approach of backward ray-tracing [7] computes the reconstructed image intensity at pm ∈ Ψ via thefollowing convolution sum over all the RX points
X
I(pm ; RX) =P (pr )g(pt , pr )g ∗ (pm , pr )pr ∈RX= |IS (pm ) + IN (pm )|,(2)
P
where IS (pm ) = pr ∈RX Λ(pm , pt , pr ) contains information
the presence of an object at pm , while IN (pm ) =P aboutP∗po′ ∈Θ Λ(po′ , pt , pr )g(po′ , pr )g (pm , pr ) is anpr ∈RXpo′ ̸=pm

interference term from objects elsewhere [4].
Definition of an RX Group: Define the RX group for an
object at location pm , RXpm ⊂ RX, as the set of all thereceiver antennas that capture the radiation scattered off of an
object placed at pm .
Problem of Surface Specularity at Lower Frequencies:
An underlying assumption for successful imaging via Eq. 2 isthat each object point within the imaging plane can scatter thetransmitted signal to the whole RX grid (i.e., has a large RX
group). In order for a point pm on the object’s surface to reflect
the incident signal to several of the antenna array elements,
the surface at pm needs to appear diffuse (i.e., rough) to theincoming wave. While this is true at very high frequencies(such as mmWave), at lower frequencies (such as WiFi) the
perturbations on the surface of an object are typically small
when compared to the wavelength. As such, the same surface
can appear much smoother and specular to the incoming wave
[1]. [1] has also established that many everyday surfaces canappear near-specular (i.e. almost mirror-like) to WiFi signals.
This then would result in a very small RX group for a surface
point (|RXpm | << |RX|, |.| denoting size of set). Thus,
relying on imaging surfaces via Eq. 2 at WiFi frequencies
will result in poor performance, as shown in the literature [4].
Edges of an object, on the other hand, interact differently
with the incoming wave. More specifically, when a wave isincident on an edge point, a cone of outgoing rays emerges
according to the Keller’s Geometrical Theory of Diffraction(GTD) [5]. This has then become the basis for a new WiFibased imaging method, via edge tracing, that we proposed in a
recent work [4]. We next briefly summarize Keller cones andhow they can be exploited for tracing edges. This then lays thefoundation for the rest of the paper, which is on understanding
the impact of many different parameters on such an imaging
system via both analysis and experimental validations.
III. K ELLER C ONES : T HE I NTERACTION OF W IRELESSS IGNALS WITH E DGES
Define an edge point as a point on an object where thedirection of the surface normal has a discontinuity. We denotethe set of edge points by E ∈ Θ, while all non-edge points
are indicated by M = Θ − E. The Geometrical Theory ofDiffraction (GTD) [5], [8] describes the interaction of an

(a)

(b)

Edge
point

Mirrorpoint
Incident
ray

Keller
cone

RX
G

RX

(c)

Conic
sectionKeller
cone
rid

Conic
sectionGridIncident
ray
RX
G

Edge
point

rid

Incidentray

Fig. 2. (a)-(b) Diffracted waves (Keller cones) off of edges with different
orientations, which intersect the RX grid in a conic section. (c) Reflected
waves off of a surface that appears near-specular (mirror like) to the wave.

electromagnetic wave incident on an edge point from a rayoptics point of view. As shown in Fig. 2 (a)-(b), when aray is incident on an edge point, the outgoing rays form acone, known as the Keller cone, with the edge as its axis
and the cone angle equal to the angle made by the incident
ray with the unit vector along the edge. For a point ϵ ∈ E,
RXϵ is subsequently given by the intersection of the Keller
cone with the RX grid, which is technically known as a conic
section, whose structure is dependent on the location of theedge point and the orientation of the edge. On the other hand,a point µ ∈ M will act near specular at WiFi frequencies,
as shown in Fig. 2 (c), resulting in a much smaller RX group(|RXµ | << |RXϵ |). See [5], [4] for more details on GTD and
Keller cones, as well as other examples of conic sections.
Edge Imaging with Wiffract: In recent work [4], we
proposed Wiffract, a WiFi-based imaging system that exploits
GTD and the resulting Keller cones to image the edges of theobjects, using only received power measurements of Eq. 1.
We here briefly summarize this approach, which then lays thefoundation for our analysis of the next sections. Readers arereferred to [4] for more details.
Interaction of the incoming waves with the edges not only
results in a bigger RX group, but also allows us to infer theedge orientation, creating the possibility of tracing the edges,
as we show next. Let hypothesis Hϕi ,pm denote the existenceof an edge at pm making an angle ϕi with the +ve x-axis. Letϕi ∈ Φ, which is a discrete set of uniformly-distributed edge
orientations in [0, π). We then define RXpm (ϕi ) as the RX
group resulting from the Keller conic section corresponding
to the edge orientation hypothesis Hϕi ,pm . We then have thefollowing edge image for pm , under hypothesis Hϕi ,pm :X
I(pm , Hϕi ,pm ) =P (pr )g(pt , pr )g ∗ (pm , pr ) .
pr ∈RXpm (ϕi )In other words, we perform the convolution only over the
RX points that carry the information about the edge hypothesis
ϕi , which effectively captures the signal term |IS |, while
rejecting the interference from the rest of the RX grid. Wethen select the hypothesis that yields the largest reconstructed
intensity I(pm ; Hϕ⋆ ,pm ) and set this as the algorithm’s confidence of the presence of an edge with orientation ϕ⋆ at pm
where ϕ⋆ (pm ) = arg maxϕi ∈Φ I(pm , Hϕi ,pm ).
Once locations with a relatively high confidence of having
an edge are identified, their inferred angles are propagated to
the rest of the locations using Bayesian information propa-

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gation. Due to space constraints, we refer the reader to [4]for more details. In this paper, we are then interested inunderstanding the impact of several different parameters on theperformance of WiFi-based edge tracing. We note that since
the analysis/experiments of Sections V and VI are focusedon a single edge, we skip the step of belief propagation forthese results. We then utilize the full Wiffract pipeline when
imaging the objects of Section VII.
IV. E XPERIMENTAL S ETUP
We next describe the technical details of our experimental
setup, shown in Fig. 3, before presenting the key results ofthe paper. We use the onboard antenna ports of two laptops(i.e., 3 antennas per WiFi card of each laptop), to emulate
an RX array for imaging. More specifically, the RX gridis synthesized by moving an array of six vertically-oriented
omni-directional antennas housed in a wooden carriage. The
carriage is horizontally translated with a NEMA 23 belt drive
system to scan the x dimension of the grid and verticallytranslated with an independent NEMA 17 motor to achieve
z axis synthesis. We have designed this motorized setup sothat the RX grid can be quickly and efficiently generated with
only two laptops, with a runtime of 2 minutes to constructa single grid. More specifically, a grid of 42 × 98 receivers
is generated by alternating each of the 7 horizontal scans (6rows per scan), with a vertical separation of 3 cm and ahorizontal separation of 1.4 cm. We use off-the-shelf halfspace panel antennas for our WiFi transmitter and log the
WiFi CSI (Channel State Information) power measurements
off of Intel 5300 NICs of the laptops using the CSI-Tool[9]. For the imaging results demonstrated henceforth, unlessotherwise stated, the default dimensions for the RX grid are
Dx = 1.36 m and Dz = 1.12 m, the TX location ispt = [3 m, 0, 0], and yI = 1 m. The imaging plane in all
the experiments is 1.4 m × 1.4 m, centered around the point[0, yI , 0] (See Fig. 1). For all the experimental results, weuse background subtraction, as used in [4]. Furthermore, sinceother work (e.g., [4] and [2]) already established through-wallsensing capabilities of WiFi, in this paper we do not focus
on imaging through walls but we note that the results can beextended to through wall imaging.
V. W HAT CONSTITUTES AN EDGE TO A WAVE ?
As discussed in Sec. III, when an electromagnetic wave isincident on a visibly-sharp edge, it generates a Keller cone ofdiffracted rays. This then raises the following question: If thecurvature of an edge is small enough with respect to thewavelength of the incident wave, would it appear as anedge to the wave? A survey of the electromagnetic theory
literature indicates that the GTD can be extended to roundededges [10]. For instance, in [10], it is concluded that an areathat does not visually look like a sharp edge can still appear as
one to the incoming wave if the radius of its curvature is lessthan half of the wavelength. This is promising when imaging
using edge tracing as it implies that we not only can image
objects that have visibly-sharp edges, but can also image alarger category of objects that have edges with small enoughFig. 3. Experimental setup – WiFi antennas of two laptops are used to emulate
an automated 2D RX array, using motors for horizontal and vertical scan.

curvatures. We next show the implication of this for our edgebased imaging system by methodically changing the curvatureof an edge, while also using two different wavelengths.
Experimental Validation – Impact of CurvatureWavelength Relationship on Keller-Cone Based Imaging:Consider an incident wave with wavelength λ, and a roundededge with a radius of curvature rc . We change the radius
methodically, from 0 cm (sharp edge) to 6 cm, as shown in
Fig. 4 (top row). For each case, we image the edge, using the
GTD-based approach of Wiffract as discussed in Sec. III, andthe setup of Sec. IV, using WiFi at both 5 GHz and 2.4 GHz.
The second row shows the imaging results for 5 GHz (subchannel at 5.32 GHz), for which the theoretically-predicted
limit for the soft edge to appear as a sharp one to the wave is
2.8 cm. As can be seen, the edge can be visibly seen in theimages up to 3 cm very well. Even for 4 cm, the edge is still
well imaged. However, the imaging quality degrades after thatas expected. This is consistent with the theories of GTD as thethreshold of λ2 is not discussed as a binary threshold but rather
as a gradual one where the edge impact is starting to diminish
afterwards [10]. The third row then shows the imaging results
at 2.4 GHz. We expect to lose resolution when reducing thefrequency from 5 GHz, as can be seen from the leftmost figurethat images the sharp edge. However, we can see that theimaging results are pretty similar for all the radii from 0 to
6 cm, since they are all below the cutoff of λ2 = 6.2 cm.
Overall, the results confirm that curvatures with small enoughradii, as compared to the wavelength, can also act as sharpedges when interacting with the incoming waves, resulting in
Keller cones that can then be exploited for imaging, as is donein Wiffract. In Sec. VII, we show how we can utilize this andimage a number of daily objects that do not have sharp edges
but have small enough curvatures. In the rest of the paper, we
use WiFi at 5GHz for imaging.
VI. E DGE V ISIBILITY A NALYSIS
Several parameters can affect the exiting Keller cone of anedge. Sample important ones include location and orientationof an edge, distance of the edge to the RX grid, location of
the transmitter, and size of the RX grid. These parameters
determine if the resulting Keller cone intersects with the RX

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Fig. 4. Experimental Results on the Impact of Curvature: (Top) Edges with different radii of curvature, (middle) Wiffract’s edge imaging results with 5 GHz
WiFi, and (bottom) Wiffract’s edge imaging results with 2.4 GHz WiFi. Quality of edge imaging starts to degrade when the radius of curvature exceeds λ/2(See the color PDF for best viewing).
0.7

100

(b)
Coverage (%)

z (m)

80

0

100

0 degree edges
45 degree edges
90 degree edges
135 degree edges

Coverage (%)

(a)

60
40
20

-0.7

-0.7

0

x (m)

0.7

0

1

Depth

2
(m)

3

Fig. 5. (a) Visibility (light) and blind (dark) regions for a vertical edge in an
imaging plane at yI = 0.6 m, with a TX at pt = [3, 0, 0]. (b) Impact of
depth (yI ) and edge orientation on the coverage, for the same imaging setup.

grid, and if so, the extent of the intersection (i.e., the number ofRX antennas affected by it). In this section, we first analyze theimpact of these parameters. We then experimentally validate
the observations with several experiments. Such a study canthen result in clear guidelines when designing the imaging
system, as we shall discuss.
We start by defining the blind region of Ψ. We say a locationin Ψ is in a blind region if the resulting Keller cone of an edge
at that location does not reach the RX grid. Similarly, the edge
is said to be in a region of visibility if its exiting Keller cone
has a non-zero overlap with the RX grid. We next show the
impact of edge orientation, the distance between the RX gridand the imaging plane, the RX grid size, and the TX location
on edge visibility.
Fig. 5 (a) shows the visibility region for a vertical edge in
yI = 0.6 m, and with the TX location at pt = [3 m, 0, 0].
We note that for the analysis of this section, the edge heightis 30 cm, small enough so it only occupies a very small
number of pixels in the imaging plane. Finally, the grid size
is Dx = 1.36 m and Dz = 1.12 m, unless stated otherwise.
The light gray part of the imaging plane denotes the visible
region, while the darker region is the blind region. As can beseen, the location of the edge can highly affect its visibility.
We next define the term coverage, as the percentage of theimaging plane that is visible. For instance, in the example ofFig. 5 (a), the coverage is 67.1%. We next use coverage asa metric to show the impact of changing system parameters.

(c)
0 degree edges
45 degree edges
90 degree edges
135 degree edges

50

0
0

(b)

(a)

0

1

2

Depth (m)

3 0

1

2

Depth (m)

30

1

2

3

Depth (m)Fig. 6. Coverage as a function of the depth, yI , for different TX locations at
r = 3 m and (a) θ = 45◦ , (b) θ = 90◦ , and (c) θ = 135◦ , and for different
edge orientations.

More specifically, Fig. 5 (b) shows the coverage as a functionof the depth (yI ), for different edge orientations, while the restof the setup stays the same. It can be seen that the coverage
decreases as the imaging plane is moved away from the RX
grid, as it is more likely for the Keller cone of the edge tomiss the grid when the edge is farther away. Moreover, thefigure shows that the coverage depends on the orientation of
the edge: a TX placed at pt = [3 m, 0, 0] illuminates verticaledges the best, while minimally illuminating horizontal edges.
We next move the TX on a semi-circle of radius r = 3 maround the RX, as shown in Fig. 1. Fig. 6 shows the coveragewhen the TX is placed at angles θ = 45◦ , 90◦ , and 135◦ on
this semi-circle, as a function of the depth yI , and for different
edge orientations. It can be seen that different TX locations
illuminate the edges of different orientations differently. For
instance, the TX at θ = 45◦ illuminates the edges of 135◦ thebest, with minimal coverage for edges at 45◦ , and a reasonable
coverage for horizonal and vertical edges. On the other hand,
the TX at θ = 90◦ illuminates the horizontal edges the best,with a minimal coverage for vertical edges, and a reasonable
coverage for edges at 45◦ and 135◦ . In general, we can seethat a TX at an angle θ best illuminates edges which make anangle θ + 90◦ with the +ve x-axis in the imaging plane. This
gives rise to a simple recipe for the number and locationsof the needed transmitters, to have a high enough coveragefor most-likely edge orientations. For instance, a survey ofdifferent objects reveal that most common edges are horizontal
and vertical, which suggests that two transmitters at θ = 0◦

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0.7

Depth = 0.6 m

Depth = 1.2 m

Depth = 1.7 m

Depth = 2 m

0.7

coverage = 67%

coverage = 59.8%

coverage = 54.5%

coverage = 50%

-0.7
0.7

0

0

-0.7
0.7

-0.7
0.7

0

0

-0.7
-0.7

1x1 m grid

0.75x0.75 m gridcoverage = 57.5%

coverage = 42.8%

0

0

-0.7
0.7

1.36x1.12 m grid00.7 -0.7

00.7 -0.7

00.7 -0.7

0

0.7

-0.7
-0.7

coverage = 67%

00.7 -0.7

00.7 -0.7

0

0.7

Fig. 7. (Top) Theoretically visible (bright) and blind (dark) regions fordifferent depths, and for two different edge locations. Experimental Validations – (Middle) Wiffract’s imaging results when placing the top edgeat different depths. Imaging quality degrades as depth increases. (Bottom)
Wiffract’s imaging results when placing the bottom edge at different depths.
The imaged edge remains visible even as depth increases. Experimental results
are consistent with the top row analysis (See color PDF for best viewing).

Fig. 8. (Top) Theoretically visible (bright) and blind (dark) regions fordifferent RX grid sizes, and for two different edge locations. Experimental
Validations – (Middle) Wiffract’s imaging results for the top edge withdifferent RX sizes. Imaging quality degrades for the smaller grid sizes as
the edge moves to the blind region. (Bottom) Wiffract’s imaging results forthe bottom edge with different RX sizes. The imaged edge remains visible
for all RX sizes but the quality degrades as the RX size decreases (See color
PDF for best viewing).

and 90◦ will be important for proper illumination.
Experimental Validation: We next experimentally verify
the aforementioned analysis by showing the impact of key
parameters on edge-based imaging quality.
1) Distance between the edge and the RX grid: Fig. 7
shows the imaging results of a series of experiments in which
we place a 30-cm sharp vertical edge at various depths of
yI = 0.6, 1.2, 1.7, and 2 m, and at two different locations. The
top row shows the theoretical visibility regions (bright color)for each distance, as well as the location of the two edges. The
middle row then shows the imaging results when placing theedge at the top location. As can be seen, the edge is imaged
well when it is closer to the RX grid but the quality degrades as
the edge is moved farther away. This is consistent with the top
row analysis as the top edge starts to move to the blind region
as the distance increases. On the other hand, for the bottom
location placement of the edge, the top row indicates that theedge should remain in the visible region even as the distanceincreases in this case. The Wiffract imaging experiments ofthe bottom row then confirm this. More specifically, we cansee that the edge is imaged well even for farther distances.
Note that, due to path loss, placing the edge farther from thegrid in general results in a weaker measured diffracted signal,
which slightly reduces the intensity of the image.
2) Impact of the RX grid size: Fig. 8 next shows the impactof the RX grid size on the imaging performance for a verticaledge placed at two different locations that are indicated in the
top figure. For these results, the depth is fixed at yI = 1 m andthe transmitter is at pt = [3 m, 0, 0]. From analysis, we expect
the visibility region to shrink as the RX grid gets smaller
since there is a higher chance of the Keller cone missing
the grid, confirmed by the top row. We can furthermore seethat when the edge is at the top location, it ends up in theblind region as the RX grid size shrinks. Finally, even whenin the visible region, as the RX grid size shrinks, we expect
the imaging quality to degrade since the coverage for eachpoint will be less (less RX points affected by the Keller cone),and the interference term will become higher. The middle row
then shows the imaging results when placing the edge at the
top location. It can be seen that while the edge is imaged
reasonably well for a larger RX grid, the imaging performancedegrades as the RX size decreases. The bottom row then shows
the imaging results when the vertical edge is at the lower
location, which is in the visible region all the time. It can beseen that the imaged edge is visible for all the sizes. However,as the RX size becomes smaller, the imaging quality degradesas the Keller cone affects less RX antennas.
3) Impact of the edge orientation: Fig. 9 shows
the imaging results for edges with different orientations(0◦ , 45◦ , 90◦ , 135◦ ), using a fixed RX grid size of 1.36 m
×1.12 m, depth of yI = 1.5 m, and TX at pt = [3 m, 0, 0].
The top row shows the analysis results, which indicates that
this transmitter location best illuminates vertical edges, as
discussed earlier. The bottom row then shows the imaging
results. As can be seen, the vertical edge is the only one
that is imaged, while the Keller cones of the edges at otherorientations did not have much overlap with the RX grid,
resulting in them not being imaged. Note that the impact ofedge orientation also depends on the edge location.
4) Impact of the TX location: Fig. 9 showed the illumination capabilities of a TX placed on the side of the RX grid,i.e. at pt = [3 m, 0, 0] (θ = 0◦ in Fig. 1). Fig. 10 next shows
the same results, but with the TX located above the RX grid at[0, −1.3 m, 1.2 m] (Due to physical limitations of the imaging
setup, this is the closest location to create θ = 90◦ of thetheoretical results of Fig. 1). The top row shows that the tested
0◦ , 45◦ , and 135◦ edges are in the visible region for this TX
placement. On the other hand, there is very little coverage

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0.7

Orientation= 0o

Orientation=45o

Orientation=90o

Orientation=135ocoverage = 46%0coverage = 46%

coverage = 0%

-0.7
0.7

coverage = 64%

0

-0.7
-0.7

0

0.7 -0.7

00.7 -0.7

00.7 -0.7

0

0.7

Fig. 9. (Top) Theoretically visible (bright) and blind (dark) regions fordifferent edge orientations, and a TX located at pt = [3 m, 0, 0]. (Bottom)
Wiffract’s imaging results. It can be seen that only the vertical edge is visibly
imaged at this location (See color PDF for best viewing).
0.7

Orientation= 0o

Orientation=45o

Orientation=90o

Orientation=135o

0

-0.7
0.7

coverage = 74%

coverage = 49%

coverage = 33%

coverage = 49%

0

-0.7
-0.7

0

0.7 -0.7

00.7 -0.7

00.7 -0.7

0

0.7

Fig. 10. (Top) Theoretically visible (bright) and blind (dark) regions fordifferent edge orientations, and a TX located at pt = [0, −1.3 m, 1.2 m].
(Bottom) Wiffract’s imaging results. It can be seen that the vertical edge is
not imaged while the horizontal edge is imaged the best by this TX location.
(See color PDF for best viewing)when illuminating vertical edges in this case. The second row
then shows the experimental results, which confirms that thevertical edge cannot be imaged while the horizontal edge isimaged the best, and the other two orientations are imaged,
but with a lesser quality. These results are consistent with theearlier analysis stating that a TX located at an angle θ on asemi-circle around the RX grid is best for imaging edges thatare at angles θ + 90◦ .
VII. I MAGING O BJECTS WITH D IFFERENT C URVATURES
We next focus on the performance of Wiffract when imaging
daily objects that do not necessarily have sharp edges, buthave sides with small curvatures. We utilize three transmitters
located at pt1 = [3 m, 0, 0], pt2 = [0, −1.3 m, 1.2 m] andpt3 = [−3 m, 0, 0], for an imaging plane at yI = 1.5 m.
We run the full Wiffract algorithm [4], briefly summarized inSec. III. More specifically, in addition to the edge hypothesis
testing of previous sections, Wiffract fuses the images of 3transmitters, after thresholding the heat maps, and performs
Bayesian information propagation on the fused image, yielding
a final edge map. Due to space constraints, we refer the readers
to Sec. 4.2 of [4] for more details on the methodology. Notethat no machine learning is used for these results. Fig. 11(Top) shows the objects: a ladder, a chair, a triangle, and a postwith a suspended box. As can be seen, these objects do nothave sharp edges but the sides have small enough curvatures

Fig. 11. Wiffract’s imaging results (whole pipeline of Sec. III) for sample
everyday objects, with a variety of curvatures. As seen, we can image theseobjects well with only WiFi received power, using fundamentals of how small
curvatures interact with incoming waves, and the resulting Keller cones.

for this wavelength. The second row then shows the final edge
imaging results of Wiffract. As can be seen, Wiffract is able
to image the edges of these objects very well, with only WiFi
received power measurements. The results further confirm theprior edge illumination analysis of Figs. 9 and 10. To the bestof our knowledge, no other approach has produced results ofthis quality when imaging with only WiFi.
VIII. C ONCLUSION
In this paper, we provided an in-depth analysis of theinteraction of wireless signals with object edges and its
impact on WiFi-based imaging systems. More specifically,
we considered the impact of curvature of a soft edge, edge
orientation, distance to the receiver grid, transmitter location,and other parameters on the resulting Keller cones and thecorresponding edge-based imaging system, via both analysis
and extensive experimentation. We also showed that soft edges
with a small enough curvature appear as sharp edges to thewave, and can be similarly imaged using edge tracing via
Wiffract, and further imaged a number of such daily objects.
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