Image Reconstruction by means of Kalman Filtering in Passive Millimetre- Wave Imaging

Transcription

1 Iage Reconstruction by eans of Kalan Filtering in assive illietre- Wave Iaging David Sith, etrie eyer, Ben Herbst 2 Departent of Electrical and Electronic Engineering, University of Stellenbosch, rivate Bag X, atieland 7602, Stellenbosch, South Africa parick@sun.ac.za 2 Division of Applied atheatics, Departent of atheatics, University of Stellenbosch, rivate Bag X, atieland 7602, Stellenbosch, South Africa herbst@sun.ac.za Corresponding author Abstract assive illietre-wave (W) iaging detects theral noise generated and reflected by etallic and nonetallic objects even in wet and foggy conditions. The use of a W iaging syste on a sall unanned aerial vehicle (UAV) offers extreely attractive possibilities for applications such as search and rescue operations, but the design is severely liited by the size of the UAV. A possible solution is a long, thin antenna array fitted under the wings of the UAV. Such an antenna has a narrow bea along the plane perpendicular to the flight path, but a very broad bea along the plane of the flight path blurs the iage, aking it difficult to accurately deterine the position of an object or to differentiate between objects situated along the plane of the flight path. This paper proposes a technique of iage reconstruction based on the Kalan filter to reconstruct an accurate iage of the target area fro such a detected signal. It is shown that given a siulated target area, populated with an arbitrary nuber of objects, the Kalan filter is able to successfully restore the iage using the easured antenna pattern to odel the scanning process and reverse the blurring effect. Keywords: Iage Reconstruction, Kalan Filter, assive illietre-wave Iaging Introduction assive illietre-wave (W) iaging is a technique that uses radio receivers to detect theral noise generated and reflected by etallic and non-etallic objects. While visual and infra-red eissions are absorbed and scattered by rain and cloud, the penetration of W eissions through atospheric constituents results in a consistent contrast between different objects fro day to night even in wet and foggy conditions, to create iages for a large range of incleent weather applications such as assistance to fog-bound airports [2]. The use of a W iaging syste on a sall unanned aerial vehicle (UAV) offers extreely attractive possibilities for applications such as search and rescue operations, but the design is severely liited by the size of the UAV, ainly in the inability to incorporate any for of reflector or echanical scanning antenna. A possible solution is a long, thin antenna array fitted under the wings of the UAV. Such an antenna has a narrow, high gain, frequency-scanned bea along the plane perpendicular to the flight path, but a very broad bea along the plane of the flight path blurs the iage, aking it difficult to accurately deterine the position of an object or to differentiate between objects situated along the plane of the flight path. This paper proposes a technique of iage reconstruction based on the Kalan filter [], a recursive filter that uses feedback control to estiate the state of a partially observed non-stationary stochastic process, to reconstruct an accurate iage of the target area fro the detected signal. Given a siulated target area, populated with an arbitrary nuber of objects, the Kalan filter is able to successfully restore the iage using the easured antenna pattern to odel the scanning process and reverse the blurring effect. In Section 2 the syste the Kalan filter is designed for is described, showing how the iage is fored and blurred. In Section 3 the atheatical odel of the Kalan filter is detailed, showing how an iproved estiate of the syste is obtained fro noisy easureents. In Section 4 the ipleented syste is described, showing how the paraeters are assigned for the given antenna pattern. In Section 5 siulated results based on the proposed technique are dissected, showing how the Kalan filter reconstructs the target area.

2 2 W Iaging Syste In this paper a W iaging syste, designed to detect objects in low-visibility conditions within the atospheric attenuation window of 30 40GHz, is proposed for fitting under the wings of a sall UAV. The syste consists of an antenna, a three-stage aplification syste, a down-conversion stage, a filtering syste, an analogue to digital conversion stage and a post-processor, as depicted in the block diagra of Fig.. RF Antenna ixer RF LNA IF LNA 2-8 IF LNA 2-8 n N LO 42 ultiplexer Filter Detector Integrator ost-rocessor Figure : Syste Block Diagra With the UAV providing otion along the plane of the flight path, the detector is only required to scan along the plane perpendicular to the flight path. The detector is a single, slotted-array, frequency-scanned, waveguide antenna with the inherent property of space-to-frequency apping, as depicted in Fig. 2. No controlling electronics and no oving parts are required, resulting in an econoical and copact syste well suited to work with the available space and power on the UAV. f L f H Figure 2: Antenna atterns at Different Frequencies The iage is built up line by line as the antenna concurrently scans the target area along the plane perpendicular to the flight path, with each orientation scanned by a bea centred at a different frequency,. Division of the frequency range f L to f H into equal-sized contiguous bandwidths, BW = f H f L, each assigned to a different pixel colun,, separates the target area into bands. Division of the flight path tie-period t 0 to t N into N easureents taken at discrete tie-intervals t = t N t 0 N, each assigned to a different pixel row, n, separates the bands into pixels. The cobination of flight-easuring and scan-filtering aps the target area s w h coordinate syste to the iage s f t coordinate syste of size N, as depicted in Fig. 3. Sweep fl N L O L w N fh h t N t 0 Figure 3: Target Area Because of the antenna construction, it displays a narrow, high gain, frequency-scanned bea along the plane perpendicular to the flight path, but a very broad bea along the plane of the flight path. When flying over the target area the iage of an object is blurred along the plane of the flight path, aking it difficult to accurately deterine the position of an object or to differentiate between objects situated along the plane of the flight path. revention of object blurring is ipossible as the size of the UAV prohibits the inclusion of a bulky reflector to focus the antenna pattern along the plane of the flight path as well. The only solution is to design a post-processor that reconstructs the target area fro the blurred iage. 2

3 3 atheatical odel A perfect iaging syste has a bea focused on a portion of landass equal in size to a pixel, resulting in a one-to-one relationship between the target area and the detected iage. However, no practical iaging syste is perfect. One of the conventional non-idealities is iage blurring, caused by an iperfectly focused lens, that soothes an iage s edges, as depicted in Fig. 4, aking it hard to distinguish between regions. As blur incorporates the surrounding area the iage is artificially increased (usually the borders are extended by one pixel) to odel the incorporation of data fro outside the target area. Conventional Blur Unconventional Blur Narrow Bea Broad Bea Figure 4: Iage Blur In the proposed application the blurring proble is agnified significantly, as there is a any-to-one relationship between the target area and the detected signal, due to the iperfectly focused bea of the antenna. Each pixel of the iage incorporates data fro a large portion of landass within the target area and a large portion of landass outside of the target area, as depicted in Fig. 4. While the conventional ethodology of iage restoration to invert the effect of the degradation holds true, conventional techniques cannot be used as they deal with localised object blurring odelled by Gaussian noise, which is insufficient to counter the global object blurring of the antenna pattern. The non-stationary, discrete-tie, linear process x k+ R n is odelled as x k+ = A k x k + B k u k + v k () where u k R l is the external control that drives the process fro state x k to state x k+, v k is the process noise, the n n atrix A k relates the state at tie step k to the state at tie step k + in the absence of both the external control u k and the process noise v k and the n l atrix B k relates the external control u k to the state x k. The state x k is only accessible fro the noise containated easureent z k R odelled as z k = H k x k + w k (2) where w k is the easureent noise and the n atrix H k relates the state x k to the easureent z k. The process noise v k and easureent noise w k are independent of each other, additive, zero-ean, white and Gaussian with noral probability distributions p(v k ) N (0,Q k ) p(w k ) N (0,R k ) (3) where Q k = E [ v k,v T ] k is the process noise covariance, Rk = E [ w k,w T ] k is the easureent noise covariance and E [ v k v T ] [ n = 0 = E wk w T ] n,n = k. The a priori estiate x k+ k R n of state x k+ is given by the expectation x k+ k = E [ x k+ Z k] (4) where x i j,i j is the estiate of the state x i using the easureents Z j = { z 0,...,z j } up to and including tie j and is obtained fro the noise-free version of the process odel of (), the a posteriori state estiate x k k and the external control u k x k+ k = A k x k k + B k u k (5) The noise-free a priori estiate z k+ k of easureent z k+ is obtained by cobining the easureent odel of (2) and the noise-free a priori state estiate x k+ k z k+ k = H k+ x k+ k (6) 3

4 The a posteriori state estiate x k+ k+ R n is obtained fro the a priori state estiate x k+ k and the weighted difference between the actual easureent z k+ and the a priori easureent estiate z k+ k x k+ k+ = x k+ k + K k+ ( zk+ z k+ k ) (7) where the n atrix K k+ is the Kalan gain. The Kalan gain K k+ is the optial linear estiator that iniises the a posteriori error covariance k+ k+. [ The a priori state error covariance k+ k = E e k+ k e T k+ k Z k ] is obtained fro the a priori state estiate x k+ k k+ k = A k k ka T k + Q k (8) where e k+ k = x k+ x k+ k is the a priori state estiation error. [ The a posteriori state error covariance k+ k+ = E e k+ k+ e T k+ k+ Z k ] is obtained fro the a posteriori state estiate x k+ k+ k+ k+ = k+ k k+ k H T k+ KT k+ K k+h k+ k+ k + K k+ ( Hk+ k+ k H T k+ + R k+) K T k+ (9) where e k+ k+ = x k+ x k+ k+ is the a posteriori state estiation error. aking the derivative of the trace of the a posteriori error covariance k+ k+ with respect to the Kalan gain K k+ equal to 0, and solving for K k+ the optial gain for the coputation of the a posteriori state estiate x k+ k+ is obtained K k+ = k+ k H T k+ ( Hk+ k+ k H T k+ + R k+) (0) which reduces the a posteriori state error covariance k+ k+ of (9) to the well known for k+ k+ = k+ k K k+ H k+ k+ k () The Kalan gain K k+ is proportional to the uncertainty in the a priori state estiate x k+ k and inversely proportional to the uncertainty in the easureent z k+. For an uncertain easureent z k+ and precise a priori state estiate x k+ k the prediction of the a posteriori state estiate x k+ k+ relies ore on the process odel of () than the easureent z k+ and the a posteriori state error covariance k+ k+ sees little reduction li K k+ = 0 x k+ k+ = x k+ k (2) k+ k 0 k+ k+ = k+ k For a precise easureent z k+ and uncertain a priori state estiate x k+ k the prediction of the a posteriori state estiate x k+ k+ relies ore on the easureent z k+ than the process odel of () and the a posteriori state error covariance k+ k+ is considerably reduced li K k+ = H R k+ 0 k+ x k+ k+ = H k+ k+ = 0 k+ z k+ Given the initial conditions of the a posteriori state estiate x k k k=0 and the a posteriori state error covariance k k k=0, the Kalan filter is obtained iteratively by predicting the a priori estiates x k+ k = A k x k k + B k u k z k+ k = H k x k+ k k+ k = A k k k A T k + Q k and coputing the Kalan gain K k+ to update the a posteriori estiates K k+ = k+ k H T ( k+ Hk+ k+ k ( H T k+ + R ) k+ x k+ k+ = x k+ k + K k+ zk+ z k+ k (5) k+ k+ = k+ k K k+ H k+ k+ k The Kalan filter works because the cobination of the a priori state estiate x k+ k, conditioned on all prior easureents Z k, and the a posteriori state estiate x k k, with state x k distribution p(x k z k ) N ( x k k, k k ) (6) into the a posteriori state estiate x k+ k+ is an iproved estiate of the state x k+. 4 ) (3) (4)

5 4 Ipleentation For this proposal the state x k is the target area seen by the antenna at tie step k, the process evolution A k odels the change of the target area fro tie step k to tie step k + as the antenna pattern is shifted by the flight of the UAV, the easureent z k is the easured output of the antenna at tie step k and the easureent evolution H k odels the easured antenna pattern. The change fro state x k to state x k+ is based entirely on the flight of the UAV, with no external control u k, thereby reducing the process odel of () to x k+ = A k x k + v k (7) There is a large overlap between the target area scanned by easureent z k and the target area scanned by easureent z k+, as depicted in Fig. 5. The extension ade to the target area by easureent z k+ is predicted, with the rest of the target area carried over fro state x k. x kk x k k Unknown Unchanged Figure 5: Correlation between Two States As the target area is assued to be static, a few slow oving objects on a stable background, there is no significant change between tie step k and tie step k + and between pixel row n and pixel row n+, the pixel rows n n=2,...,n of the a posteriori state estiate x k k are shifted unchanged into the pixel rows n n=,...,n of the a priori state estiate x k+ k using the state evolution A k A k A k 0 0 A k = where A k = (8) A k where pixel row N of state x k+ is predicted as equal to pixel row N of state x k, the a posteriori state estiate x k k is initialised as x k=0 k k = 0, a A k is required for each pixel colun and the pixel coluns of state x k are lexicographically ordered into one colun for ultiplication with the state evolution A k. The easureent evolution H k is the cobined effect of the easured 2D antenna patterns, Ant 2D. Each row of the easureent evolution H k relates one frequency coponent of easureent z k to state x k, as depicted in Fig. 6. As the antenna has a narrow bea along the plane perpendicular to the flight path, only pixel colun is significant affected by Ant 2D. As the antenna has a very broad bea along the plane of the flight path, any pixels within pixel colun are significant affected by Ant 2D. This blurs the iage, aking it difficult to deterine the position of objects or to differentiate between objects situated along the flight path. Antenna attern easureent Evolution N,, N, N,, N, The three covariance atrices are initialised as Figure 6: easureent odel k=0 k k = A k ε A T k Q k k=0 = A k ε 2 A T k (9) R k k=0 = H k ε 3 H T k where ε, ε 2 and ε 3 reflect the degree of uncertainty with regards to the state, the easureent and the process. 5

6 5 Results Specific target areas are siulated to test the ability of the Kalan filter to odel the flight process and to reove the errors in the easureent process. The input paraeters are gradually increased fro idealised values to the expected values in order to siplify the optiisation process. As a first experient, the ability of the Kalan filter to odel the flight process is tested for a target area containing only a single frequency coponent and with a easureent evolution H k based on antenna easureents using a reflector that concentrates the antenna pattern Ant 2D into a ain bea, as depicted in Fig. 7. This idealisation reoves the interfering effect of neighbouring frequency coponents and the blurring effect of the antenna pattern to siplify the proble to just one involving the odelling of the flight process. Antenna attern easureent Evolution N,, N, N,, N, Figure 7: Idealised easureent odel The response to the change between pixel row n and pixel row n+ is not quick enough, resulting in a delayed response in the posteriori state estiate x k k to the state x k, as depicted in Fig. 8. An iproved response is obtained by adding a velocity paraeter, x k, to the state x k that calculates the change between pixel row N and pixel row N. osition odel osition-velocity odel Delayed Response Iproved Response Figure 8: Idealised Single Frequency Target Area For this position-velocity odel pixel row N of state x k+ is predicted as the su of pixel row N of state x k and the change between pixel row N and pixel row N of state x k using the state evolution A k A k A k 0 0 A k = where A k = (20) A k where the change between pixel row N and pixel row N of state x k+ is equated with the change between pixel row N and pixel row N of state x k. As there is no correlation between the velocity paraeter, x k, of the state x k and the easureent z k, the easureent evolution H k is altered to the for depicted in Fig. 9. Antenna attern easureent Evolution N,, N,, N, x, N Figure 9: Idealised easureent odel using osition-velocity odel 6

7 A second experient is perfored to test the ability of the Kalan filter to odel the flight process for the sae target area, but this tie with a easureent evolution H k based on antenna easureents using a reflector that concentrates the antenna pattern Ant 2D into a 0 ain bea, as depicted in Fig. 0. The idealisation of the easureent odel is lessened to incorporate odelling of the blurring effect of the antenna pattern. The increased width of the ain bea aps onto a larger surface than the ain bea, requiring a larger easureent evolution H k to odel the relationship between the state x k and the easureent z k. Antenna attern easureent Evolution N,, N,, N, x, N Figure 0: artially Idealised easureent odel The response to the change between pixel row n and pixel row n+ of state x k is spread over a nuber of pixels proportional to the width of the ain bea, resulting in a blurred response in the posteriori state estiate x k k to the state x k, as depicted in Fig.. An iproved response is obtained by increasing the nuber of tie steps calculated per Kalan filter loop. Uncertainty eriod osition odel Extended Response osition-velocity odel Iproved Response Figure : artially Idealised Single Frequency Target Area For this ulti-easureent odel the state x k is extended by c pixel rows, where c is the nuber of extra tie steps calculated per Kalan filter loop, which extends the size of each A k by c in the state evolution A k and the rows within the easureent evolution H k are repeated c ties, but with a one pixel offsets due to the shift in focus fro tie step k to tie step k +, as depicted in Fig. 2. Antenna attern easureent Evolution N,, N,, N, x, N Figure 2: artially Idealised easureent odel using ulti-easureent odel The size of the easureent evolution H k has a direct effect on the nuber of tie steps the Kalan filter takes to adapt to the odel. The larger the area the ain bea of the antenna pattern Ant 2D aps onto, the ore pixels that need to be predicted concurrently in the initialisation of the posteriori state estiate x k k k=0 and the longer the uncertainty period of the a posteriori state estiate x k k. The uncertainty period for the 0 ain bea is longer than for the ain bea, as depicted in Fig.. The reason for the iproved response of the ulti-easureent odel is accredited to the increased size of the Kalan gain K k. For a single tie step per Kalan filter loop and a single frequency coponent target area, the easureent z k is a atrix resulting in a atrix Kalan gain K k that can only globally rectify all the pixels of state x k. When the nuber of tie steps is increased per Kalan filter loop, the easureent z k increases in size resulting in a Kalan gain K k approaching the size of the easureent odel H k that can rectify the pixels of state x k individually. Only when a large nuber of tie steps are used per Kalan filter loop does K k H k+ when R k+ 0. 7

8 The final set of experients test the ability of the Kalan filter to reove the errors in the easureent process using the easureent evolution H k of Fig. 2 and a full frequency range target area containing a single central high intensity object surrounded by a low intensity background, as depicted in Fig. 3. The last idealisation is reoved to incorporate odelling of the interfering effect of neighbouring frequency coponents. The large size of the easureent evolution H k results in a long uncertainty period. H k Figure 3: artially Idealised Full Range Target Area with Single Object The variance in absolute gain between the coponents of the antenna pattern Ant 2D returns a false ulti-level object and background, while the partial blurring of the object to neighbouring bands is accredited to non-ideal slope of the ain bea. The Kalan filter is able to reduce both of these inaccuracies once the long uncertainty period has past, as depicted in Fig. 4. easureent Uncertainty eriod Figure 4: for artially Idealised Full Range Target Area with Single Object The ability of the Kalan filter to reove the errors in the easureent process is retested using the sae easureent evolution H k of Fig. 2, but this tie with a full frequency range target area containing a single central high intensity object bordered by two oderately high objects and surrounded by a low intensity background, as depicted in Fig. 5. H k Figure 5: artially Idealised Full Range Target Area with ultiple Objects The ability of the Kalan filter to reduce the variance in absolute gain between the coponents of the antenna pattern Ant 2D and the infringeent of the ain bea into neighbouring bands is not affected by the nuber of objects within the target area. The blurring effect of the antenna pattern Ant 2D erges the closely spaced objects into a single object, with a false higher intensity at the location of the two border objects. The Kalan filter is able to response quick enough to a change between pixel row n and pixel row n+ of state x k, resulting in the objects being individually sharpened and detached fro each other, as depicted in Fig. 6. easureent Uncertainty eriod Figure 6: for artially Idealised Full Range Target Area with ultiple Objects 8

9 6 Conclusions Advances in UAV technology have created the option of introducing passive illietre-wave iaging capability into an autonoous vehicle for incleent weather conditions. However, the sall size of the UAV places restrictions on the design of the syste that lead to a non-ideal antenna pattern. With the inability to incorporate any easure to focus the antenna pattern before iage creation, a technique is needed to restore the iage after iage creation. Conventional iage restoration processes deal with localised object blurring odelled by Gaussian noise, which is insufficient to counter the ore global object blurring of the antenna pattern, and are designed for stationary stand-alone iages. This paper proposes a new unconventional technique based on the Kalan filter. The Kalan filter uses the unfocused antenna pattern to odel the blur and is designed for non-stationary processes. For each tieinterval the Kalan filter akes a prediction of the detected signal using the easured antenna pattern. The coparison between the predicted signal and the detected signal is used to generate an accurate iage of the target area. The Kalan filter is able to correct for unequal gain between frequency coponents, reduces blur into other frequency coponents and reduces blur into other tie intervals and to separate closely spaced objects for a partially focused bea easuring a siulated target area. It still reains to be proven whether the Kalan filter is functional for an unfocused bea easuring an actual target area. References [] R.E. Kalan, A New Approach to Linear Filtering and rediction robles. Transactions of the ASE Journal of Basic Engineering, 82 (Series D), 35 45, arch, 960. [2] L. Yuriji and. Shoucri and. offa, assive illieter-wave Iaging, IEEE icrowave agazine, 39 50, Septeber,