This paper proposes a joint reconstruction algorithm for compressed correlated images that are given under the form of linear measurements. We consider the particular problem where one image is selected as the reference image and it is used as the side information for decoding the compressed correlated images. These compressed images are given under the form of random measurements that are further quantized and entropy coded. The joint decoder estimates the correlation model based on the geometric transformation of features captured by a structured dictionary. We observe that the high frequency components are not efficiently captured in the estimated image, when the correlation information is used alone for image prediction. Hence, we propose a reconstruction strategy that uses the information in the measurements to recover the missing visual information in the predicted image. The reconstruction is based on an optimization algorithm that enforces the reconstructed image to be consistent with the quantized measurements. We further add additional constraints to ensure that the reconstructed image is close to the image predicted from the correlation estimation. The non-linearities introduced due to quantization are considered on both correlation and reconstruction algorithms in order to improve the performance. Experimental results demonstrate the benefit of the reconstruction algorithm as it brings improved coding performance especially at high rate and outperforms independent coding solutions based on JPEG 2000.