Bitmap gravity set to center not filling the screen even though the image is big enough

I have a RelativeLayout in my Android project. This has it's background set to a Bitmap:

<?xml version="1.0" encoding="utf-8"?>
<bitmap xmlns:android="http://schemas.android.com/apk/res/android"
    android:gravity="center"
    android:src="@drawable/splash2" />

splash2 is a PNG image sized 2560x1440 pixels. I haven't set the background of the layout directly to the image, because the default scale mode (or gravity) is fill which stretches the image to fit the screen. With center it should take correct size image from the center and show it unscaled. In case of a vertical 1080x1920 screen, it should take that big piece and center it in the layout.

However, I have a problem. The image is bigger than any screen out in the market today. Still, with my Nexus 7, which has a 1920x1080 screen, it has borders around the image. The layout is set to full screen. The image is shrinked vertically.

How do I fix this?

ImageView's scale type centerCrop was what I wanted. Unfortunately I couldn't specify this property for bitmaps. I changed my splash screen layout to FrameLayout and added an ImageView and TextView overlapping each other. This way I was able to achieve what I wanted.

<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    tools:context=".SplashScreen"
    android:orientation="vertical" >

    <ImageView
        android:id="@+id/imageView1"
        android:layout_width="match_parent"
        android:layout_height="match_parent"
        android:src="@drawable/splash2"
        android:scaleType="centerCrop" />

     <RelativeLayout
        android:layout_width="match_parent"
        android:layout_height="match_parent" >

        <TextView
            android:id="@+id/roadSignName"
            android:layout_width="fill_parent"
            android:layout_height="wrap_content"
            android:layout_marginLeft="20dp"
            android:layout_marginRight="20dp"
            android:layout_centerHorizontal="true"
            android:layout_centerVertical="true"
            android:background="#FFF"
            android:gravity="center_vertical|center_horizontal"
            android:padding="10dp"
            android:text="My program"
            android:textAppearance="?android:attr/textAppearanceLarge"
            android:textColor="#000000"
            android:textSize="40sp" />
    </RelativeLayout>
</FrameLayout>